Elastic graphs

ELASTIC GRAPHS

arXiv:1607.00340v1 [math.MG] 1 Jul 2016

DYLAN P. THURSTON Abstract. An elastic graph is a graph with an elasticity associated to each edge, which may be viewed as a network made out of ideal rubber bands. The rubber bands may be stretched on a target space, giving an elastic energy. We characterize when a homotopy class of maps from one elastic graph to another is loosening, i.e., decreases this elastic energy for all possible targets. This fits into a more general framework of energies for maps between graphs.

Contents 1. Introduction 2. Basic notions and sub-multiplicativity 3. Reduced and taut maps 4. Weak graphs and Lipschitz energy 5. Harmonic maps 6. Minimizing embedding energy Appendix A. General graph energies Appendix B. Relation to resistor networks References

1 10 14 26 31 37 50 57 59

1. Introduction Question 1.1. When is one network of rubber bands looser than another? To make Question 1.1 precise, we give some definitions. Definition 1.2. In this paper, a graph is a topologist’s graph, a finite 1-dimensional CW complex, with multiple edges and self-loops allowed. Maps between graphs are continuous maps between the underlying topological spaces. In particular, they need not send vertices to vertices. For convenience, we will generally work with PL maps with respect to some (fixed) PL structure on the graph. A marked graph is a pair pΓ, M q, where Γ is a graph and M Ă Γ is a finite subset of marked points (possibly empty). A map f : pΓ1 , M1 q Ñ pΓ2 , M2 q between marked graphs is required to send marked points to marked points (i.e., f pM1 q Ă M2 ). Homotopy is considered within the space of such maps. In particular, the restriction of f to a map from M1 to M2 is fixed. Date: June 25, 2016. 2010 Mathematics Subject Classification. Primary 37E25; Secondary 58E20, 05C21. 1

2

THURSTON

a b c

v

d

b a c

x y

v

w z

d

Figure 1. Local schematic pictures of the equilibrium condition for harmonic maps. Top: A vertex maps to an edge. Bottom: A vertex maps to a vertex. Definition 1.3. A length graph K “ pΓ, `q is a graph in which each edge e has a positive length `peq; this gives a natural metric on the underlying topological space, which is required to be PL with respect to the underlying PL structure. If f : K1 Ñ K2 is a PL map between length graphs, then |f 1 | : K1 Ñ Rě0 is the local derivative with respect to the metrics. It is locally constant, with jump discontinuities. Definition 1.4. A elastic graph G “ pΓ, αq is a graph Γ in which each edge e has a positive elastic constant αpeq, which we will also view as the length of the edge. For G an elastic graph, K a length graph, and f : G Ñ K a PL map, the Dirichlet energy of f is ż (1.5) Dirpf q :“ |f 1 pxq|2 dx. xPΓ

The Dirichlet energy of a (marked) homotopy class rf s is defined to be (1.6)

Dirrf s :“ inf Dirpgq. gPrf s

It is easy to see that a minimizer g for Dirrf s must be constant-derivative: |g 1 | is constant on each edge of g (Definition 4.5). If g is constant-derivative, we then have ÿ `pgpeqq2 (1.7) Dirpgq “ , αpeq ePEdgepΓq where `pgpeqq is the length of the image of the edge e in a natural sense (Definition 2.2). This is the Hooke’s law energy of g, where each edge e is an ideal spring with resting length 0 and spring constant given by αpeq. Physically, you could think about stretching a rubber band graph shaped like G over a system of pipes shaped like K. Harmonic maps are maps that locally minimize the energy (1.5) or (1.7) within their homotopy class. Intuitively, a map is harmonic if it is constant-derivative and each vertex is at equilibrium, in the sense that the force pulling in one direction is less than or equal to the total force pulling it in other directions. For instance, if a

ELASTIC GRAPHS

3

vertex v of G maps to the middle of an edge of K with three edges on the left and one edge on the right as on top of Figure 1, the net force to the left must equal the net force to the right: |f 1 paq| ` |f 1 pbq| ` |f 1 pcq| “ |f 1 pdq|. On the other hand, if a vertex v of G maps to a vertex w of K as on the bottom of Figure 1, we no longer have an equality. Instead, the net force pulling v into any one of the three edges incident to w can’t be greater than the total force pulling it into the other two edges. This gives three triangle inequalities: |f 1 paq| ` |f 1 pbq| ď |f 1 pcq| ` |f 1 pdq| |f 1 pcq| ď |f 1 paq| ` |f 1 pbq| ` |f 1 pdq| |f 1 pdq| ď |f 1 paq| ` |f 1 pbq| ` |f 1 pcq|. We can always find harmonic maps in any homotopy class of maps to a target length graph. Example 1.8. If G and K are both tripods with their ends marked, with elastic weights and lengths, respectively,

G“

1

1

K“

,

3 3 1 1 then the map f : G Ñ K that minimizes Dirichlet energy maps the vertex of G to a point 1{5 of the way to a vertex along one of the edges of K: 1 3

f

ÝÑ

4/5 .

3

Up to a factor of 2, the net force pulling the vertex of G upwards is 4{5, the same as the net force pulling it downwards. The total Dirichlet energy is Dirpf q “ Dirrf s “ p4{5q2 ` p6{5q2 {3 ` p6{5q2 {3 “ 8{5. On the other hand, if the elastic weights on the legs of G are p1, x, xq for any 0 ă x ď 2, then the harmonic representative has the central vertex of G mapping to the central vertex of K. We can think of Dirichlet energy as a function of the target lengths (or more precisely the target lengths and the homotopy class). We next compare these functions. Definition 1.9. Given a homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs, we say that rφs is (weakly) loosening if, for all marked length graphs K and marked maps f : G2 Ñ K, Dirrf ˝ φs ď Dirrf s.

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THURSTON

For a finer invariant, define the Dirichlet stretch factor to be SFDir rφs :“ sup

(1.10)

K,f

Dirrf ˝ φs , Dirrf s

where the supremum runs over all marked length graphs K and all marked homotopy classes rf s, so that rφs is loosening iff SFDir rφs ď 1. Definition 1.11. For φ : G1 Ñ G2 a PL map between marked elastic graphs, the embedding energy of φ is ÿ (1.12) Embpφq :“ ess sup |φ1 pxq|. yPG2

xPφ´1 pyq

We are taking the essential supremum over points y P G2 , ignoring in particular vertices of G2 , images of vertices of G1 , and points where φ´1 pyq is infinite, all of which have measure zero. For a homotopy class rφs, define Embrφs :“ inf Embpψq.

(1.13)

ψPrφs

Example 1.14. If G1 and G2 are both tripods with marked ends, with elastic weights p1, 3, 3q and p1, 1, 1q, then the map φ that minimizes Embpφq is not the map from Example 1.8, but instead sends the vertex of G1 to a point 2{5 of the way along an edge of G2 , with Embpφq “ 3{5: 3/5

1 5/3

φ

5/3

ÝÑ

.

Our answer to Question 1.1 says that Embrφs “ SFDir rφs, and in particular the homotopy class rφs is loosening exactly when Embrφs ď 1. Before stating the theorem, we give another characterization of loosening, in terms of maps to an elastic graph G (rather than maps from G). Definition 1.15. A marked one-manifold is a (not necessarily connected) one-manifold C with boundary, where the marked points are BC. A marked curve on a marked graph Γ is a marked one-manifold C and a marked map c : C Ñ Γ. (Thus, it is a union of loops on G and arcs between marked points of G.) A marked curve is reduced if it is PL and has no backtracking: on each component of C, c is either constant or has a perturbation that is locally injective. In each homotopy class of curves there is a reduced representative. If pC, cq is a curve on G, then for y P Γ, define nc pyq to be the size of c´1 pyq. If c is reduced and y is on an edge e of Γ, nc pyq depends only on e except for isolated points, so we may write nc peq. A marked weighted curve is a marked curve in which each component Ci has a positive weight wi . For a marked weighted curve on Γ, nc pyq is the weighted count of c´1 pyq.

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Definition 1.16. The extremal length of a reduced curve c on an elastic graph G is ÿ ELrcs “ ELpcq :“ αpeqnc peq2 . ePEdgepGq

The extremal length of a homotopy class of marked curves is the extremal length of any reduced representative. Definition 1.17. For rφs : G1 Ñ G2 a homotopy class of maps between elastic marked graphs, the extremal length stretch factor is (1.18)

SFEL rφs :“ sup C,c

ELrφ ˝ cs ELrcs

where the supremum runs over all curves pC, cq on G1 . We are now ready to state the main goal of this paper. Theorem 1. For rφs : G1 Ñ G2 a homotopy class of maps between marked elastic graphs, (1.19)

Embrφs “ SFDir rφs “ SFEL rφs.

Furthermore, there is ‚ a PL map ψ P rφs realizing Embrφs; ‚ a marked length graph K and a PL map f : G2 Ñ K realizing SFDir , in the sense that Dirrf ˝ φs Dirpf ˝ ψq SFDir rφs “ “ ; Dirrf s Dirpf q and ‚ a marked weighted curve pC, cq on G1 realizing SFEL , in the sense that c and ψ ˝ c are both reduced and SFEL rφs “

ELpψ ˝ cq . ELpcq

Note that Embrφs is naturally defined as an infimum (over the homotopy class), while SFDir rφs and SFEL rφs are naturally defined as suprema (over all possible targets or curves). As such, Equation (1.19) helps us compute these quantities, by sandwiching the target value. In the course of the proof in Section 6, we also give an algorithm that produces, simultaneously, the map ψ P rφs minimizing Emb, the pair pK, f q maximizing the ratio of Dirichlet energies, and the reduced curve pC, cq maximizing the ratio of extremal lengths.

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THURSTON

Example 1.20. Example 1.14 fits into the following sequence of maps realizing SFEL and SFDir : c

φ

g

C ÝÑ G1 ÝÑ G2 ÝÑ K 1

1

c

ÝÑ

1

1

φ

ÝÑ

g

ÝÑ

2

3 3 1 1 1 1 Here c is the evident curve map, φ is the map from Example 1.14, and g is the map that takes the vertex of G2 to the vertex of K. Specifically, we have 4`1`1 Dirrg ˝ φs 36{25 ` 27{25 ` 27{25 ELrφ ˝ cs “ “ Embpφq “ 3{5 “ “ . ELrcs 4`3`3 Dirrgs 4`1`1 We can furthermore generalize the target spaces in Theorem 1 considerably. Definition 1.21. Let G by a marked elastic graph and let X be a marked length space (a length space with a distinguished finite set of points). Let f : G Ñ X be a Lipschitz map. Define the Dirichlet energy of f by ż Dirpf q :“ |f 1 pxq|2 dx xPG 1

where |f pxq| is the best local Lipschitz constant of f in a neighborhood of x. For a homotopy class of marked maps, define Dirrf s :“ inf Dirpgq. gPrf s

In this generality, we certainly have no guarantee that minimizers for Dirrφs exist. (See [KS93, EF01] for some cases where minimizers do exist.) In the course of proving Theorem 1 we also prove the following. Theorem 11 . For rφs : G1 Ñ G2 a homotopy class of maps between marked elastic graphs, Dirrf ˝ φs Embrφs “ sup , Dirrf s X,f where the supremum runs over all length spaces X and all homotopy classes of marked maps f : G2 Ñ X with 0 ă Dirrf s. We next put Theorem 1 in a broader context of graphical energies. There is a family of energies between weighted graphs, elastic graphs, and length graphs: ` (1.22)

Weighted graph W WR

? EL

Elastic graph G ? Emb

? Dir

Length graph K Lip

ELASTIC GRAPHS

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(Weighted graphs (Definition 2.3) are a mild generalization of weighted curves.) The label on an arrow gives the appropriate energy of a map between the given types of graphs. ELpf q, Embpf q and Dirpf q are as defined above, except that we take the square root for reasons to be explained shortly, and EL has been extended from curves to maps from general weighted graphs in a natural way (Equation (2.6)). We make some additional definitions. ‚ For f : W Ñ K a map from a weighted graph to a length graph, `pf q is the weighted length of the image of f : ÿ (1.23) `pf q :“ wpeq ¨ `pf peqq. ePEdgepW q

‚ For f : K1 Ñ K2 a PL map between length graphs, Lippf q is the best global Lipschitz constant for f : (1.24)

Lippf q :“ ess sup|f 1 pxq|. xPK1

‚ For f : W1 Ñ W2 a PL map between weighted graphs, the weight ratio is the maximum ratio of weights: ř xPf ´1 pyq wpxq (1.25) WRpcq :“ ess sup . wpyq yPW2 A key fact is that these energies are submultiplicative, in the sense that the energy of a composition of two maps is less than or equal to the product of the energies of the two pieces. To state this uniformly, we make some further definitions. Definition 1.26. For p P t1, 2, 8u, a p-conformal graph Gp is ‚ for p “ 1, a weighted graph; ‚ for p “ 2, an elastic graph; and ‚ for p “ 8, a length graph. For p, q P t1, 2, 8u with p ď q and f : Gp1 Ñ Gq2 a PL map from a p-conformal graph G1 to a q-conformal graph G2 , define Eqp pf q by a 1 E11 pf q :“ WRpf q E21 pf q :“ ELpf q E8 pf q :“ `pf q a a 2 E22 pf q :“ Embpf q E8 pf q :“ Dirpf q 8 E8 pf q :“ Lippf q.

In each case, for a homotopy class rf s of maps, define Eqp rf s :“ inf Eqp pgq. gPrf s

Let p, q, r P t1, 2, 8u with p ď q ď r. Suppose we have a sequence of maps f

g

Gp1 ÝÑ Gq2 ÝÑ Gr3

8

THURSTON

between marked graphs of the respective conformal type. Then (1.27)

Erp pg ˝ f q ď Eqp pf qErq pgq

(1.28)

Erp rg ˝ f s ď Eqp rf sErq rgs.

See Proposition 2.17 for a more detailed statement, spelling out the cases. More generally, p-conformal graphs and energies Eqp can be defined uniformly for real numbers p, q with 1 ď p ď q ď 8. See Appendix A. Theorem 1 can be interpreted as saying that some cases of Equation (1.28) are tight, as follows. Definition 1.29. Let p, q P t1, 2, 8u with p ď q. If f : Gp1 Ñ Gq2 is a PL map between marked graphs of the respective conformal type and Eqp pf q “ Eqp rf s, we say that f is an energy minimizer (in its homotopy class). Let p, q, r P t1, 2, 8u with p ď q ď r. If we have a sequence of maps f

g

Gp1 ÝÑ Gq2 ÝÑ Gr3 of PL maps between graphs of the respective conformal type, we say that this sequence is tight if g ˝ f is an energy minimizer and Equation (1.27) is sharp: Erp rg ˝ f s “ Erp pg ˝ f q “ Eqp pf qErq pgq. We may similarly say that a longer sequence of maps is tight. f

g

Lemma 1.30. Let Gp1 ÝÑ Gq2 ÝÑ Gr3 be a sequence of maps. If the sequence is tight, then f and g are also energy minimizers. Conversely, if f and g are energy minimizers and Erp rg ˝ f s “ Eqp rf sErq rgs, then the sequence is tight. Proof. In a general sequence of maps, we have inequalities: Erp rg ˝ f s ď Erp pg ˝ f q ď Eqp pf qErq pgq Erp rg ˝ f s ď Eqp rf sErq rgs ď Eqp pf qErq pgq. The two conditions in the lemma assert that the inequalities in the top line or in the bottom line are equalities. In either case, the inequalities in the other line must be equalities as well.  In this language, Theorem 1 says that, for every homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs, there is a corresponding tight sequence c

ψ

f

C ÝÑ G1 ÝÑ G2 ÝÑ K where ψ P rφs, pC, cq is a weighted curve on G1 , and pK, f q is a map from G2 to a length graph. In particular, ψ is an energy-minimizer for Emb. In the course of the proof, we will also characterize which maps minimize Emb. In the course of the paper, we also prove several other tightness results.

ELASTIC GRAPHS

9

‚ In Section 3, we prove Theorem 2, which gives a version of the max-flow/min-cut theorem in the context of maps between graph. This gives a characterization of which maps minimize Ep1 (for any p). ‚ In Section 4, we prove Theorem 3, which shows that for any homotopy class rφs : K1 Ñ K2 of maps between marked length graphs, there is a tight sequence ψ

c

C ÝÑ K1 ÝÑ K2 where ψ P rφs. This is a theorem of White, used by Bestvina [Bes11]: the minimal Lipschitz stretch factor between metric graphs equals the maximal ratio by which lengths of curves are changed. In that section we also extend the graphs we are looking at to allow weak graphs, where lengths are allowed to be zero, and prove the corresponding tightness result in that setting (Theorem 31 ). ‚ In Section 5, we prove Theorem 4, which shows that for any homotopy class rf s : G Ñ K of maps from a marked elastic graph to a marked length graph, there is a tight sequence g

c

C ÝÑ G ÝÑ K where g P rf s. Theorem 4 also characterizes which maps g minimize Dirpgq (the harmonic maps). Again, we extend the results to the setting of weak graphs (Theorem 41 ). ‚ Section 6 has the proof of the main results, Theorems 1 and 11 . ‚ Appendix A defines a family of energies Eqp pf q for any 1 ď p ď q ď 8 and any PL map f between graphs with suitable extra structure. It gives a brief sketch of Theorem 5, which is the most general statement: for any 1 ď p ď q ď 8 and homotopy class of maps rφs : G Ñ H between marked graphs with the appropriate extra structure, we can find ψ P rφs and a tight sequence c

ψ

f

C ÝÑ G ÝÑ H ÝÑ K. The equality Embrφs “ SFEL rφs in Theorem 1 is particularly important for applications, in light of the connection between extremal length on surfaces and conformal embeddings [KPT15]. In forthcoming work, we will use this to give criteria for when one degenerating family of surfaces conformally embeds in another. These embeddings of (degenerating) surfaces can in turn be used to give a positive characterization of post-critically finite rational maps among branched self-covers of the sphere [Thu16], giving a converse to an earlier characterization by W. Thurston [DH93]. 1.1. Prior work. Harmonic maps between manifolds and singular spaces have been studied for a long time, and there is a great deal of work on various cases [GS92, KS93, Wol95, Pic05, inter alia]. In particular, Eells and Fuglede proved that in every homotopy class of maps between suitable (marked) Riemannian polyhedra there is a harmonic representative [EF01, Theorem 11.2], which is a large part of our Theorem 4. Their theory is much more general, and also considerably more delicate. On the other hand, the question of when one network is “looser” than another as in Definition 1.9 appears to be new. In the related context of resistor networks, there has

10

THURSTON

been much attention devoted to when two resistor networks are electrically equivalent; see Appendix B for the similarities and differences. The definition of embedding energy in Definition 1.11 also appears to be new, although it is in a sense dual to Jeremy Kahn’s notion of domination of weighted arc diagrams [Kah06]. Specifically, the criteria mentioned above for embedding degenerating families of surfaces has substantial overlap with Kahn’s work, as will be explained in the sequel. 1.2. Acknowledgements. This project grew out of joint work with Kevin Pilgrim, and owes a great deal to conversations with him and with Jeremy Kahn. This work was partially supported by NSF grants DMS-1358638 and DMS-1507244. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. 2. Basic notions and sub-multiplicativity As mentioned already, all graphs are assumed to have a PL structure, metrics are assumed to be PL with respect to this structure, weights are assumed to be piecewise-constant, and maps between graphs are assumed to be PL. All the energies extend naturally to a wider class of maps between graphs; however, the exact wider class of maps depends on the energy, and it is more convenient to stick to PL maps. The downside is that the existence of minimizers of the energies is not obvious (since the space of PL maps is not locally compact in any reasonable sense). We will prove existence of energy minimizers in each case of interest. We start by defining weighted graphs and energies for maps from them. Definition 2.1. If f : Γ1 Ñ Γ2 is a PL map, the regular points of f are the points in Γ1 that are in the interior of a segment on which f is linear with non-zero derivative, and the singular points are all other points, namely vertices of Γ1 , preimages of vertices of Γ2 , points where the derivative changes, and segments on which f is constant. Similarly, the singular values in Γ2 are the images of singular points, and the regular values are the rest of Γ2 . There are only finitely many singular values. Definition 2.2. As in the introduction, a length graph K is a pair pΓ, `q of a graph and an assignment of a length `peq ą 0 to edge e of Γ; we call ` a metric on Γ. The space of all metrics on Γ is denoted L` pΓq. Given a PL map f : Γ1 Ñ pΓ2 , `q to a length graph, we can pull-back the metric on Γ2 to a metric f ˚ ` on Γ1 , assigning to each edge e P EdgepΓ1 q the total length traced out by f peq. We also write `pf peqq for f ˚ `peq. Definition 2.3. A weighted graph is a graph W “ pΓ, wq with a piecewise-constant non-negative weight function w on each edge. We may (implicitly) subdivide W so that w is constant on each edge, in which case we write wpeq for e an edge of W . The space of all weights on Γ that are constant on each edge is denoted WpΓq, and W ` pΓq is the subspace where the weights are positive.

ELASTIC GRAPHS

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Definition 2.4. For W a weighted graph, Γ a graph, and c : W Ñ Γ a PL map, the multiplicity function nc : Γ Ñ R is defined at regular values of c by ÿ nc pyq :“ wpxq. xPc´1 pyq

This may also be written nw c or c˚ w if we need to make the dependence on w explicit. We will never care about the value of nc at non-regular values. If in addition Γ has more structure we have an energy. ‚ If Γ is also a weighted graph, the weight ratio is (2.5)

WRpcq :“ ess sup yPΓ

nc pyq . wpyq

‚ If Γ is an elastic graph, the extremal length of c is ż (2.6) ELpcq :“ nc pyq2 dy, yPΓ

where we view each edge e of G as having a measure of total mass αpeq. (This generalizes Definition 1.16.) ‚ If Γ is a length graph, the weighted length of c, defined in Equation (1.23), can also be written (Lemma 2.12) ż (2.7) `pcq “ nc pyq dy. yPΓ

In Section 3 we will show the existence of taut maps in each homotopy class that simultaneously minimize nc pyq for all regular y P Γ. As such, taut maps are automatically minimizers of WR, EL, and `, independent of the structure on Γ. If c is taut, nc is constant on each edge of Γ and Equation (2.6) reduces to ÿ ELpcq “ nc peq2 ¨ αpeq ePEdgepGq

In addition to these energies, the energies Emb, Dir, and Lip were defined in Section 1. For all of these energies, if we wish to make explicit the dependence of the energy on the geometric structure, we use superscripts for the structure on the w w 1 domain and subscripts for the structure on the range. Thus we write WRw w2 , ELα , `` , Embαα12 , Dirα` , or Lip``12 . We can also give another formula for Dirichlet energy. Definition 2.8. For G an elastic graph, Γ either a length graph or an elastic graph, and f : G Ñ Γ a PL map, the filling function Fillf : Γ Ñ R is defined at regular values of f by ÿ (2.9) Fillf pyq :“ |f 1 pxq|. xPf ´1 pyq

In particular, Equation (1.12) says that Embpf q “ ess supy Fillf pyq.

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THURSTON

Lemma 2.10. For f : G Ñ K a PL map from an elastic graph to a length graph, ż ż 1 2 (2.11) Dirpf q “ |f | dx “ Fillf pyq dy. xPΓ

yPK

There is a corresponding formula for `. Lemma 2.12. For c : W Ñ K a PL map from a weighted graph to a length graph, ż ż 1 wpxq|c pxq| dx “ nc pyq dy. (2.13) `pcq “ xPW

yPK

Proof of Lemmas 2.10 and 2.12. Elementary change of variables. φ



ψ

Lemma 2.14. For G1 ÝÑ G2 ÝÑ Γ a sequence of PL maps between graphs, where G1 and G2 are elastic graphs and Γ is a length graph or an elastic graph, we have ˆ ˙ ÿ 1 Fillψ˝φ pzq “ |ψ pyq| Fillφ pyq ď Fillψ pzq max Fillφ pyq . ´1 yPψ

yPψ ´1 pzq φ

pzq

c

Lemma 2.15. Let W1 ÝÑ W2 ÝÑ Γ be a sequence of PL maps, with W1 and W2 weighted graphs. Then for almost every z P Γ, nc˝φ pzq ď WRpφqnc pzq. f

φ

Lemma 2.16. Let Γ ÝÑ K1 ÝÑ K2 be a sequence of PL maps, with K1 and K2 length graphs. Then for almost every x P Γ, pφ ˝ f q1 pxq

ď |f 1 pxq| Lippφq.

Proof of Lemmas 2.14, 2.15 and 2.16. Immediate from the definitions.



We now turn to sub-multiplicativity, Equations (1.27) and (1.28). For concreteness (and because the square roots get a little confusing), we list all 10 cases individually. Proposition 2.17. The energies from (1.22) are sub-multiplicative, in the following sense. For i P t1, 2, 3u, let Wi be marked weighted graphs, Gi be marked elastic graphs, and Ki be marked length graphs. Then, if we are given marked PL maps between these spaces as specified on each line, we have the given inequality. φ

ψ

WRpψ ˝ φq ď WRpφq WRpψq

φ

c

ELpc ˝ φq ď WRpφq2 ELpcq

φ

f

`pf ˝ φq ď WRpφq Lippf q

c

φ

ELpφ ˝ cq ď ELpcq Embpφq

(2.18)

W1 ÝÑ W2 ÝÑ W3 :

(2.19)

W1 ÝÑ W2 ÝÑ G :

(2.20)

W1 ÝÑ W2 ÝÑ K :

(2.21)

W ÝÑ G1 ÝÑ G2 :

(2.22)

W ÝÑ G ÝÑ K :

(2.23)

W ÝÑ K1 ÝÑ K2 :

c

c

f

φ

`pf ˝ cq2 ď ELpcq Dirpf q `pφ ˝ cq ď Lippφq`pcq

ELASTIC GRAPHS

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φ

ψ

Embpψ ˝ φq ď Embpφq Embpψq

φ

f

Dirpf ˝ φq ď Embpφq Dirpf q

f

φ

Dirpφ ˝ f q ď Lippφq2 Dirpf q

φ

ψ

Lippψ ˝ φq ď Lippφq Lippψq.

(2.24)

G1 ÝÑ G2 ÝÑ G3 :

(2.25)

G1 ÝÑ G2 ÝÑ K :

(2.26)

G ÝÑ K1 ÝÑ K2 :

(2.27)

K1 ÝÑ K2 ÝÑ K3 :

(When there is only one graph of a particular type, we omit the subscript.) Each inequality still holds if we take homotopy classes on both sides. Proof. Equations (2.18), (2.19), and (2.20) follow from Lemma 2.15 and Equations (2.5), (2.6), and (2.13), respectively. Equations (2.23), (2.26), and (2.27) follow from Lemma 2.16 and Equations (2.13), (1.5), and (1.24), respectively. For Equation (2.21), we have ż ELpφ ˝ cq “ nφ˝c pyq2 dz zPG2

˙2 nc pyq dz

ˆ ÿ

ż “ zPG2

φpyq“z

ˆ ÿ

ż

˙ˆ ÿ ˙ 1 nc pyq {|φ pyq| |φ pyq| dz 2

ď zPG2

1

φpyq“z

φpyq“z

ż nc pyq2 dy

ď Embpφq yPG1

“ Embpφq ELpcq, ř using the definition of EL; the equality nφ˝c pyq “ φpxq“y nc pxq at regular values; the Cauchy-Schwarz inequality; the definition of Embpφq and a change of variables between G2 and G1 ; and the definition of EL again. For Equation (2.22), we have ˙2

ˆż 2

`pf ˝ cq “

nf ˝c pzq dz zPK

˙2

ˆż 1

“ yPG

ˆż ď

nc pyq|f pyq| dy ˙ ˆż 2 nc pyq dy

yPG

˙ 1

2

|f pyq| dy

yPG

“ ELpcq Dirpf q, using Lemma (2.12), a change of variables from K to G, the Cauchy-Schwarz inequality, and the definitions of EL and Dir. Equation (2.24) follows from Lemma 2.14.

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For Equation (2.25), we have ż Fillf ˝φ pzq dz

Dirpf ˝ φq “ zPK

ż ď żzPK

´ ¯ Fillf pzq max Fillφ pyq dz f pyq“z

Fillf pzq Embpφq dz

ď zPK

“ Embpφq Dirpf q, using Lemma 2.10, Lemma 2.14, the definition of Embpφq, and Lemma 2.10 again. For each of these equations, to replace concrete functions by homotopy classes, take representatives f, g of the two homotopy classes whose energy is within a factor of  of the optimal value, in the sense that Eqp rf s ď Eqp pf q ď Eqp rf sp1 ` q, and similarly for g. Then Erp rg ˝ f s ď Erp pg ˝ f q ď Eqp pf qErq pgq ď Eqp rf sErq rgsp1 ` q2 . Since  can be chosen as small as desired, we are done.



These arguments give us one direction of Theorem 1. Corollary 2.28. For any homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs, SFDir rφs ď Embrφs and SFEL rφs ď Embrφs. Proof. For any marked length graph K and homotopy class rf s : G2 Ñ K, by the homotopy version of Equation (2.25) we have Dirrf s Embrφs Dirrf ˝ φs ď “ Embrφs. Dirrf s Dirrf s Since K and rf s were arbitrary, it follows that SFDir rφs. The argument that SFEL rφs ď Embrφs is exactly parallel.  3. Reduced and taut maps We next turn to notions of efficiency of maps between graphs. We first have a weak notion depending on no extra structure (“reduced”), and then develop more a notion that depends on a weight structure (“taut”). 3.1. Reduced maps. By analogy with the standard notion of a reduced (cyclic) word in a group or on a graph, we define notions of reduced maps between marked graphs. Definition 3.1. A map f : Γ1 Ñ Γ2 between marked graphs is edge-reduced if, on each edge e of Γ1 , f is either constant or has a perturbation that is locally injective. Definition 3.2. For a graph Γ and point x P Γ, a direction d at x is a germ of a PL map from Rě0 to Γ starting at x, considered up to PL reparametrization. There is always a zero direction, the germ of a constant map. Points on edges of Γ have two non-zero directions, and vertices have as many non-zero directions as their valence.

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If f : Γ1 Ñ Γ2 is a PL map and d is a direction at x P Γ1 , then f pdq is a direction at f pxq. Definition 3.3. A PL map f : Γ1 Ñ Γ2 between marked graphs is strictly reduced if it is locally injective on the edges of Γ1 and, at each non-marked vertex v of Γ1 , there are directions d1 and d2 at v so that f pd1 q and f pd2 q are distinct and non-zero. More generally, pick y P Γ2 and let Z Ă Γ1 be a component of f ´1 pyq. A direction from Z is a point x P Z and a direction d from x that points out of Z (so that f pdq ‰ 0). We say that Z is a dead end for f if Z does not have a marked point and there is exactly one direction in the set DpZq :“ t f pdq | d a direction from Z u. (If DpZq “ H, then Z is necessarily an entire connected component of Γ1 .) We say that f is reduced if it has no dead ends. If f is not reduced, there is a natural reduction operation at a dead end Z, where we modify f by pulling the dead end in the direction in DpZq until it hits another feature: a vertex of Γ2 or the boundary of a domain of linearity. Proposition 3.4. If f : Γ1 Ñ Γ2 is any map between marked graphs, then repeated reduction makes f into a reduced map. In particular, there is a reduced map in every homotopy class. Proof. Proceed by induction on the number of linear segments of f with non-zero derivative. This is reduced by each reduction operation.  Proposition 3.5. For p, q P t1, 2, 8u with 1 ď p ď q ď 8, reduction does not increase Eqp , and strictly decreases Eqp if p ă q. Proof. Clear from the definitions.



As a result of Propositions 3.4 and 3.5, when looking for energy minimizers we can restrict our attention to reduced maps. 3.2. Taut maps and flows. We now add some more structure, and correspondingly get a more restrictive condition on which maps can be energy-minimizing. We will consider maps from weighted graphs, and in particular how to minimize Eq1 for any q. Definition 3.6. Let c : W Ñ Γ be a PL map from a marked weighted graph W to a marked graph Γ. We defined the multiplicity nc : Γ Ñ Rě0 in Definition 2.4. To define a multiplicity for homotopy classes, for y in the interior of an edge of Γ set nrcs pyq :“ inf nd pyq. dPrcs

The infimum is taken over maps for which nd pyq is defined. Then nrcs pyq depends only on the edge containing y. We say that c is taut if nc “ nrcs almost everywhere. We say that c is locally taut if every point y P Γ has a regular neighborhood N so that nc cannot be reduced by homotopy of c on c´1 pN q. (A regular neighborhood of y is a tubular neighborhood that has no singular points except possibly at y. In particular, it has no other vertices of Γ.)

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It will require work to show that taut maps exist, but if they do exist they have good properties. Lemma 3.7. A taut map from a positive weighted graph is reduced. Proof. Reduction reduces nc .



Proposition 3.8. Let c : W Ñ Γ be a taut map from a marked weighted graph W to a marked graph Γ. If Γ is weighted, then WRpcq is minimal in rcs. If Γ has an elastic structure, then ELpcq is minimal in rcs. If Γ has a length structure, then `pcq is minimal in rcs. Proof. Clear from the definitions, since the energies are monotonic in nc . When Γ is a length graph, we use Lemma 2.12.  Example 3.9. Let W be a marked weighted graph with two marked vertices s and t. Let I be the interval r´1, 1s with the endpoints marked, and let f : W Ñ I be a map with f psq “ ´1 and f ptq “ 1. Then f is taut iff each edge is mapped monotonically and, for each regular value y P I, the set of edges containing f ´1 pyq is a minimal cut-set for W , a set of edges of W that separates s from t and has minimal weight among all such sets. The max-flow/min-cut theorem says that in the context of Example 3.9 the total weight of a minimal cut-set for W is equal to the maximum flow from s to t through the edges of t. We will show that taut maps exist, and that they satisfy a generalization of the max-flow/min-cut theorem; see Theorem 2 and Corollary 3.29 below. To give the strongest statement, we make some definitions and preliminary lemmas first. Lemma 3.10. A marked weighted curve c : C Ñ Γ on a marked graph Γ, with weights that are positive and constant on the components of C, is taut iff it is reduced. Proof. One direction is Lemma 3.7. For the other direction, use the standard fact that, given a non-trivial free homotopy class of maps from S 1 to Γ, a strictly reduced representative is unique up to reparametrization of the domain (which does not change nc ). The same is true for homotopy classes of maps from I to Γ relative to the endpoints.  Definition 3.11. If c : W1 Ñ W2 is a PL map between weight graphs, we say that W2 carries pW1 , cq if WRpcq ď 1. If W2 carries pW1 , cq, then we say that a point y of W2 is saturated if nc pyq “ wpyq. Similarly, a subset of a weighted graph is saturated if almost every point in it is saturated. One notion of a “flow” on a weighted graph W is a taut weighted curve carried by W . These curves are a little awkward to work with directly. As such, we will also define and work with a more general type of flow from train tracks. Definition 3.12. A sequence of non-negative numbers pxi qki“1 is said to satisfy the triangle inequalities if, for each i, xi is no larger than the sum of the remaining numbers: ÿ (3.13) xi ď xj . 1ďjďk i‰j

ELASTIC GRAPHS

17

This implies that k ‰ 1, and if k “ 2 then x1 “ x2 . If k ą 3 and one of these inequalities is an equality, then there is exactly one i so that xi is equal to the sum of the remainder. That xi is said to dominate the rest. Definition 3.14. A train track structure τ on a graph Γ is, for each point x of Γ, a partition of the non-zero directions from x into equivalence classes, called the gates at x, with at least two gates at each non-marked point. A reduced curve on a train track graph makes valid turns if, at each point the curve passes through, the incoming and outgoing directions are in different gates. If Γ is weighted, the weight of a gate is the sum of the weights of the edges corresponding to the directions that make it up. A weighted train track T is a tuple pΓ, w, τ q of a marked weighted graph pΓ, wq with a train track structure τ so that, at each non-marked vertex, the weights of the gates satisfy the triangle inequalities. (If we don’t want to impose the triangle inequalities, we may refer to a weighted graph with a train track structure.) If the weight of one gate g at a vertex of a weighted train track dominates the others, we can smooth the vertex, changing the train track structure so that there are only two gates, one with the directions from g and one with all the other directions. A graph Γ with no unmarked univalent ends has a discrete train track structure δΓ , in which two different directions are never equivalent. By default we use the discrete train track structure on a graph. A weighted graph for which the discrete train track structure satisfies the triangle inequalities is said to be balanced. If f : pΓ1 , τ1 q Ñ pΓ2 , τ2 q is a map between train track graphs that is locally injective on the edges of Γ1 , we say that f is a train-track map if, for each vertex v of Γ1 and directions d1 and d2 at v, ` ˘ ` ˘ (3.15) d1 „τ1 d2 ðñ f pd1 q „τ2 f pd2 q . More generally, we say that f is a train-track map if it has arbitrarily small perturbations f so that f is locally injective on the edges of Γ1 and satisfies Equation (3.15). If f : Γ1 Ñ Γ2 is a strictly reduced map, the train track of f is the unique train-track structure τ pf q on Γ1 so that f is a train-track map with respect to the train track structures τ pf q on Γ1 and δΓ2 on Γ2 . Concretely, ` ˘ ` ˘ d1 „τ pf q d2 ðñ f pd1 q “ f pd2 q . It follows immediately from the definitions that a composition of train-track maps is a train-track map. We can now state the main goal of this section. Theorem 2. Let f : W Ñ Γ be a PL map from a marked weighted graph to a marked graph. Then there is a taut map in rf s. Furthermore, the following conditions are equivalent. (1) The map f is taut. (2) The map f is locally taut. (3) The graph W carries a marked weighted train track pT, tq, with t : T Ñ W , so that f ˝ t is a train-track map and nf ˝t “ nt . We may choose pT, tq so that T is a subgraph of W and t saturates every edge of W on which f is not constant.

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(4) The graph W carries a marked weighted curve pC, cq, with c : C Ñ W , so that f ˝ c is taut and nf ˝c “ nf . We may choose pC, cq so that c saturates every edge of W on which f is not constant. As a preliminary step towards Theorem 2, we relate train tracks and curves. Proposition 3.16. Any weighted train track T “ pΓ, w, τ q carries a marked weighted curve pC, cq that saturates T and so that c : C Ñ T is a train-track map. We can choose pC, cq so that each component of C runs over each edge of Γ at most twice. Proof. Let T 1 Ă T be the sub-train-track of T in which all edges of zero weight have been deleted. Let T 2 be T 1 modified by smoothing all vertices in which one gate dominates the others. Let YardpT 2 q Ă VertpT 2 q be the set of non-marked vertices of T 2 with at least three gates. We will proceed by induction on |EdgepT 2 q| ` |YardpT 2 q|. The base case is when T 1 and T 2 are empty. Otherwise, choose any oriented edge ~e0 of T 2 , and find a path (forward and backwards) from ~e0 within T 2 , always making valid turns. Since there are at least two gates at each non-marked vertex, we can always find a successor edge for ~ei , unless we hit a marked vertex. Since there are only finitely many oriented edges in T , we will eventually either find a path of edges between marked points or see a repeat and find a cyclic loop of edges. Consider the marked curve pC1 , c1 q that runs over the cycle or path. For  ą 0, let w peq :“ wpeq ´ nc1 peq. Then for  sufficiently small, w gives a weighted train-track structure on T 2 : ‚ for each edge e, wpeq ą 0 by construction of T 1 so w peq ě 0; and ‚ for each vertex v, the construction of T 2 ensures that the triangle inequalities at v continue to hold. Let 1 be the maximum value of  so that pΓ, w1 , τ q is a weighted train track. Let w1 “ w1 and T1 “ pΓ, w1 , τ q. Consider the derived train tracks T11 (deleting edges of weight 0) and T12 (smoothing dominating vertices). By the choice of 1 , there is either ‚ an edge e of T 2 with w1 peq “ 0 but wpeq ‰ 0, or ‚ a vertex v with a gate g so that w1 pgq dominates the other gates at v but wpgq does not. In the first case, |EdgepT12 q| ă |EdgepT 2 q|. In the second case, |YardpT12 q| ă |YardpT 2 q|. In either case, by induction T1 carries a marked weighted curve pC2 , c2 q that saturates T1 . Then c2 \ c1 : C2 \ C1 Ñ T is the desired curve c from the statement, where we give C1 the weight 1 . If we make a cyclic loop as soon as we see a repeated oriented edge, the components of C run over each (unoriented) edge at most twice.  Lemma 3.17. If f : W Ñ Γ is a map from a marked weighted graph to a marked graph, and pC, cq is a marked weighted curve carried by W so that f ˝ c is taut and nf ˝c “ nf , then f is taut. Proof. If g : W Ñ Γ is any other map in rf s, then ng ě ng˝c ě nrf ˝cs “ nf ˝c “ nf ,

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19

using the fact that W carries c (and Lemma 2.15), the definition of nrf ˝cs as an infimum over the homotopy class, and the hypothesis.  Lemma 3.18. A train-track map from a marked weighted train track to a marked train track is taut. Proof. Immediate from Proposition 3.16 and Lemma 3.17.



3.3. Local models. To prove Theorem 2, we will first analyze the situation locally in a regular neighborhood of a singular value in Γ. This case reduces to studying maps from a graph with k marked points to a k-leg star graph. Let Stark be the star graph with k legs, with a central vertex s˚ , marked endpoints s1 , . . . , sk , and edges rsi , s˚ s. A k-marked graph is a graph with k marked vertices tvi uki“1 (in order). There is a canonical homotopy class of marked maps from any k-marked graph to Stark , taking vi to si . The goal of this section is an analogue of Theorem 2 in this context. Proposition 3.19. Let W be a k-marked weighted graph. Then there is a taut map in the canonical homotopy class of maps to Stark . Furthermore, the following conditions are equivalent. (1) The map f is taut. (2) The graph W carries a marked weighted train track pT, tq so that f ˝ t is a train-track map and nf ˝t “ nf . (3) The graph W carries a marked weighted curve pC, cq so that f ˝ c is taut and nf ˝c “ nf . Definition 3.20. A cut of a graph Γ is a partition of the vertices of the graph into two disjoint subsets: VertpΓq “ S \ S. The corresponding cut-set cpSq “ cpSq is the set ofředges that connect S to S. If Γ has weights w, the weight of the cut is wpSq :“ ePcpSq wpeq. Two cuts S1 \ S1 and S2 \ S2 are nested if they are disjoint or one is contained in the other : S1 X S2 “ H, S1 Ă S2 , or S2 Ă S1 . (It doesn’t matter if we replace S1 with S1 and/or replace S2 with S2 .) Let W be a k-marked weighted graph. A vi -cut is a subset Si Ă VertpGq so that Si X tv1 , . . . , vk u “ tvi u. A vertex cut is a vi -cut for some i. A minimal vi -cut is one with minimal weight. Let mincuti pwq be the weight of a minimal vi -cut. Lemma 3.21. If S1 \ S1 and S2 \ S2 are two cuts on the same weighted graph, then wpS1 q ` wpS2 q ě wpS1 X S2 q ` wpS1 Y S2 q. Proof. This is the standard sub-modular property of cuts, and is easy to prove.



Lemma 3.22. If W is a k-marked weighted graph and Si and Si1 are two minimal vi -cuts, then Si X Si1 and Si Y Si1 are also minimal vi -cuts. Proof. The sets Si X Si1 and Si Y Si1 are vi -cuts, so by minimality they all have weight at least as large as mincuti pwq “ wpSi q “ wpSi1 q. Lemma 3.21 gives the other inequality. 

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Lemma 3.23. If W is a k-marked weighted graph, Si is a minimal vi -cut, and Sj is a minimal vj cut for j ‰ i, then Si zSj is also a minimal vi -cut and Sj zSi is also a minimal vj -cut. Proof. Si zSj is a vi -cut and Sj zSi is a vj -cut. By minimality, their weights are at least as large as mincuti pwq and mincutj pwq, respectively. Applying Lemma 3.21 to Si and Sj gives the other inequalities.  Definition 3.24. If W is a k-marked weighted graph, we say that an edge of W is slack if it has non-zero weight and is not contained in any minimal vertex cut. W is minimal if it has no slack edges. See Figure 2 for the next two lemmas. Lemma 3.25. If W “ pΓ, wq is a weighted graph with k marked vertices, then there is a set of weights w0 ď w on Γ so that W0 “ pΓ, w0 q is minimal and so that for all i, mincuti pwq “ mincuti pw0 q. Proof. We prove the lemma by induction on the number of slack edges. If W is not minimal, pick any slack edge e0 of W . For 0 ď k ď wpe0 q, define a modified set of weights by # k e “ e0 wte0 ÞÑ kupeq :“ wpeq otherwise. Let k0 be minimal so that 0 ď k0 ď wpe0 q and, for all i, mincuti pwte0 ÞÑ k0 uq “ mincuti pwq. By the choice of k0 , the edge e is not slack with respect to wte0 ÞÑ k0 u, and so by induction we can find a desired set of weights w0 ď wte0 ÞÑ k0 u ď w on Γ so that pΓ, w0 q is minimal.  Lemma 3.26. Let W “ pΓ, wq be a k-marked weighted graph that is minimal and let S Ă PpPpVertpW qqq be a nested set of minimal vertex cuts. Then there is a nested set of minimal vertex cuts T Ą S so that every edge of W with non-zero weight is contained in at least one cut-set of a cut in T . (Here PpXq is the power set of X.) Ť Proof. We proceed by induction on the size of EdgepW qz SPS cpSq. If there is at least one such edge e1 with non-zero weight, it is contained in some minimal vertex cut-set cpS1 q by minimality of W . Then, by repeatedly applying Lemmas 3.22 and 3.23, we can find another minimal vertex cut S11 so that e P cpS11 q and S11 is nested with respect to all S P S. By induction, S Y tS11 u can be completed to the desired set T .  Proof of Proposition 3.19. We first prove For each i, pick Ť the existence of a taut map. 1 1 a minimal vi -cut Si , and let Si “ Si z j‰i Sj . By Lemma 3.23, tSi u forms a nested collection of minimal vi -cuts.ŤDefine f : W Ñ Stark by mapping all vertices in Si to si , all vertices in VertpGqz i Si to s˚ , and all edges to reduced paths connecting their endpoints. (Thus f is constant on any edge not in any of the cpSi q.) Then f is taut since the Si are minimal.

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v3 9

v3

15

9

16 17

2

2

10

5 14 13

v1

8

3

11

4 1

6 7

15

10 3

18 12 11

14

v2

v1

4 1

8

3

6 7

2

9 11

v2

Figure 2. Finding minimum cuts and maximum flows near a vertex. Left: a weighted graph W with three marked vertices, with the weights indicated. Minimal cuts that separate the vertices are marked; these are used to construct a taut map from W to Star3 . Right: a weighted graph W0 carried by W , with minimal weights that have the same mincuti for each i. The minimal cuts on W have been extended to a complete nested family of minimal cuts. These are used to construct a train track carried by W . Proposition 3.16 and Lemma 3.10 tell us that (2) implies (3), and Lemma 3.17 tells us that (3) implies (1), so it remains to prove that (1) implies (2). Suppose that f is taut. Use Lemma 3.25 to find weights w0 ď w so that pΓ, w0 q is minimal. Let Γ0 Ă Γ consist of those edges e for which w0 peq ‰ 0, let W0 “ pΓ0 , w0 q, and let t : W0 ãÑ W be the inclusion. We next construct a set of cuts from the initial map f . For each y P rs˚ , si s Ă Stark that is a regular value for f , consider the vertex cut Sy “ tv P VertpΓq | f pvq P ry, si su. Then Sy is a minimal vertex cut, since f is taut. Furthermore, if z is another regular value of f (on the same edge of Stark or a different one), then Sy and Sz are nested. Let S “ t Sy | y P Stark , y regular u, considered as a nested system of cuts on W0 . By Lemma 3.26, S can be completed to a nested system of cuts T on W0 that contains every edge in at least one cut-set. For each i, let the distinct vi -cuts in T be tvi u “ Si,0 Ĺ Si,1 Ĺ ¨ ¨ ¨ Ĺ Si,ni . For each i and for 0 ď j ď ni , pick distinct points xi,j P rsi , s˚ s Ă Stark with xi,0 “ si and with the xi,j in order. Then define a map g : W0 Ñ Stark on vertices by $ ’ s v “ vi ’ & i gpvq “ xi,j v P Si,j zSi,j´1 ’ ’ Ť %s v P VertpW qz S. ˚

0

SPT

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THURSTON

(These cases are exclusive since T is nested.) On an edge e of W0 , define gpeq to be a reduced path connecting its endpoints. Since T is a complete set of cuts, no edge maps to a single point. Let τ0 “ τ pgq. Then pW0 , τ0 q is a weighted train track: the relevant triangle (in)equalities are implied by the fact that the Si,j are all minimal cuts. By an appropriate choice of the xi,j , the train-track map g can be made to be an arbitrarily small perturbation of f ˝ t, so f ˝ t is also a train-track map. If e is an edge on which the original map f is not constant, then e is not slack and wpeq “ w0 peq, so nf ˝t “ nf as desired.  3.4. General flows. We now use these local models to prove Theorem 2. Definition 3.27. Let f : W Ñ Γ be a PL map from a marked weighted graph W to a marked graph Γ. For y P Γ, we define a local model for f at y. Let N Ă Γ be a closed regular neighborhood of y. Then the local model for f at y is the map f ´1 pN q{„ ÝÑ N, where ‚ N is considered as a marked graph (equivalent to Stark ) with marked points at BN ; ‚ „ is the equivalence relation that identifies two points in f ´1 pBN q if they map to the same point in BN ; and ‚ f ´1 pN q{„ is considered as a k-marked weighted graph, with weights inherited from W . Lemma 3.28. Let rf s : W Ñ Γ be a homotopy class of maps from a marked weighted graph W to a marked graph Γ. Then there is a locally taut map in rf s. Proof. Suppose f is any PL representative for its homotopy class. We can modify f to send vertices to vertices without increasing nf , as follows. For each edge e of Γ, look at the division of e into intervals according to the value of nf . Pick one of these intervals on which nf is minimal (among the values of nf that appear), and spread out this interval by a homotopy until it covers e. This gives us an initial map f0 . If f0 is not locally taut, then there is some vertex v of Γ for which the local model of f0 at v is not taut. By Proposition 3.19, there is a taut map in the homotopy class of the local model. Let f01 be f0 with the map replaced by its taut model near v. There is at least one edge e of Γ with a segment near v on which nf01 ă nf0 . Homotop f01 as above, spreading out segments of minimal multiplicity, to construct a map f1 that sends vertices to vertices with nf1 ď nf0 everywhere and a strict inequality on e. Now repeat the process, with f1 in place of f0 , reducing nf at each stage. Our initial representative f0 gives an upper bound on nfi peq for all e. Since there are only finitely many linear combinations of the non-zero weights of edges of W to get a value less than this bound, and at each step nf strictly decreases on at least one edge, the process terminates in a locally taut map in a finite number of steps.  Proof of Theorem 2. We first prove the equivalence of the four conditions on taut maps. (1) implies (2) is obvious.

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To see that (2) implies (3), suppose f is locally taut. Then for each singular value y P Γ, the local model for f at y carries a weighted train-track compatible with f by Proposition 3.19. Define a weighted train-track T and map t : T Ñ W by assembling these local models following the pattern of W , leaving W unchanged outside of the local models. Then pT, tq is the desired train track carried by W . Proposition 3.16 shows that (3) implies (4). Lemma 3.17 shows that (4) implies (1). Now if rf s is any homotopy class, by Lemma 3.28 there is a locally taut element of rf s, which by the above equivalences is also globally taut. The statements about saturation follows immediately from the definitions.  3.5. Connection to max-flow/min-cut. We can connect up Theorem 2 and Proposition 3.19 to a statement that looks more like the classical max-flow/min-cut problem (Example 3.9). For simplicity of the statements, we restrict to the local setting of Proposition 3.19. Given a graph Γ with k marked points tvi uki“1 , define a flow from vi to vj to be a marked weighted curve pC, cq on Γ, with each component a marked interval with one endpoint mapping to vi and the other mapping to vj . Such a flow has multiplicities nc on each edge as usual, as well as a total weight wpcq, the sum of the weights of all components of C. Corollary 3.29. Let W be a weighted graph with k marked points tvi uki“1 . Then we can find ‚ for i “ 1, . . . , k, a vi -cut Si , and ‚ for i, j “ 1, . . . , k with i ă j, a flow cij “ cji from vi to vj so that ‚ the collection of all flows cij is carried by W : for each edge e of Γ, ÿ (3.30) ncij peq ď wpeq, iăj

‚ the total flow into a vertex equals the weight of the cut: for each i, ÿ (3.31) wpcij q “ wpSi q. j‰i

In this situation, for each i, wpSi q is minimal and

ř j‰i

wpcij q is maximal.

Proof. Let f be a taut map in the canonical homotopy class of maps from W to Stark , as given by Proposition 3.19. By that proposition, W carries a marked weighted curve pC, cq so that f ˝ c is taut and nf ˝c “ nf . The only non-trivial homotopy classes of marked curves on Stark are given by paths connecting si to sj for some i ‰ j. Let the flow cij be the flow given by those components of pC, cq that connect vi to vj . Then the fact that W carries pC, cq is equivalent to Equation (3.30). On the other hand, for each i, let yi be a regular value on rs˚ , si s, and let Si be Syi as defined in the proof of Proposition 3.19. Then the equality nf ˝c pyq ř “ nf pyq is equivalent to Equation (3.31). Minimality of wpSi q and maximality of j‰i wpcij q both follow from tautness, and indeed are easy to deduce from Equation (3.31). 

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3.6. Strongly reduced maps. We introduce a stronger notion of “reduced”, for maps between general marked graphs, without a weight structure. Definition 3.32. A map f : Γ1 Ñ Γ2 between marked graphs is strongly reduced if it is taut for some choice of positive weights on Γ1 . Lemma 3.33. If f : Γ1 Ñ Γ2 is strongly reduced, then for each edge e of Γ1 , either f is constant on e or else there is a curve pC, cq on Γ1 that runs over e and so that both c and f ˝ c are reduced. Furthermore, pC, cq can be chosen to be connected and run over each edge of Γ1 at most twice. Proof. The first part is a consequence of the fact that (1) implies (4)) in Theorem 2. For the second part, use Proposition 3.16 to guarantee that the relevant connected component of pC, cq runs over each edge at most twice.  Definition 3.34. Given an edge-reduced PL map f : Γ1 Ñ Γ2 between marked graphs, the combinatorial type of f consists of the following discrete data. (1) For each vertex v of Γ1 , whether f pvq is on an vertex or in the interior of an edge of Γ2 , and which vertex or edge it lies on. (2) For each oriented edge ~e of Γ1 , the edge-path of f p~eq, the reduced sequence of oriented edges of Γ2 that f p~eq travels over. This edge-path may start and/or end with a partial edge, depending on whether the endpoints of ~e map to vertices or edges. There are degenerate cases when ~e maps to a single vertex or stays within the interior of a single edge. The combinatorial type of f determines the homotopy class of rf s. Proposition 3.35. Let rf s : Γ1 Ñ Γ2 be a homotopy class of maps between marked graphs. Then there are only finitely many combinatorial types of maps appearing for strongly reduced maps in rf s. Proof. Let g P rf s be strongly reduced, and let ~e be an oriented edge of Γ1 . We must show that there is a finite set of possible edge-paths recording f p~eq. For this, we apply Lemma 3.33. Either f is constant on e (with only finitely many combinatorial possibilities), or else there is a curve pC, cq running over e as in the statement. Since pC, cq runs over each edge of Γ1 at most twice, there are only finitely many possibilities for it. Furthermore, rf ˝ cs is determined by rf s and rcs, so there are only finitely many edge-paths in Γ2 arising from rf ˝ cs. The edge-path of f p~eq must be a sub-path of the edge-path of f ˝ c, and again there are only finitely many possibilities.  Remark 3.36. Proposition 3.35 is false if we look at reduced maps rather than strongly reduced maps. (The map f can be “spun” around a taut cycle of Γ1 .) 3.7. Restricting the domain and range. When considering whether maps are taut, we can ignore edges of weight 0. Lemma 3.37. Let f : W Ñ Γ be a map from a marked weighted graph to a marked graph. Let W0 Ă W be a subgraph so that, for e R EdgepW0 q, we have wpeq “ 0. Then f is taut iff the restriction f |W0 : W0 Ñ Γ is taut.

ELASTIC GRAPHS

Proof. Edges of weight 0 do not effect nf .

25



Definition 3.38. A homotopy class rf s : Γ1 Ñ Γ2 of marked maps is essentially surjective if every element of rf s is surjective. Remark 3.39. If Γ1 and Γ2 are connected with no marked points and no unmarked 1-valent vertices, then if rφs : Γ1 Ñ Γ2 is π1 -surjective it is essentially surjective. If f is not (essentially) surjective, then tautness of f is equivalent to tautness of the map with restricted range. Lemma 3.40. Let f : W Ñ Γ be a map from a marked weighted graph to a marked graph. Suppose the image of f is contained in a subgraph Γ0 Ă Γ. Let Γ˚ be Γ with all edges not in Γ0 collapsed, with κ : Γ Ñ Γ˚ the collapsing map. Then the following are equivalent: (1) f is taut, (2) the map with restricted range f : W Ñ Γ0 is taut, and (3) the map κ ˝ f : W Ñ Γ˚ is taut. Proof. Recall from Theorem 2 that being taut is equivalent to being locally taut. If you replace “taut” with “locally taut” in each of the three conditions, they are easily seen to be equivalent.  3.8. Continuity. Finally, let rf s : Γ1 Ñ Γ2 be a homotopy class of maps between marked graphs. We are interested in how nrf s varies as the weights on Γ1 vary. Define Nrf s : WpΓ1 q Ñ WpΓ2 q by Nrf s pwq :“ nw rf s . (There is usually no single g P rf s that is taut as a map from pΓ1 , wq for all w.) Lemma 3.41. Let rf s : Γ1 Ñ Γ2 be a homotopy class of maps between marked graphs. Then Nrf s is continuous. Proof. Pick w0 P WpΓ1 q and e0 P EdgepΓ2 q, and let K “ Nrf s pw0 qpe0 q. Suppose that w P WpΓ1 q is some set of weights with |wpeq ´ w0 peq| ă  for all e. To show continuity of Nrf s near w0 , we will give upper and lower bounds on Nrf s pwqpe0 q in terms of K and . To get an upper bound, let f0 P rf s be a taut map from pΓ1 , w0 q to Γ2 . Pick a regular value y P e0 of f0 , and let k “ |f0´1 pyq|. Then since f0 P rf s, w0 Nrf s pwqpe0 q ď nw f0 pyq ď nf0 pyq ` k “ K ` k,

as desired. To get a lower bound, let E1 be t e P EdgepΓ1 q | w0 peq ‰ 0 u. Let ν “ minePE1 w0 peq. Make sure  ă ν, and let M “ K{pν ´ q. Let g : pΓ1 , wq Ñ Γ2 be a taut map in rf s. Now nw g pe0 q can be written as an integer linear combination of weights of edges: ÿ nw pe q “ ae wpeq. 0 g ePEdgepΓ1 q

26

THURSTON

If nw g pe0 q ě K, we are done. Otherwise, we must have ÿ pe q ě ae wpeq nw 0 g

ř ePE1

ae ď M , and so

ePE1

ě ´M  `

ÿ

ae w0 peq

ePE1

ÿ

“ ´M  `

ae w0 peq

ePEdgepΓ1 q

ě ´M  ` K. 0 The last inequality uses the fact that nw g pe0 q ě K, which follows from the definition w0  of K as nrf s pe0 q.

Lemma 3.42. Let rf s : Γ1 Ñ Γ2 be a homotopy class of maps between marked graphs. Then Nrf s is piecewise-linear. Proof. Given a set of weights w P WpΓ1 q, the value of Nrf s pwq on an edge e can be computed by computing a taut representative g P rf s and finding the set g ´1 pxq for x a generic point on e close to one of the endpoints. It is easy to see that g ´1 pxq is determined by the combinatorial type of g, which by Proposition 3.35 is restricted to a finite set of possibilities. For each possibility, ng peq is an integer linear combination of weights on Γ1 , as desired.  4. Weak graphs and Lipschitz energy We now turn to minimizing the Lipschitz stretch factor of maps between graphs. Theorem 3 (White). Let rφs : K1 Ñ K2 be a homotopy class of maps between marked length graphs. Then there is a representative ψ P rφs and a connected marked curve c : C Ñ K1 so that the sequence c

ψ

C ÝÑ K1 ÝÑ K2 is tight. In particular, Liprφs “ Lippψq “

`rψ ˝ cs `rψ ˝ cs “ sup , `rcs `rcs c:CÑK1

where the supremum runs over all connected non-trivial curves. We get the same quantity if we take the supremum over curves that cross each edge of K1 at most twice. A version of Theorem 3 appears in papers by Bestvina [Bes11, Proposition 2.1] and Francaviglia-Martino [FM11, Proposition 3.11], and is attributed to Tad White (unpublished). Both papers work in the context of outer space and so assume that φ is a homotopy equivalence, although that assumption is never used in the proof. The extension to marked graphs is also immediate. It turns out we need a further extension of Theorem 3. In order to prove Theorem 1, we will need to understand the behavior of energies on the boundary of L` pΓq. This leads us to a notion of weak graphs, in which some edges have length 0. The definition

ELASTIC GRAPHS

27

Γ2

fr

Γ1

κ v

f

Γ˚2

Figure 3. A weak graph map where an exact lift is impossible. On the right is a weak graph Γ2 (in which the red edge is a null edge) and the collapsed graph Γ˚2 . The map f : Γ1 Ñ Γ˚2 maps the edge of Γ1 forward and backward over the edge of Γ˚2 , so that the inverse image of the vertex v is a single point. The local lift fr: Γ1 Ñ Γ2 maps the edge of Γ1 around the null loop. of maps between weak graphs requires a little care, since we want to remember homotopy information that may not be present at the level of the collapsed graph. Definition 4.1. A weak graph is a graph Γ in which certain edges are designated as null edges; the union of the null edges forms the null graph. The collapsed graph Γ˚ of Γ is the graph obtained by identifying each null edge to a single point. There is a natural collapsing map κ : Γ Ñ Γ˚ . Observe that κ is a homotopy equivalence iff the null graph is a forest. As in the introduction, a length graph K is a pair pΓ, `q of a graph and an assignment of a length `peq ą 0 to edge e of Γ. A weak length graph is such a pair where `peq is allowed to be 0. In this case, we consider Γ to be a weak graph, where the null edges are the edges of length 0. The lengths `peq determine a pseudo-metric on Γ and a metric on Γ˚ . The space of all weak lengths on Γ is LpΓq Ą L` pΓq. If Γ1 and Γ2 are weak graphs, a PL weak map from Γ1 to Γ2 is a pair pfr, f q of a PL map f : Γ˚1 Ñ Γ˚2 between the collapsed graphs, together with a map fr: Γ1 Ñ Γ2 that is a local homotopy lifting of f , in the followingŤsense. Pick disjoint regular neighborhoods Nv˚ of each vertex v of Γ˚2 . Let N2˚ “ v Nv˚ Ă Γ˚2 , N1˚ “ f ´1 pN2˚ q, ´1 ˚ ˚ r´1 pN2 q and that the N2 “ κ´1 2 pN2 q, and N1 “ κ1 pN1 q. Then we require that N1 “ f diagram Γ1 (4.2)

fr

κ1

Γ˚1

Γ2 κ2

f

Γ˚2

commute up to a homotopy that is the identity on Γ1 zN1 . We can not in general require that fr is an exact lift of f , as those do not always exist; see Figure 3. When we speak about an energy of a weak map pfr, f q between weak graphs in the sense of Definition 4.1 (with some additional structure, as appropriate for the

28

THURSTON

particular energy), we mean the energy of the map f between collapsed graphs, using the same analytic expression as usual. This applies to all of the energies we have introduced so far, including the Dirichlet, Lipschitz, and embedding energies. On the other hand, by the homotopy class of the weak map we mean the homotopy class rfrs of the map between the uncollapsed graphs. If the null graph has non-trivial π1 , then rf s will have less information than rfrs, as in Figure 3. We consider two weak maps pfr, f q and pr g , gq to be equal if f “ g as functions and fr and gr are locally homotopic. We next generalize the notions of reduced maps. Definition 4.3. Let pfr, f q : pΓ1 , Γ˚1 q Ñ pΓ2 , Γ˚2 q be a weak marked map. Suppose we have point y P Γ˚2 , a regular neighborhood Ny˚ of y, and a component Z of ´1 κ´1 pNy˚ qq that is not an entire component of Γ1 . Then we say that Z is a dead 1 pf end of f if the local lift fr can be changed by a marked homotopy so that frpZq does ´1 ˚ not intersect κ´1 2 pyq. Concretely, we have the following, where Ny “ κ2 pNy q and ´1 Fy “ κ2 pyq. (1) If Z contains a marked point x, then Z is not a dead end, since frpxq is a marked point in Fy . (2) If there are two points x1 , x2 P BZ so that frpx1 q and frpx2 q are distinct points of BNy , then Z is not a dead end, since the image of a path between x1 and x2 necessarily passes through Fy . (3) If there are points x1 , x2 P BZ (possibly identical) so that frpx1 q and frpx2 q are the same point BNy and there is a path γ from x1 to x2 in Z so that frpγq is non-trivial in π1 pNy , frpx1 qq, then Z is not a dead end. (4) Otherwise, Z is a dead end. (The above conditions could be stated more uniformly with fundamental groupoids.) If pfr, f q has no dead ends, we say that it is reduced. As with non-weak maps, we can reduce dead ends: if Z is a dead end mapping to y P Γ˚2 , modify f so that f pκ2 pZqq misses y and continuing to pull until you reach another singular value of f . Proposition 4.4. Every weak map is homotopic to a reduced map. Reduction does not increase Eqp for any p ď q. Proof. As in Proposition 3.4, repeatedly reduce dead ends. As in Proposition 3.5, energies are clearly not increased.  Theorem 31 . Let rφs : K1 Ñ K2 be a homotopy class of maps between weak marked length graphs, so that K1 has at least one non-null edge and so that no null cycle in K1 maps to a non-null cycle in K2 . Then there is a weak map ψ P rφs and a connected marked curve c : C Ñ K1 with `rcs ą 0 so that the sequence c

ψ

C ÝÑ K1 ÝÑ K2

ELASTIC GRAPHS

29

is tight. In particular, Liprφs “ Lippψq “

`rψ ˝ cs `rψ ˝ cs “ sup , `rcs `rcs c:CÑK1 `rcsą0

where the supremum runs over all connected curves c of positive length in K1 . We get the same quantity if we take the supremum over connected curves of positive length that cross each edge of K1 at most twice. To prove Theorem 31 , we introduce a finite-dimensional approximation to the space of all maps from K1 to K2 . We first treat the case of ordinary (non-weak) maps. Definition 4.5. A PL map f : K1 Ñ K2 between marked length graphs is constantderivative if it is edge-reduced and |f 1 | is constant on each edge of K1 . A constantderivative map is determined by its combinatorial type (Definition 3.34) plus a bounded amount of continuous data: for each vertex v of Γ1 that maps to the interior of an edge e of Γ2 , record where f pvq is along e. Thus we can assemble the constantderivative maps in rf s into a finite-dimensional polyhedral complex PL˚ rf s, with dimension given by the number of non-marked vertices of Γ1 . PL˚ rf s is locally finite. For D ą 0 a constant, let PL˚ rf sďD Ă PL˚ rf s be the subspace of maps g P PL˚ rf s with Lippgq ď D. Lemma 4.6. For any homotopy class rf s : K1 Ñ K2 of maps between marked length graphs and any D ą 0, the space PL˚ rf sďD is a compact polyhedral complex. Proof. The bound on |g 1 peq| gives a bound on how many edges of K2 the image of an edge e of K1 can cross. Therefore only finitely many different cells of PL˚ rf s intersect PL˚ rf sďD .  We now turn to versions of Definitions 3.34 and 4.5 for weak maps. Definition 4.7. The combinatorial type of an edge-reduced weak map pfr, f q : Γ1 Ñ Γ2 between weak marked graphs consists of the following discrete data. ‚ For each vertex v of Γ˚1 , whether f pvq is on a vertex or in the interior of an edge of Γ˚2 , and which vertex or edge it lies on. ‚ The homotopy class of fr as a map from Γ1 to Γ2 . As in Definition 3.34, the homotopy class of fr is conveniently recorded by picking, for each vertex v of Γ1 , a lift of f pκ1 pvqq to a point frpvq P Γ2 , and then recording, for each oriented edge ~e of Γ2 , the reduced sequence of (partial) edges that f p~eq passes over. But this is not uniquely specified by the homotopy class of fr: different choices of lifts of f pκ1 pvqq will lead to different sequences of edges in f p~eq. The combinatorial type of the map only depends on the homotopy type of fr, not on the auxiliary choices. Definition 4.8. A weak PL map pfr, f q : K1 Ñ K2 between weak marked length graphs is constant derivative if it is edge-reduced and |f 1 | is constant on each non-null edge of K1 . As before, we can assemble the constant-derivative maps in rf s into a finite-dimensional complex PL˚ rf s (with dimension bounded by the number of vertices of K1˚ ), and if we give an upper bound on Lippf q we get a subcomplex PL˚ rf sďD .

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THURSTON

Lemma 4.9. For any homotopy class rf s : K1 Ñ K2 of maps between weak marked length graphs and any constant D ą 0, the space PL˚ rf sďD is a compact polyhedral complex. Proof. As in Lemma 4.6.



Proof of Theorem 3 1 . We follow the previous proofs [Bes11, FM11], adapted to the setting of weak maps between marked graphs. First, we may assume a weak map in rφs of minimal Lipschitz energy is in PL˚ rφs. The condition that no null cycle of K1 maps to a non-null cycle of K2 guarantees that there is some map pψr0 , ψ0 q P PL˚ rφs of finite Lipschitz energy. Then any optimizer r ψq must be in PL˚ rφsďLippψ q . By Lemma 4.9 we are minimizing a continuous pψ, 0 r ψq P rφs be a weak function over a compact set, so the minimum is achieved. Let pψ, map realizing Liprφs. By Proposition 4.4, we may suppose that ψ is reduced. Next we prove the existence of a connected marked curve c exhibiting the Lipschitz stretch of ψ. For each non-null edge e of K1 , say that it is ‚ a tension edge if |ψ 1 peq| is the maximal value, Lippψq, and ‚ a slack edge if |ψ 1 peq| ă Lippψq. Assume that the set of tension edges is minimal among maps with minimal Lippψq. To find the desired curve, we will find a suitable sequence of oriented edges p~ei qN i“1 of K1 passing only over tension and null edges. Lemma 4.10. In the setting above, let ~e0 be an oriented tension edge of K1 . Then we can find a reduced sequence of edges ~e1 , . . . , ~ek on K1 so that either ‚ the ~ei for i “ 1, . . . , k are null edges and ~ek ends at a marked point, or ‚ the ~ei for i “ 1, . . . , k ´ 1 are null edges and ~ek is a tension edge. In either case, the local model for pψp~e0 q, . . . , ψp~ek qq is also reduced. Proof. Let C1 be the connected component of the null graph of K1 that contains the end of ~e0 . The collapsing map κ1 maps C1 to a point x. Let y “ ψpx1 q and C2 be the relevant component of κ´1 2 pyq. As such, C2 may be a point on an edge of K2 , a vertex of K2 , or a connected null subgraph of K2 . We proceed by cases on C1 and C2 , parallel to the cases in Definition 4.3. (1) If C1 has a marked point x, connect the end of ~e0 to x through null edges. (2) If there are is another tension edge f~ of K1 incident to C1 so that rp~e0 q and f~ are mapped by ψ ˝ κ to different directions in K2˚ , connect ~e0 to f~ by a reduced path in K1 . (Here rp~eq is the reverse of the oriented edge ~e.) (3) If π1 pC1 q maps non-trivially to π1 pC2 q, connect ~e0 to rp~e0 q by a reduced loop in C1 mapping to a non-trivial loop in C2 . (4) If none of the above cases apply, we get a contradiction, as follows. Partition the non-null edges incident to C1 according to which direction they map to from y. (Put each edge with |ψ 1 | “ 0 into its own part.) Since we are not in cases (1) or (3), we can displace y P K2˚ in any direction, reducing |ψ 1 | on one group of edges at the cost of increasing it on all other groups. Since we are not

ELASTIC GRAPHS

31

in case (2), we can reduce |ψ 1 p~e0 q| so e0 is no longer a tension edge without creating another tension edge, contradicting the assumption that the set of tension edges of ψ was minimal. In any of cases (1)–(3), by construction the sequence of edges in K1 is reduced, and the path in K2 falls into one of the cases of Definition 4.3 and so is reduced as a weak map.  We can now complete the proof of Theorem 31 . Start with any tension edge ~e0 of K1 . (There is at least one since not all edges of K1 have length 0.) Use Lemma 4.10 to construct a non-backtracking chain of tension and null edges forward and backward from ~e0 . Either the chain ends in a marked point both forward and backward, or there is a repeated oriented edge in the chain. In either case we can extract a connected marked curve c (either an interval or a cycle) so that both c and ψ ˝ c are (weakly) reduced. Thus Lippψq “ `pψ ˝ cq{`pcq “ `rψ ˝ cs{`rcs. If we make the null paths as efficient as possible and close off to make a cycle the first time that an oriented edge is repeated in the chain, then c will cross each (unoriented) edge of K1 at most twice, as in Proposition 3.16.  Remark 4.11. The proof of Theorem 31 is similar to the proof of Proposition 3.16. However, the presence of null edges introduces extra complications in Theorem 31 , while in Proposition 3.16 connected curves do not suffice without a further argument. 5. Harmonic maps 5.1. Definition and existence. In this section, we give a concrete description of harmonic maps and prove their existence, making the intuitive description of harmonic maps from the introduction more precise. See Section 1.1 for a summary of prior work. Definition 5.1. Let G “ pΓ, αq be a marked elastic graph and let K be a marked length graph. Fix a homotopy class rf s : G Ñ K. Recall from Definition 4.5 that the space PL˚ rf s of constant derivative maps is the space of PL maps for which |f 1 | is constant on each edge of G.1 To any constant-derivative map f , we can associate the tension-weighted graph Wf “ pΓ, |f 1 |q; that is, the weight of an edge e is |f 1 peq|. Then we say that f is harmonic if it is constant-derivative and the associated map f : Wf Ñ K is taut. The condition of tautness is in general not obvious to check. We can simplify the condition in this case to look more like the intuitive inequalities in the introduction. Proposition 5.2. Let f : G Ñ K be a constant-derivative map from a marked elastic graph to a marked length graph. Let G0 Ă G be the subgraph of G on which |f 1 | ‰ 0, and let f0 be the restriction of f to G0 . Then f is harmonic iff pWf0 , τ pf0 qq forms a marked weighted train track according to Definition 3.14. Here, τ pf0 q is the train track of f0 as defined in Definition 3.14. Another interpretation of Proposition 5.2 is that f is harmonic iff, at each non-marked vertex v 1We

extend the definition slightly to allow the domain to be an elastic graph.

32

THURSTON

of Γ, the non-zero values of |f 1 pdq| as d ranges over directions at v satisfy the triangle inequalities when grouped according to the image direction f pdq. Proof. By Lemma 3.37, we may restrict attention to G0 . Then the map f is harmonic iff the map f0 : Wf0 Ñ K is taut. By Theorem 2, there is a weighted train track T with underlying graph G0 so that f0 ˝ t is a train-track map. (We use the fact that f0 is not constant on any edge of G0 .) We must therefore have T “ pWf0 , τ pf0 qq.  Theorem 4. Let rf s : G Ñ K be a homotopy class of maps from a marked elastic graph to a marked length graph. Then there is a harmonic map in rf s. Furthermore, the following conditions are equivalent. (1) (2) (3) (4)

The The The The

map f is a global minimum for Dir. map f is a local minimum for Dir. map f is harmonic. natural map ι : Wf Ñ G is part of a tight sequence f

ι

Wf ÝÑ G ÝÑ K. (5) There is a weighted curve pC, cq on G that forms a tight sequence f

c

C ÝÑ G ÝÑ K. Proof. By definition, (1) implies (2). To show that (2) implies (3), let f be a local minimizer for Dirpf q within rf s; we wish to show f is harmonic. The fact that f is constant-derivative is one of the first results in calculus of variations. So we only need to show the triangle inequalities at the vertices. Let v be a non-marked vertex of G of valence k, and let d1 , . . . , dk be the directions of K at f pvq. For small  ą 0, let fi, be the function f modified on the vertices by moving f pvq a distance  the direction di , and extended to the edges so that fi, is still constant-derivative. By hypothesis Dirpf q ď Dirpfi, q. We then have ÿ ` ˘ Dirpfi, q “ Dirpf q ` ´2|f 1 pdq| ` 2 {αpdq dPf ´1 pdi q

ÿ

ÿ

`

`

˘ 2|f 1 pdq| ` 2 {αpdq

j‰i dPf ´1 pdj q

ÿ `

2 {αpdq.

dPf ´1 p0q

Here, the sums run over directions d at v that so that f pdq is di , dj , or 0, respectively. To be a local minimum, we must have ˘ˇˇ d` Dirpfi, q ˇ ě 0. d “0 This gives the i’th triangle inequality, as desired.

ELASTIC GRAPHS

33

To see that (3) implies (4), suppose that f is harmonic. Then f ˝ ι is taut and therefore energy-minimizing, by Definition 5.1. Furthermore, ÿ ELpιq “ αpeq|f 1 peq|2 “ Dirpf q ePEdgepΓq

`pf ˝ ιq “

ÿ

|f 1 peq| `pf peqq “ Dirpf q,

ePEdgepΓq

so the energies are multiplicative, as desired. Proposition 3.16 tells us that (4) implies (5). Lemma 1.30 tells us that (4) or (5) implies (1). Finally, to prove existence, suppose g P PL˚ rf s is any constant-derivative map. Then Dirrf s ď Dirpgq. For each edge e, this gives an upper bound on possible values of |f 1 peq| for any minimizer; let D be a bound for all edges. Then Dir is a continuous function on the compact complex PL˚ rf sďD , and therefore has a global minimum, which is harmonic by the above arguments.  5.2. Harmonic maps to weak graphs. As for Lipschitz energy, we need a further generalization of Theorem 4 to allow for the target to be a weak length graph. Definition 5.3. Let pfr, f q : Γ1 Ñ Γ2 be weak map between marked weak graphs, and suppose there is a weight structure w1 on Γ1 . Then we say that f is taut if there exists a local lift fr1 of f so that fr1 is a taut map. (Recall that in a weak map pfr, f q, the local lift fr is only defined up to (local) homotopy.) Definition 5.4. Let G “ pΓ, αq be a marked elastic graph and let K be a marked weak length graph. We say that a weak map pfr, f q : G Ñ K is harmonic if it is constant-derivative and the weak map Wf Ñ K from the tension-weighted graph is taut. Note that we do not allow G to be a weak graph. The definition could be extended to this case with some more work (since |f 1 peq| is not defined when e is a null edge and `pf peq “ 0), but we will not need it. Theorem 41 . Let rf s : G Ñ K be a homotopy class of maps from a marked elastic graph to a marked weak length graph. Then there is a harmonic weak map in rf s. Furthermore, the following conditions are equivalent. (1) The weak map f is a global minimum for Dir. (2) The weak map f is a local minimum for Dir. (3) The weak map f is harmonic. (4) There is a weighted curve pC, cq on G that forms a tight sequence c

f

C ÝÑ G ÝÑ K. Proof. This is largely the same as the proof of Theorem 4. The only significant change is the proof that item (2) implies item (3), which we now do. Let pr g , gq P PL˚ rf s, and

34

THURSTON

suppose that g is not harmonic in the sense of Definition 5.4. Then we will find a modification to prove that Dirpgq is not a local minimum. The idea is similar to the earlier proof, except that the possible local modifications are more complicated, so we will use the technology of taut maps developed in Section 3. Fix  sufficiently small (to be specified). For v a vertex of K ˚ , let Nv˚ be the ˚ -neighborhood of v. We assume that small enough so that the NŤ v are disjoint Ť  is ˚ ˚ ´1 ˚ regular neighborhoods. Ť Let N “ v Nv , Nv “ κ pNv q Ă K, N “ v Nv , Mv “ ´1 ˚ g pNv q Ă G, and M “ v Mv . We will use M and N for our neighborhoods in the definition of a weak map. On the restriction of gr to each neighborhood gr : Mv Ñ Nv , we can set up a local map of marked graphs (marking BMv and BNv ), analogously to the constructions in the proof of Theorem 2. Choose gr to be taut on each neighborhood. By assumption, gr is not locally taut everywhere on K. The failure to be taut can only happen on BN . Let y P BNv be one point where gr is not locally taut, and let Z be gr´1 pyq, minus any isolated points in the middle of edges in gr´1 pyq. The edges incident to Z can be divided into “inside” edges, that proceed into the local neighborhood towards v, and “outside” edges, that proceed away from v. (For simplicity of notation assume that no edge has both ends incident to Z.) Since gr is not locally taut at y, the total tension of the inside edges is not equal to the total tension of the outside edges. We must have ÿ ÿ (5.5) |g 1 peq| ă |g 1 peq|, e inside

e outside

as gr restricted to Mv is taut. g , gq so that h maps each vertex Let pr h, hq P PL˚ rf s be the small modification of pr in Z to y and agrees with g on all other vertices. In h the length of the image of each inside edge was increased by , and the length of the image of each outside edge was decreased by . Then we have ÿ `pgpeqq ` 2 ÿ ´`pgpeqq ` 2 ` αpeq αpeq e inside e outside ˆ ÿ ˙ ÿ “ |f 1 peq| ´ |f 1 peq| ` Op2 q,

Dirphq ´ Dirpgq “

e inside

e outside

which is negative for  sufficiently small by Equation (5.5). Thus Dirphq ă Dirpgq and f is not a local minimum for Dir. The rest of the proof is unchanged: (1) implies (2) by definition, and (4) implies (1) by properties of tight sequences. For (3) implies (4), if f is harmonic, by definition of weak harmonic maps and Theorem 2 there is a weighted curve pC, cq on G so that f c nc “ |f 1 |. Then C ÝÑ G ÝÑ K is tight. Finally, existence follows from compactness of PL˚ďD rf s (Lemma 4.9).  We can improve the local lifts in a weak harmonic map a little.

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35

Definition 5.6. In a weak map pfr, f q : Γ1 Ñ Γ2 , we say that the local lift fr is vertex-precise if, for every vertex v of Γ1 , κ2 pfrpvqq “ f pκ1 pvqq on the nose (with no need for homotopy). Proposition 5.7. If pfr, f q : G Ñ K is a harmonic weak map, then fr can be chosen to be vertex-precise and taut from the tension-weighted graph. Proof. By definition of harmonic weak maps, we can find a taut initial lift fr. If fr is not vertex-precise, pick some vertex v so that f pvq ‰ κpfrpvqq. Then frpvq lies on an edge e incident to κ´1 pf pvqq. Since fr is taut, nfr is constant on e. We can thus push frpvq into κ´1 pf pvqq without changing f or n r. Repeat this until fr is vertex-precise.  f

5.3. Uniqueness and continuity. Harmonic maps are not in general unique in their homotopy class. For instance, if the target length graph is a circle, then composing a harmonic map with any rotation of the circle gives a harmonic map. However, we will show that the length of the image of an edge in a harmonic map is unique. In fact, we show that the lengths are unique in a larger set. Definition 5.8. For Γ a marked graph, K a weak marked length graph, and rf s : Γ Ñ K a homotopy class of maps, a relaxed map r with respect to rf s is an assignment of a length rpeq to each edge e of Γ, so that, for any taut marked weighted curve pC, cq on Γ, ÿ (5.9) nc peqrpeq ě `rf ˝ cs. ePEdgepΓq

A relaxed map r naturally gives a weak length metric on Γ. Let Rlxrf s Ă LpΓq be the set of relaxed maps. We write Rlx` rf s if we need to make precise the dependence on ` P LpKq. Although a relaxed map is not, in fact, any sort of map, the following three lemmas show how relaxed maps are related to actual maps. Lemma 5.10. If rf s : Γ1 Ñ Γ2 is a homotopy class, r P LpΓ1 q, and ` P LpΓ2 q, then r P Rlx` rf s iff there is a map g P rf s with Lipr` pgq ď 1. Proof. This is Theorem 31 .



Lemma 5.11. If f : Γ Ñ K is a PL map, then r “ f ˚ ` is a relaxed map with respect to rf s. Proof. Let pC, cq be a marked curve on Γ. Give Γ a weight structure by defining wpyq “ nc pyq for regular values y P Γ. Then WRw pcq “ 1 and the lemma follows from Equation (2.20) of Proposition 2.17 (which was easy).  Lemma 5.12. If rf s : Γ Ñ K is a homotopy class of maps from a marked graph Γ to a weak marked length graph K and r P Rlxrf s, then there is a PL map g P rf s so that r “ g ˚ `.

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THURSTON

Proof. Lemma 5.10 gives a PL map g0 : pΓ, rq Ñ K in rf s with Lippg0 q ď 1. That is, `pg0 peqq ď rpeq for each edge e of Γ. Define g by modifying g0 : for each edge e on which `pg0 peqq ă rpeq, make the length of the image of e longer by introducing some zigzag folds.  Lemma let c be (1) a (2) a (3) a (4) a

5.13. We get the same definition of relaxed map if, in Equation (5.9), we weighted train track, weighted curve, connected curve with weight 1, or connected curve with weight 1 that crosses each edge at most twice.

Proof. The types of curve-like objects are progressively more restrictive, so we need to show that the existence of a structure of one type implies the next. Condition (1) implies condition (2) by Proposition 3.16. Condition (2) implies condition (3) by additivity of the inequality in Eq. (5.9) over connected components. Condition (3) implies condition (4) by Lemma 5.10 and Theorem 31 or, more simply, by taking any connected curve, looking for a maximal segment with no repeated oriented edges, doing cut-and-paste if necessary, and then using additivity over the connected components.  Lemma 5.14. For rf s : Γ1 Ñ Γ2 a homotopy class of maps between marked graphs and ` P LpΓ2 q, Rlx` rf s is a closed, non-compact, convex polytope defined by finitely many inequalities, each inequality depending linearly on `. Proof. This follows from condition (4) of Lemma 5.13, as there are only finitely many curves crossing each edge at most twice, and Equation (5.9) cuts out a closed half-space for each such curve. Scaling a relaxed map by a factor λ ą 1 gives another relaxed map, so Rlxrf s is not compact.  If r P Rlxrf s is a relaxed map where the domain is an elastic graph pΓ, αq, we can define the Dirichlet energy of r in a natural way: ÿ rpeq2 (5.15) Dirα prq :“ . αpeq ePEdgepΓq Indeed, Equation (5.15) makes sense for any r P LpΓq. We can now give the uniqueness statement. Proposition 5.16. If f : G Ñ K is a harmonic map from a marked elastic graph to a weak marked length graph, then the relaxed map r “ f ˚ ` uniquely minimizes Dirichlet energy in Rlxrf s. Proof. Let G “ pΓ, αq. The function Dirα is strictly convex on LpΓq, and its sub-level sets are compact. As such, Dirα achieves a unique minimum on the closed convex set Rlxrf s. Since f was harmonic, it minimizes Dirpf q in rf s; since every point in Rlxrf s gives the lengths of an actual map in rf s, the minimizer in Rlxrf s must be f ˚ `. 

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37

In light of Proposition 5.16, we can think of Dirichlet energy and the length of edges of the harmonic minimizer as functions on LpKq. Definition 5.17. For rf s a homotopy class of maps from a marked elastic graph pΓ1 , αq to a marked graph Γ2 , define maps Dirrf s : LpΓ2 q Ñ R Hrf s : LpΓ2 q Ñ LpΓ1 q by setting Dirrf s p`q to be minimizing Dirα prq.

Dirα` rf s

and Hrf s p`q to be the relaxed map r P Rlx` rf s

Lemma 5.18. Let rf s : G Ñ Γ2 be a homotopy class of maps from a marked elastic graph to a marked graph. Then Dirrf s and Hrf s are continuous on LpΓ2 q, with Dirrf s piecewise-quadratic and Hrf s piecewise-linear. Proof. As ` P LpΓ2 q varies, the closed convex set Rlx` rf s varies continuously by Lemma 5.14. Since Dirα is strictly convex, both the value and location of the minimum of Dirα on Rlx` rf s depend continuously on `. Furthermore, since Dirα is quadratic on LpΓq, the value of the minimum of Dirα on Rlx` rf s is a piecewise-quadratic function of ` and the location of the minimum is a piecewise-linear function of `. (The particular quadratic or linear function depends on the face of Rlx` rf s containing the minimum.)  Remark 5.19. An alternate way to see that Dirrf s is piecewise-quadratic and that Hrf s is piecewise-linear is to note that they are quadratic or linear, respectively, for a fixed combinatorial type of a harmonic representative, and only finitely many combinatorial types of harmonic maps can appear. The combinatorial type corresponds to the face of Rlx` rf s containing Hrf s p`q. 6. Minimizing embedding energy 6.1. Characterizing minimizers: λ-filling maps. We now turn to Theorem 1, starting with a characterization of which maps can appear as the optimizer for Embrφs. Definition 6.1. A strip graph S “ pΓ, w, `q is a marked graph Γ with weights w P WpΓq and lengths ` P LpΓq, so that wpeq ‰ 0 iff `peq ‰ 0. The strip graph is positive if all lengths and weights are positive, and is balanced if w is balanced (satisfies triangle inequalities at the vertices). There is an associated marked weighted graph WS “ pΓ, wq and weak marked length graph KS “ pΓ, `q. We say that S is compatible with an elastic graph GS “ pΓ, αq if `peq “ wpeqαpeq for each edge e. (If S is positive, then GS is unique.) A strip graph also has an area ÿ AreapSq :“ `peqwpeq. ePEdgepΓq

Lemma 6.2. A balanced strip graph S gives a tight sequence of maps WS ÝÑ GS ÝÑ KS ÝÑ KS˚ .

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THURSTON

(Recall from Definition 4.1 that KS˚ is KS with the null edges collapsed.) Proof. The fact that the map WS Ñ KS is taut (hence energy-minimizing) is exactly the balanced condition on w. The map Ws Ñ KS˚ is then taut by Lemmas 3.37 and 3.40. We also have `pWS Ñ KS˚ q “ ELpWS Ñ GS q “ DirpGS Ñ KS q “ AreapSq LippKS Ñ KS˚ q “ 1, so the energies are multiplicative, as desired.



Definition 6.3. Let S1 “ pΓ1 , w1 , `1 q and S2 “ pΓ2 , w2 , `2 q be two marked balanced strip graphs. Write G1 for GS1 , etc. (This may involve the choice of a compatible elastic graph.) For a map φ : Γ1 Ñ Γ2 , we also write, e.g., φW G for the associated map from the weighted graph pΓ1 , w1 q to the elastic graph pΓ2 , `2 {w2 q. Then if λ ą 0 is a real number and S2 is positive, φ : Γ1 Ñ Γ2 is λ-filling if ‚ φW is taut; ‚ lengths are preserved: φK K is a local isometry; and ‚ weights are scaled by a factor of λ: for every regular value y P Γ2 , ÿ (6.4) w1 pxq “ λw2 pyq. xPφ´1 pyq W 1 In other words, nw φ “ λw2 and in particular WRpφW q “ λ. In this case, we also say the associated map φG G between marked elastic graphs is λ-filling.

Lemma 6.5. Suppose φ : S1 Ñ S2 is a λ-filling map. Then FillφGG is identically equal to λ. In particular, EmbpφG G q “ λ. Proof. Since φ is constant on the 0-length edges of S1 , we may assume that S1 is positive as well. Since φK K is length-preserving and αi peq is defined to be `i peq{wi peq for i “ 1, 2, it follows that, for regular points x P G1 , 1 |φG G pxq| “ w1 pxq{w2 pφpxqq.

The result follows from Equation (6.4).



Lemma 6.6. A λ-filling map φ : S1 Ñ S2 gives a tight sequence of maps φG

G W1 ÝÑ G1 ÝÑ G2 ÝÑ K2 .

Proof. The composite W1 Ñ K2 is taut by assumption. The energies of the various maps are ELpW1 ÝÑ G1 q “ AreapS1 q φG

G EmbpG1 ÝÑ G2 q “ λ

DirpG2 ÝÑ K2 q “ AreapS2 q “ AreapS1 q{λ `pW1 ÝÑ K2 q “ AreapS1 q. This is multiplicative, as desired.



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39

Proposition 6.7. Let φ : G1 Ñ G2 be a map between marked elastic graphs G1 and G2 . If there is a λ-filling map ψ P rφs, then Embrφs “ SFDir rφs “ SFEL rφs “ Embpψq “ λ. Proof. Immediate from Lemmas 6.5, 6.6 and 1.30.



Thus, λ-filling maps are optimizers for Emb. If there were always a λ-filling map in rφs, then we would be done with Theorem 1. Unfortunately this is not always true. Example 6.8. Let G1 and G2 both be the join of two circles, with elastic constants p2, 2q and p1, 4q, respectively, and let φ : G1 Ñ G2 be the constant-derivative map which is the identity on π1 and maps the vertex to the vertex: 2 2 1 4 φ ÝÑ . Then Embpφq “ 2. Furthermore, SFEL rφs “ 2, by considering a loop on the left of G1 . Thus Embrφs “ Embpφq “ 2. There is no λ-filling map in the homotopy class rφs. Definition 6.9. Let S1 “ pΓ1 , w1 , `1 q and S2 “ pΓ2 , w2 , `2 q be two marked balanced strip graphs, with S2 positive, let λ ą 0 be a real number, and let Γ0i Ă Γi be subgraphs with induced strip structures Si0 “ pΓ0i , wi , `i q. Then a map φ : S1 Ñ S2 is partially λ-filling if (1) Γ01 “ φ´1 pΓ02 q; (2) the restriction of φ to a map from S10 to S20 is λ-filling; (3) φ is everywhere length-preserving; and (4) outside of S10 and S20 , the map φ scales weights by strictly less than λ: for every regular value y P S2 zS20 , ÿ w1 pxq ă λw2 pyq. xPφ´1 pyq

We call S10 and S20 the filling subgraphs of S1 and S2 . There is a naturally associated marked weighted graph W10 “ pS10 , w1 q and a marked length graph K20˚ which is pΓ2 , `2 q with all the non-filling edges collapsed to points. Note that we do not think of K20˚ as a weak graph. Lemma 6.10. If φ : S1 Ñ S2 is a partially λ-filling map then, with notation as in Definition 6.9, there is a tight sequence φG

G W10 ÝÑ G1 ÝÑ G2 ÝÑ K20˚ .

Proof. The version of Lemma 6.5 in this context says that FillφGG pyq “ λ if y is a regular point in S20 and FillφGG pyq ă λ if y is a regular point in S2 zS20 . We then have ELpW10 Ñ G1 q “ AreapS10 q EmbpG1 Ñ G2 q “ λ DirpG2 Ñ K20˚ q “ AreapS20 q “ AreapS10 q{λ `rW10 Ñ K20˚ s “ `pW10 Ñ K20˚ q “ AreapS10 q,

40

THURSTON

which is multiplicative, as desired. To see that the map W10 Ñ K20˚ is taut, note that the map S10 Ñ S20 is λ-filling, which in particular implies that W10 Ñ Γ02 is taut. By Lemma 3.40 applied to W10 Ñ Γ2 , this implies that W10 Ñ K20˚ is taut.  Proposition 6.11. For rφs : G1 Ñ G2 a homotopy class of maps between marked elastic graphs, there are compatible strip structures on G1 and G2 and map ψ P rφs so that ψ is partially λ-filling. Proof of Theorem 1, using Proposition 6.11. Immediate with Lemmas 6.10 and 1.30.  The bulk of this section will be devoted to the proof of Proposition 6.11. 6.2. Iterating to optimize embedding energy. One approach to proving Proposition 6.11 would be to study those maps that locally minimize the embedding energy, analogously to the proof of Theorem 3. From a local minimum, you can extract lengths and weights to form the desired strip structure. We will take a different approach, one that also suggests an algorithm to actually compute the embedding energy. From a homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs, we will give an explicit iteration that has a fixed point at a partially λ-filling map. To motivate the iteration, we first give an analogue in the setting of vector spaces. First, recall that a norm on a finite-dimensional vector space V defines a dual norm on the dual space V ˚ . This is the minimal norm that satisfies, for all v P V and v ˚ P V ˚ , (6.12)

xv ˚ , vy ď kvkkv ˚ k.

Equation (6.12) is tight in the sense of Proposition 2.17, namely, for every non-zero v P V there is a non-zero v ˚ P V ˚ so that Equation (6.12) is an equality. If in addition kv ˚ k “ kvk, we say that v and v ˚ support each other. (The hyperplane corresponding to v ˚ is parallel to a supporting hyperplane for the unit ball in V .) We will suppose that the norms are strictly convex, which implies that supporting vectors are unique: equality in Equation (6.12) implies that v ˚ is a positive multiple of the supporting vector of v. a Example 6.13. If the norm on V comes from an inner product, with kvk “ xv, vy, the inner product gives a canonical linear isomorphism V Ñ V ˚ that takes v to its supporting vector. Now suppose we given an isomorphism φ : V Ñ W between two finite-dimensional vector spaces, with a strictly convex norm on each. We wish to find the operator norm kφkV,W by finding a non-zero vector v P V that maximizes the ratio NRpvq :“

kφpvqkW . kvkV

Algorithm 6.14. To attempt to maximize NRpvq, pick an initial vector v0 P V , and set i “ 0. (1) Let wi P W be φpvi q. (2) Find a supporting vector wi˚ P W ˚ for wi .

ELASTIC GRAPHS

41

(3) Let vi˚ P V ˚ be φ˚ pwi˚ q. (4) Find a supporting vector vi`1 P V for vi˚ , increase i by 1, and return to Step (1). This gives a sequence of vectors vi P V , vi˚ P V ˚, wi P W , and wi˚ P W ˚ . We may also consider the corresponding sequence in the respective projective spaces. The candidate for maximizing NR on P V is limiÑ8 rvi s, if it exists. Let Iterφ : V Ñ V be the composition of the steps in Algorithm 6.14, and let P Iterφ : P V Ñ P V be the corresponding map on projective spaces. Lemma 6.15. NRpvq weakly increases under Iterφ . That is, NRpvq ď NRpIterφ pvqq. If we have equality, then v is a projective fixed point of Iterφ . Proof. With the notation from Algorithm 6.14, we wish to prove that kwi k{kvi k ď kwi`1 k{kvi`1 k. Repeatedly apply Equation (6.12), using it as an equality for vectors that support each other. kwi k kwi˚ k “ xwi˚ , wi y “ xwi˚ , φvi y “ xφ˚ wi˚ , vi y “ xvi˚ , vi y kvi˚ k kvi`1 k “ xvi˚ , vi`1 y “ xφ˚ wi˚ , vi`1 y “ xwi˚ , φvi`1 y “ xwi˚ , wi`1 y kwi k kvi`1 k xvi˚ , vi y xwi˚ , wi`1 y “ kvi k kwi`1 k kvi k kwi˚ k kwi`1 k kvi˚ k kvi˚ kkvi k kwi˚ kkwi`1 k “ 1. ď kvi k kwi˚ k kwi`1 k kvi˚ k If we have equality, then xvi˚ , vi y “ kvi˚ kkvi k and so vi is a multiple of the supporting vector for vi˚ , namely vi`1 .  Corollary 6.16. A vector rv0 s P P V that maximizes NRpv0 q is a fixed point for P Iterφ . If v0 is an attracting fixed point for the P Iterφ , then kφvk{kvk has a local maximum at v0 . Example 6.17. In the setting of Example 6.13, if the norms on V and W come from inner products, then Iterφ is φ˚ φ (using the canonical identifications V – V ˚ and W – W ˚ ) and the iteration reduces to power iteration: find the maximum eigenvector of M “ φ˚ φ by repeatedly applying M . For given φ, this almost always converges to an eigenvector of maximal eigenvalue. The convergence rate is determined by the ratio between the two largest distinct eigenvalues. Example 6.18. Consider the case when φ is the identity on Rn and k¨k2 is the standard inner product. Then k¨k1 is defined by its unit norm ball B1 Ă Rn . The supporting vector of v P BB1 is the tangent hyperplane to B1 . Up to scale, Algorithm 6.14 alternates between taking a tangent hyperplane to B1 , finding the closest point to the origin on the tangent hyperplane, and projecting to B1 , as in Figure 4. We now return to the actual case of interest, which we can think of as finding a length graph that realizes SFDir . For a marked elastic graph G “ pΓ, αq, we think of maps f : G Ñ K from G to a marked length graph K as dual to maps c : W Ñ G from a marked weighted graph W to G, with pairing given by xc, f y :“ `rf ˝ cs. More concretely, elements of LpΓq and WpΓq give such maps. The norms on LpΓq and

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THURSTON

Figure 4. Examples for iteration on vector spaces. On the left, B1 comes from a quadratic norm as in Example 6.17. The right shows an example where Algorithm 6.14 does not converge to the maximum of NR. WpΓq are given by the square roots of Dirichlet energy (Equation (5.15)) and extremal length (Equation (2.6)), respectively, and there is a natural duality map DG : WpΓq Ñ LpΓq DG pwqpeq :“ αpeqwpeq, corresponding to taking the supporting vector. The duality works best when the weights w P WpΓq are balanced. Fortunately this will be automatic. Algorithm 6.19. Given a homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs G1 “ pΓ1 , α1 q and G2 “ pΓ2 , α2 q, pick a generic initial set of lengths `0 P LpΓ2 q, and let K0 be the marked length graph pΓ2 , `0 q. We will write Di for DGi . Let f0 : G2 Ñ K0 be the identity on the level of graphs. Set i “ 0 and consider the following iteration. (1) Find a harmonic representative gi of the composite map rφ ˝ fi s : G1 Ñ Ki . For each edge e of Γ1 , let mi peq “ `pgi peqq. Thus mi “ Hrf s p`i q, with Hrf s from Definition 5.17. (2) Set wi “ D1´1 pmi q P WpΓ1 q, so that wi peq is |gi1 peq|, the tension weight on e. Let Wi be the weighted graph pΓ1 , wi q, and let di : Wi Ñ G1 be the identity on the level of graphs. (3) Set vi “ Nrφs pwi q, the push-forward of wi . Since gi is harmonic and thus taut from the tension-weighted graph, we also have ÿ vi pyq “ wi pxq. xPgi´1 pyq

Let ci : Wi Ñ G2 be gi ˝ di . (4) Set `i`1 “ D2 pvi q P LpΓ2 q, let Ki`1 “ pΓ2 , `i`1 q, and let fi`1 : G2 Ñ Ki`1 be the identity on the level of graphs. Increase i by 1 and return to Step (1).

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Schematically, Algorithm 6.19 iterates around the following loop. LpΓ2 q (6.20)

D2

WpΓ2 q

Hrφs

Nrφs

LpΓ1 q

D1´1

WpΓ1 q.

We have the following diagram of spaces and maps, in which ‚ rows are tight sequences in the sense of Lemma 1.30; ‚ the dashed lines are only defined up to homotopy; and ‚ the regions commute up to homotopy. ci´1

Wi´1

G2

fi

Ki

rφs

(6.21)

Wi

di

gi

G1

Ki

rφs

Wi

ci

G2

fi`1

Ki`1

Let Iterφ : LpΓ2 q Ñ LpΓ2 q be the composition Iterφ “ D2 ˝ Nrφs ˝ D1´1 ˝ Hrφs . All of the maps involved are piecewise-linear and linear on rays in the respective spaces (using Lemmas 3.42 and 5.18), so Iterφ is as well. We allow some of the lengths or weights to vanish (i.e., we include the boundary of LpΓi q and WpΓi q in the maps). The maps Di˘1 , Nrφs , and Hrφs all extend continuously to this case, using the theory of weak maps for the case of Hrφs . If φ is essentially surjective, then ` ‰ 0 implies Iterφ p`q ‰ 0, so we have a map on projective space P Iterφ : P LpΓ2 q Ñ P LpΓ2 q. P LpΓ2 q is a compact ball, so the existence of a fixed point is automatic. The iteration is parallel to the iteration on vector spaces, with V Ø LpG2 q vi Ø fi

W Ø LpG1 q wi Ø gi

vi˚ Ø ci

wi˚ Ø di .

For ` P LpΓ2 q, let DRp`q be the ratio of Dirichlet energies Dirα` 1 rf ˝ φs{ Dirα` 2 rf s, and w for w P WpΓ1 q, let ERpwq be the ratio of extremal lengths ELw α2 rφ ˝ ds{ ELα1 rds. Remark 6.22. The only computationally-expensive step in computing Iterφ is Step (1), finding the harmonic equilibrium. Lemma 6.23. Iterφ increases the objective functions for both Dirichlet energy and extremal length: with the notation from Algorithm 6.19, for i ě 0, DRp`i q ď DRp`i`1 q

ERpwi q ď ERpwi`1 q

Furthermore, if we have equality in either case, then `i is a projective fixed point of Iterφ .

44

THURSTON d

g

i i Proof. This is parallel to Lemma 6.15. In Diagram (6.21), the row Wi ÝÑ G1 ÝÑ Ki f c i`1 i is tight since gi is harmonic and wi “ |gi1 |, while the row Wi ÝÑ G2 ÝÑ Ki`1 is tight because ci is taut and `i`1 “ D2 pvi q. We thus have

Dirrgi s ELrdi s “ `rgi ˝ di s2 “ `rfi ˝ φ ˝ di s2 “ `rfi ˝ ci s2 Dirrfi`1 s ELrci s “ `rfi`1 ˝ ci s2 “ `rfi`1 ˝ φ ˝ di s2 “ `rgi`1 ˝ di s2 `rfi ˝ ci s2 `rgi`1 ˝ di s2 Dirrgi s Dirrfi`1 s “ Dirrfi s Dirrgi`1 s Dirrfi s ELrdi s Dirrgi`1 s ELrci s Dirrfi s ELrci s Dirrgi`1 s ELrdi s ď “ 1. Dirrfi s ELrdi s Dirrgi`1 s ELrci s c

i If we have equality, then `rfi ˝ ci s2 “ Dirrfi s ELrci s and so by Lemma 1.30 Wi ÝÑ fi`1 fi ci G2 ÝÑ Ki is a tight sequence, in addition to Wi ÝÑ G2 ÝÑ Ki`1 . In the proof of Equation (2.22), we used the Cauchy-Schwarz inequality, so in a tight sequence f c W ÝÑ G ÝÑ K, |f 1 | is proportional to nc . Thus `i and `i`1 must be multiples of each other, and so `i is a projective fixed point of Iterφ , as desired. The proof for the inequality on EL is parallel:

`rgi`1 ˝ di s2 `rfi`1 ˝ ci`1 s ELrci s ELrdi`1 s “ ď 1. ELrdi s ELrci`1 s ELrdi s Dirrfi`1 s ELrci`1 s Dirrgi`1 s



Now consider the lengths in r`s P P LpΓ2 q that maximize DRp`q. By Lemma 6.15, r`s is a fixed point for P Iterφ . Conversely, any fixed point of P Iterφ in the interior of P LpΓ2 q is a global maximum for DR, as follows. Proposition 6.24. Let rφs : G1 Ñ G2 be an essentially surjective homotopy class of maps between marked elastic graphs and suppose r`s is a fixed point of P Iterφ in the interior of P LpΓ2 q. Let K “ pΓ2 , `q, let f : G2 Ñ K be the identity on the level of graphs, and let g : G1 Ñ K be a harmonic representative of φ ˝ f . Then f ´1 ˝ g : G1 Ñ G2 is λ-filling. Proof. At the fixed point, the tight sequences from Diagram (6.21) collapse to G2

d

(6.25)

W

rφs

c

g

f

K

G1 where the top and bottom are tight sequences and the two triangles commute up to homotopy. Since r`s is in the interior of P LpΓ2 q, K is a positive length graph and f is invertible. Set ψ “ f ´1 ˝ g; by hypothesis, ψ P rφs. We will show that ψ is λ-filling.

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45

To be concrete, we are given ` P LpΓ2 q. Choose m, w, v, and λ so m “ Hrφs p`q P LpΓ1 q w “ D1´1 pmq P WpΓ1 q

(6.26)

λv “ Nrφs pwq P WpΓ2 q ` “ D2 pvq P LpΓ2 q.

(Since ` is only a projective fixed point, we need to insert a scaling factor into the cycle.) Then we have balanced strip graphs S1 “ pΓ1 , w, mq and S2 “ pΓ2 , v, `q. The definition of Hrφs ensures that ψ preserves lengths, and the definition of Nrφs and the fact that a harmonic map is taut from the tension-weighted graph ensures that ψ scales weights by a factor of λ.  Corollary 6.27. Any fixed point r`s for P Iterφ in the interior of P LpΓ2 q is a global maximum for DRp`q. Proof. Immediate from Propositions 6.24 and 6.7.



6.3. Behavior on the boundary. We now turn to understanding boundary fixed points of P Iterφ . Here the behavior is more subtle: a boundary fixed point need not be a global maximum of DR. Example 6.28. In Example 6.8, P LpΓ1 q is an interval, and both endpoints of the interval are fixed points for P Iterφ . One endpoint is attracting (and the maximum of DR) and the other is repelling (and the minimum of DR). However, we can still extract much useful structure from a boundary fixed point. Proposition 6.29. Let rφs : pΓ1 , α1 q Ñ pΓ2 , α2 q be an essentially surjective homotopy class of maps between marked elastic graphs, and suppose that r`s P BP LpΓ2 q is a fixed point of P Iterφ with multiplier λ. Then there are decompositions of Γi into complementary subgraphs Γi “ ∆i Y Σi , along with a map ψ : Γ1 Ñ Γ2 so that ‚ the edges of Σ2 are those edges e with `peq “ 0; ‚ ψp∆1 q Ă ∆2 and ψpΣ1 q Ă Σ2 ; and ‚ the restriction of ψ to a map from p∆1 , α1 q to p∆2 , α2 q is λ-filling. The map ψ above is almost partially λ-filling (with Γ0i “ ∆i ), except it need not satisfy condition (4) of Definition 6.9. Proof. Define m, w, v and λ from ` by Equation (6.26). There are tight sequences similar to Diagram (6.25), where we now have weak harmonic maps pfr, f q and pr g , gq:

(6.30)

W

rφs

c

Γ1

fr

Γ2

d

gr

f g

K κ

K ˚.

Choose the local lift gr so that gr ˝ c is vertex-precise and taut, as guaranteed by Proposition 5.7.

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THURSTON

As in the statement, let Σ2 Ă Γ2 be the subgraph consisting of edges e on which v and ` are 0, i.e., the null edges of K. Let ∆2 be the complementary subgraph. Let Σ1 Ă Γ1 be gr´1 pΣ2 q. We first check that Σ1 is a subgraph of Γ1 , all of whose edges have weight 0 in w. Since gr ˝ c is taut, v “ Nrrgs pwq{λ. Consider an edge e of Γ1 whose interior intersects Σ1 . ‚ If gr is constant on e, then κpeq is a point and mpeq “ 0, so wpeq “ 0 as well. ‚ If gr is not constant on e, consider some regular value y P Σ2 X grpeq. Then, r is a vpyq “ 0 “ nw gr pyq, so wpeq “ 0. Thus mpeq “ 0 and `pgpeqq “ 0. Since g vertex-precise lift, this implies that grpeq Ă Σ2 . In either case wpeq “ 0 and e Ă Σ1 . Let ∆1 be the complementary subgraph to Σ1 . By definition of Σ1 , we have grp∆1 q Ă ∆2 and grpΣ1 q Ă Σ2 . Since grpΣ1 q Ă Σ2 and κ is injective on Σ2 , we can modify the local lift gr so that it is an exact lift of g on Σ1 , with no homotopy required. (In fact at this point gr is a global exact lift of g.) Let ψ be this choice of gr, considered as a map from G1 to G2 . Let ψ|∆ be the restriction of ψ to a map from ∆1 to ∆2 . We must show that ψ|∆ is λ-filling. More precisely, we will show that ψ|∆ : p∆1 , w, mq ÝÑ p∆2 , v, `q is λ-filling as a map between strip graphs. As in Proposition 6.24, the definition of m as mpeq “ `pgpeqq “ `pψpeqq ensures that ψ is length-preserving. We chose gr to be taut, and by restricting first the domain to ∆1 by Lemma 3.37 and then the range to ∆2 by Lemma 3.40, we see that ψ|∆ is taut. The definition of v then tells us that ψ|∆ scales weights by λ.  In the setting of Proposition 6.29, let ψ|Σ be restriction of ψ to a map between the collapsed subgraphs Σ1 and Σ2 . For i “ 1, 2, let Pi1 “ Σi X ∆i be the vertices shared between the two subgraphs, and let Pi be the union of Pi1 with the vertices of Σi that were already marked. We view ψ|Σ as a map of marked elastic graphs pΣ1 , P1 , α1 q Ñ pΣ2 , P2 , α2 q, and consider the problem of finding Embrψ|Σ s in its own right. Proposition 6.31. In the above setting, suppose that there is a fixed point r`0 s of P Iterψ|Σ with multiplier λ0 ą λ. Then ` is not a local maximum of DRp`q. Proof. From the projective fixed point `0 P LpΣ2 q, construct m0 P LpΣ1 q, w0 P WpΣ1 q, v0 P WpΣ2 q, and λ P Rě0 by Equation (6.26). Extend `0 , m0 , w0 , and v0 to Γi by zero on edges in ∆i . Let ψ0 : pΣ1 , P1 q Ñ pΣ2 , P2 q be the map constructed from `0 by Proposition 6.29. In particular, ψ0 is (weakly) harmonic as a map from pΣ1 , α1 q to pΣ2 , `0 q. Define a new map h : Γ1 Ñ Γ2 by # ψpxq x P ∆1 (6.32) hpxq :“ ψ0 pxq x P Σ1 . Since we pinned the vertices in ∆i XΣi in ψ|Σ , the map h is continuous. By construction, h is weakly harmonic as a map from pΓ1 , α1 q to pΓ2 , `q. We claim that h is also harmonic

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47

if we perturb `. For small , consider the modified lengths ` “ ` ` `0 P LpΓ2 q. Let h be h considered as a map from pΓ1 , α1 q to pΓ2 , ` q. Claim 6.33. For  sufficiently small, h is a (weak) harmonic map. Proof. For simplicity, we suppose that `0 is non-zero on every edge, so that pΓ2 , ` q is a length graph and h is an ordinary map (not weak). The case when `0 has some zeroes could, for instance, be treated by induction. The tensions for h in the edges of Γ1 are given by w P WpΓ1 q. Let w P WpΓ1 q be the tension weights of h . Concretely, # wpeq e P Edgep∆1 q w peq “ w0 peq e P EdgepΣ1 q. We must check that h is still taut as a map from pΓ1 , w q to Γ2 . By Proposition 5.2, we must check that w satisfies the train-track triangle inequalities at the vertices of Γ1 . For vertices in ∆1 zΣ1 and Σ1 z∆1 , the triangle inequalities follow from the fact that ψ and ψ0 are harmonic, respectively. For vertices in ∆1 X Σ1 , the triangle inequalities follow from Lemma 6.34 below, where the ai ’s are the weights of the incident groups of edges of ∆1 and the bi ’s are the weights of the incident groups of edges of Σ1 .  Lemma 6.34. If pa1 , . . . , an q P Rn` satisfies the triangle inequalities and pb1 , . . . , bm q P Rm ` is another vector, then, for all  sufficiently small, pa1 , . . . , an , b1 , . . . , bm q satisfies the triangle inequalities. Proof. Elementary.



Returning to the proof of Proposition 6.31, now that we know that h is harmonic we can show DRp` q ą DRp`q: DRp` q “

Dirph q Dirpgq ` 2 Dirpg0 q Dirpgq “ ą “ λ “ DRp`q. 2 2 Dirpf q `  Dirpf0 q Dirpf q `  Dirpf0 q Dirpf q



We can now prove the existence of a partially λ-filling map. Proof of Proposition 6.11. First, if rφs is not essentially surjective, we can restrict to the essential image of rφs, the image of a taut representative of rφs with respect to any weight structure on G1 . After doing this, P Iterφ is defined. We proceed by induction on the size of Γ1 . Given the homotopy class rφs, let ` P LpΓ1 q be a global maximum of DR (which exists since P LpΓ1 q is compact). By Lemma 6.23, ` is a fixed point of P Iterφ ; let λ be its multiplier. If ` is in L` pΓ1 q, we are done by Proposition 6.24. Otherwise, consider the subgraphs ∆i and Σi given by Proposition 6.29, let ψ0 : ∆1 Ñ ∆2 be the map constructed there, and let rφ1 s : pΣ1 , P1 q Ñ pΣ2 , P2 q be the restricted map. Since Σ1 is a proper subgraph of Γ1 , by induction we can find a partially λ1 -filling map ψ1 P rφ1 s for some λ1 ě 0.

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THURSTON

Now assemble ψ0 and ψ1 to a single map ψ : Γ1 Ñ Γ2 by Equation (6.32). Then ψ is λ-filling on ∆1 and partially λ1 -filling on Σ1 . By Proposition 6.31, λ1 ď λ, so ψ is partially λ-filling on all of Γ1 .  Remark 6.35. The map constructed above has a stronger “layered” structure, where Γ1 and Γ2 are divided into layers, each with its own filling constant. Specifically, there are properly nested subgraphs Γ1 “ Σ01 Ľ Σ11 Ľ ¨ ¨ ¨ Ľ Σn1 “ H Γ2 “ Σ02 Ľ Σ12 Ľ ¨ ¨ ¨ Ľ Σn2 , a map ψ : Γ1 Ñ Γ2 in rφs, and a sequence of filling constants λ0 ą λ1 ą ¨ ¨ ¨ ą λn “ 0, with the following properties. (1) ψ preserves Σi : for 1 ď i ď n, ψpΣi1 q Ă Σi2 . (2) ψ preserves Σi zΣi`1 : for k P t1, 2u and 0 ď i ď n ´ 1, let ∆ik be the graph i i whose edges are in Σik zΣi`1 k . Then ψp∆1 q Ă ∆2 . (3) ψ is λi -filling on ∆i : for 0 ď i ď n, let ψi be the restriction of ψ to a map from ∆i1 to ∆i2 , where for i ą 0 we additionally mark the vertices in ∆i1 X ∆i´1 and 1 i´1 i in ∆2 X ∆2 . Then ψi is λi -filling. 6.4. General targets. We turn to Theorem 11 , allowing more general length space targets. First we need a proof of Equation (2.25) that works in this setting. Lemma 6.36. Let G1 and G2 be elastic graphs, let X be a length space, let φ : G1 Ñ G2 be a piecewise-linear map, and let f : G2 Ñ X be a Lipschitz map. Then Dirpf ˝ φq ď Embpφq Dirpf q. Proof. We compute ż Dirpf ˝ φq “ xPG1

ż “ yPG2

|pf ˝ φq1 pxq|2 dx ˆ ÿ ˙ 1 1 2 |φ pxq||f pyq| dy xPφ´1 pyq

ď Embpφq Dirpf q, For the second equality we do a change of variables from G1 to G2 , using dx “ |φ1 pxq| dy. (Any portions of G1 where φ is constant and so φ´1 pyq is uncountable do not ş contribute to the şintegrals.) For the last inequality we use the obvious bound that a ¨ b dy ď ess sup a ¨ b dy for a, b non-negative functions.  Proof of Theorem 1 1 . Suppose Embpφq is minimal within the homotopy class rφs and that Dirpf q is within a multiplicative factor of  of the optimal value. Then Dirrf ˝ φs ď Dirpf ˝ φq ď Embpφq Dirpf q ď Embrφs Dirrf sp1 ` q. Since we can choose  as small as we like, this gives one inequality of the desired equality. The other direction comes already from Theorem 1, where we gave a length graph so that the embedding energy was the ratio of Dirichlet energies. 

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6.5. Algorithmic questions. Given a homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs, we have proved that Iterφ has a projectively fixed set of lengths ` P LpG2 q maximizing DRp`q and giving a partially λ-filling map in rφs. The lengths maximizing DRp`q need not be unique. Question 6.37. What is the structure of the subset of LpG2 q on which DR reaches its maximum value? For instance, is it a convex subset of LpG2 q? Can it be a proper subset of the interior of LpG2 q? Example 6.38. Let G1 and G2 both be the join of two circles, as in Example 6.8, with all elastic weights equal to 1, and let φ be the identity map. Then Iterφ is the identity and DR is constant on LpG2 q. As for convergence, we would like to say that Algorithm 6.19 “works”, in the sense that iterating P Iterφ always converges to a point with the maximum value of DR (which thus also computes Embrφs). The presence of non-maximum fixed points of Iterφ means that this does not always happen. (In dynamical terms, DR is only weakly increased by Iterφ , not strictly increased; so DR is not quite a Lyapunov function for this discrete dynamical system.) But we can make some statements. Proposition 6.39. Algorithm 6.19 gives a sequence of lengths `i P LpG2 q that converge projectively to a fixed point for P Iterφ . Proof. By Lemma 6.23, DRp`i q weakly increases, with an upper bound; thus DRp`i q has a limit, and the r`i s have an accumulation point r`8 s with DRpIterφ p`8 qq “ DRp`8 q. By Lemma 6.23 again, r`8 s is a projective fixed point for P Iterφ , and therefore the r`i s limit to r`8 s, without the need to pass to a subsequence.  Lemma 6.40. For rφs : G1 Ñ G2 a homotopy class of maps between marked elastic graphs, the set t DRp`q | ` a fixed point of P Iterφ u is finite. Proof. Let ` P LpG2 q be a projective fixed point for Iterφ . Proposition 6.29 gives a λ∆ -filling map on subgraphs φ|∆ : ∆1 Ñ ∆2 with DRp`q “ λ∆ . By Proposition 6.7, λ∆ is Embrφ|∆ s and thus depends only on the subgraphs ∆i . Since there are only finitely many subgraphs, there are only finitely many values of DRp`q that can arise from projective fixed points.  Proposition 6.41. For φ : G1 Ñ G2 a homotopy class of maps between marked elastic graphs, there is an open subset of P LpG2 q on which P Iterφ converges to a maximum of DR. Proof. Let λ be the maximum value of DR on P LpG2 q. By Lemma 6.40, there is an  ą 0 so that there are no fixed points of P Iterφ in DR´1 pλ ´ , λq. Then by Proposition 6.39, P Iterφ converges to a maximum of DR on DR´1 pλ ´ , λs.  Question 6.42. Does Algorithm 6.19 converge to a maximum of DR for an open dense set of initial data? A few words are in order on why Question 6.42 is not as easy as it may appear. If r`1 s is a fixed point of P Iterφ with DRp`1 q ă Embrφs, then by Proposition 6.31 `1 is

50

THURSTON

not a local maximum for DR. If Iterφ were linear, that would imply the set that is attracted to r`1 s does not contain an open set. Since Iterφ is only PL, the situation is more complicated. For instance, Iterφ can map an open subset of L` pG2 q to a small subset of BLpG2 q, since harmonic maps can generically map vertices to vertices; this small subset of BLpG2 q could then potentially be attracted to r`1 s. We can nevertheless improve Algorithm 6.19 to always find Embrφs. Algorithm 6.43. Given an essentially surjective homotopy class rφs : G1 Ñ G2 of maps between marked elastic graphs, to find Embrφs, pick arbitrary non-zero initial lengths `0 P LpG2 q and iterate Algorithm 6.19 to get a sequence of lengths `i . Since Iterφ is piecewise-linear, each set of lengths `i is in a closed domain of linearity Di Ă LpG2 q. Since there are only finitely many domains of linearity, there must be some i and k so that Di`k “ Di . By standard linear algebra techniques, we can see if pIterφ qk has a projective fixed point in Di . By Proposition 6.39, the `i converge to a projective fixed point, so eventually the linear algebra check will succeed, giving projectively fixed lengths `8 with multiplier λ8 . If `8 is non-zero on every edge, we are done by Proposition 6.24. Otherwise, apply Proposition 6.29 to extract a map ψ|Σ : pΣ1 , P1 , α1 q Ñ pΣ2 , P2 , α2 q between simpler graphs, with its own embedding energy λΣ (which we can find by induction). If λΣ ă λ8 , we have found a partially λ8 -filling map and are done. Otherwise, by Proposition 6.31, we can find nearby lengths `10 with DRp`10 q ą DRp`8 q ě DRp`0 q. In this case, repeat the algorithm, with `10 in place of `0 . By Lemma 6.40, eventually we will find the true maximum value λ of DRp`q, and thus the true value of Embrφs. In practice, Algorithm 6.43 appears to run quickly, at least for small examples. The additional steps to continue past a projective fixed point have been unnecessary. Theoretically, there is no reason to expect it to always perform well. In particular, in the closely related case of pseudo-Anosov maps, Bell and Schleimer have recently given examples where the analogous algorithm is slow [BS15]. (The convergence rate is determined by the ratio of the two leading eigenvalues of the corresponding matrix, and they constructed examples were that ratio is small.) Appendix A. General graph energies As suggested by the notation in Definition 1.26, the energies above fit into a more general framework, which we now summarize. We start with a notion of p-conformal graphs, generalizing weighted graphs, elastic graphs, and length graphs. We can take several different points of view: A p-conformal graph can be interpreted as ‚ a graph with a p-weight αpeq on each edge, which we interpret as a metric; ‚ an equivalence class of strip graph pΓ, w, `q under a rescaling operation; or ‚ an equivalence class of spaces with a length and a measure pX, `, µq under a rescaling interpretation.

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We start with the metric interpretation, since the notation is most standard, although the formulas may appear unmotivated. Definition A.1. For p P p1, 8s, a p-conformal graph Gp “ pΓ, αq is a graph with a positive p-weight αpeq on each edge e, which we will interpret as a metric. For p “ 1, a 1-conformal graph is a weighted graph. We will sometimes find it convenient to also think about a metric graph as a 1-conformal graph (implicitly with weight 1), but in this case the energies will be independent of the metric. Definition A.2. For f : Gp Ñ K a PL map from a p-conformal graph to a length graph, define E p pf q :“ kf 1 kp,G .

(A.3)

(We take the Lp norm of |f 1 | with respect to α, and use an additional subscript to make clear where the norm is being evaluated.) If furthermore f is constant-derivative and 1 ă p ă 8, then ˜ ¸1{p ÿ `pf peqqp . E p pf q “ αpeqp´1 ePEdgepGq For f : W Ñ Gp a PL map from a weighted graph to a p-conformal graph, define Ep pf q :“ knf kp,G ,

(A.4)

where p “ p{pp ´ 1q is the Hölder conjugate of p. For f : Gp Ñ H q a PL map from a p-conformal graph to a q-conformal graph with 1 ď p ď q ď 8 and p ă 8, define Fillp pf q : H q Ñ Rě0 ÿ Fillp pf qpyq :“ |f 1 pxq|p´1

(A.5) (A.6)

xPf ´1 pyq

`

Eqp pf q :“ kFillp pf qkq{q´p,H

(A.7)

˘1{p

.

If p ă q, this is ˆż Eqp pf q

(A.8)

p

Fill pf qpyq

“

1 1´p{q

˙ p1 ´ 1q dαpyq

.

H

Energies of homotopy classes are defined as an infimum as usual: Ep rf s :“ inf Ep pgq gPrf s

Eqp rf s :“ inf Eqp pgq gPrf s

E p rf s :“ inf E p pgq. gPrf s

Remark A.9. As with the earlier energies, in each case Eqp pf q naturally extends to a wider class of graph maps than PL maps, with the precise class of maps depending on p and q. Proposition A.10. For PL maps f as above, Ep pf q “ Ep1 pf q and, if p ă 8, E p pf q “ p E8 pf q. Proof sketch. Change of variables. 8 In light of Proposition A.10, we define E8 pf q to be E 8 pf q “ Lippf q.



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THURSTON

Proposition A.11. For 1 ď p ď q ď 8 and φ : Gp Ñ H q a PL map, Eqp pφq “ sup K,f

E p pf ˝ φq E q pf q

where the supremum runs over all metric graphs K and PL maps f : H q Ñ K with non-zero energy. If p ă q ă 8, equality holds exactly when |f 1 | is proportional to ` p ˘1{pq´pq Fill pφq . Proof sketch. We may as well assume that K has the same underlying graph as H (with varying metric) and f is the identity as a graph map. Then we are essentially picking |f 1 | as a piecewise-constant function on H. We have ˆż ˙1{p p 1 p 1 p E pf ˝ φq “ f pφpxqq |φ pxq| dx G

“

˙1{p p ` ˘ p f pyq Fill pφqpyq dy

ˆż H

1

` ˘1{p ď kf kq kFillp pφqkq{pq´pq 1

“ E q pf qEqp pφq, using Hölder’s inequality in the form kabk1 ď kakq{p kbkq{pq´pq . The equality statement follows from the equality condition in Hölder’s inequality.  f

g

Proposition A.12. Given 1 ď p ď q ď r ď 8 and a sequence Gp1 ÝÑ Gq2 ÝÑ Gr2 of maps between a marked p-conformal graph G1 , a marked q-conformal graph G2 , and a marked r-conformal graph G3 , Erp pg ˝ f q ď Eqp pf qErq pgq Erp rg ˝ f s ď Eqp rf sErq rgs. Proof sketch. The first equation is an immediate consequence of Proposition A.11. The second equation follows as in the proof of Proposition 2.17.  Taut maps automatically minimize Ep within their homotopy class, as in Proposition 3.8. We can also minimize E p within a homotopy class. Proposition A.13. Let rf s : Gp Ñ K be a homotopy class of maps from a marked p-conformal graph to a marked length graph. Then there is a constant-derivative PL map g P rf s so that E p pgq “ E p rf s. Proof sketch. For p “ 8, this is Theorem 3. For p ă 8, the form of E p guarantees that an energy minimizer will be constant-derivative. This then reduces the space of possibilities to the compact polyhedron PL˚ rf sďD for suitable D, which in turn guarantees there will be a minimizer.  A map that minimizes E p pf q within rf s is called a p-harmonic map. Theorem 41 may also be extended to give a local characterization of p-harmonic maps, including cases where the target is weak. Specifically, rf s : Gp Ñ K is p-harmonic iff it is

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constant-derivative and the map Wf Ñ K is taut, where Wf is the tension-weighted graph with wpeq “ |f 1 peq|p´1 . There are also versions of stretch factors: for rφs : Gp Ñ H q a homotopy class of marked maps, define E p rf ˝ φs Ý Ñ (A.14) SFpq rφs :“ sup E q rf s K,f Eq rφ ˝ cs Ð Ý (A.15) SFpq rφs :“ sup , Ep rcs W,c where in (A.14) we maximize over marked length graphs K and PL maps f : H q Ñ K and in (A.15) we maximize over all marked weighted graphs (or curves) on Gp . Theorem 5. For 1 ď p ď q ď 8 and rφs : Gp Ñ H q a homotopy class of maps from a marked p-conformal graph to a marked q-conformal graph, there is a map ψ P rφs, a marked weight graph W , a marked weak length graph K, and a tight sequence of marked maps ψ f c W ÝÑ Gp ÝÑ H q ÝÑ K. In particular, E p pf ˝ ψq Ý Eq pψ ˝ cq Ð Ñ Ý Eqp pψq “ Eqp rφs “ “ SFpq rφs “ “ SFpq rφs. q E pf q Ep pcq Theorem 5 at first looks similar to Proposition A.11, but is more analogous to Theorem 1. Proof sketch. The proof is quite similar the proof of Theorem 1 given in Section 6. For p “ 1, the tautness results of Section 3 give the result, while for q “ 8 this is Proposition A.13. So assume that 1 ă p ď q ă 8. For Gp a p-conformal graph with 1 ă p ă 8, there is an invertible “duality” map p DG : WpGq Ñ LpGq, defined by setting ` p ˘ (A.16) DG pwq peq “ αpeqwpeq1{pp´1q . Then, for a homotopy class as in the statement of Theorem 5, define an iteration Iterφ : LpΓ2 q Ñ LpΓ2 q as follows. (1) For ` P LpΓ2 q, define K “ pΓ2 , `q, find a p-harmonic representative g of rid ˝ φs : Gp Ñ K, and set mpeq “ `pgpeqq P LpΓ1 q. (2) Set w “ pD1p q´1 pmq P WpΓ1 q, so pΓ1 , wq is the tension-weighted graph of g. (3) Set v “ Nrφs pwq P WpΓ2 q. (4) Set Iterφ p`q “ D2p pvq P LpΓ2 q. If p “ q, Iterφ descends to a map on projective spaces P Iterφ : P LpΓ2 q Ñ P LpΓ2 q, and one has to do an analysis of the possible boundary fixed points, entirely parallel to Section 6.3. If p ă q, then Iterφ is not linear on rays, and we do the iteration on LpΓ2 q itself. In this case, there are never attracting fixed points on the boundary, and we end up constructing strip graphs compatible with Gp and H q in the sense of Definition A.17 below, along with a 1-filling map between them. 

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THURSTON

We now turn to the first alternate definition of Eqp . Definition A.17. For p P r1, 8q, a p-conformal rescaling of a positive strip graph pΓ, w, `q changes the weight, length, and area by pw, `, Areaq ÞÑ pλp´1 w, λ`, λp Areaq where λ : EdgepΓq Ñ R` is a positive rescaling factor on each edge. (The identity Area “ w ¨ ` is preserved.) Also define an 8-conformal rescaling by scaling w and Area by the same factor while keeping ` fixed. We write pΓ, w1 , `1 q ”p pΓ, w2 , `2 q if the two strip structures are related by a p-conformal rescaling. For p P p1, 8s, we say that a graph pΓ, αq with given p-weights αpeq is compatible with a positive strip structure pΓ, w, `q if pΓ, w, `q ”p pΓ, 1, αq, or equivalently if `peq “ αpeqwpeq1{pp´1q . Thus we may also think of a p-conformal graph as an equivalence class of ”p . This naturally extends to arbitrary (not necessarily positive) strip structures by requiring that, for each edge e, `peqp´1 “ αpeqp´1 wpeq. Also define a 1-conformal graph to be compatible with a strip structure if the weights agree. For p “ 1, 2, or 8, a p-conformal graph is the same as a weighted graph, an elastic graph, or a length graph, respectively. The duality map Dp from Equation (A.16) is natural from this definition. Suppose we have a p-conformal graph pΓ, αq. Then for w P W ` pGq, the lengths Dp pwq are the unique values so that pΓ, w, Dp pwqq ”p pΓ, 1, αq. Definition A.18. Let S1 “ pΓ1 , w1 , `1 q and S2 “ pΓ2 , w2 , `2 q be two marked strip graphs (not necessarily balanced), with S2 positive. For λ ą 0, a map f : S1 Ñ S2 is weakly λ-filling if it satisfies the conditions of Definition 6.3, except for the condition that pΓ1 , w1 q Ñ Γ2 is taut. That is, we require that ‚ f is a local isometry from pΓ1 , `1 q to pΓ2 , `2 q and ‚ for regular y P Γ2 , ÿ w1 pxq “ λw2 pyq. xPf ´1 pyq

Definition A.19. For 1 ď p ď q ď 8, let Gp1 be a p-conformal graph, Gq2 be a p q-conformal graph, and f : Gp1 Ñ Gq2 a PL map. Then Eq,strip pf q is defined in the following way. (1) If p ă q, then there are unique strip graphs S1 and S2 , compatible with subdivisions of Gp1 and Gq2 respectively, so that f is weakly 1-filling as a map from S1 to S2 . Then (A.20)

p Eq,strip pf q “ AreapS2 q1{p´1{q .

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(2) If p “ q ă 8, there are in general no strip structures as above. Instead, take any strip structures pΓ1 , w1 , `1 q and pΓ2 , w2 , `2 q as above so that f is length-preserving, and define the energy to be the maximum ratio of weights: 1 nw pyq p . (A.21) Ep,strip pf q “ ess sup f w2 pyq yPΓ2 This is independent of the choices of strip structures. Proposition A.22. For φ : Gp1 Ñ Gq2 a PL map from a p-conformal graph to a p q-conformal graph, Eqp pφq “ Eq,strip pφq. Proof sketch. This is closely related to Proposition A.11. For p ă q, subdivide G2 so that Fillp pφq is constant on each edge. Then for e an edge of G2 , construct the strip structure pΓ2 , w2 , `2 q compatible with G2 so that ` ˘1{pq´pq `2 peq “ α2 peq ¨ Fillp pφqpeq . This determines `1 by the condition that φ be length-preserving, and w1 and w2 by the compatibility condition. It is elementary to check that φ is 1-filling with respect p to these strip structures and then verify that Eqp pφq “ Eq,strip pφq. The case p “ q is easier, as you can choose `2 arbitrarily.  For the final variation on the definition of the energies, we in principle allow more general spaces than graphs. Definition A.23. For 1 ď p ď 8, a p-conformal space is (loosely) a tuple pX, `, µq of a space X, a length metric ` on X, and a measure µ on X, up to rescaling by pX, `, µq ”p pX, λ`, λp µq for a suitable rescaling function λ : X Ñ Rą0 . Write rpX, `, µqsp for an equivalence class of ”p . There are analytic subtleties in Definition A.23 in, e.g., how to define the rescaling and exactly which metrics are allowed; we do not attempt to address them in this paper. But note that oriented conformal n-manifolds M n give examples of n-conformal spaces: given a conformal class of (Riemannian) metrics on M , pick a base metric g in the conformal class, and set ` and µ to be distance with respect to g and the Lebesgue measure of g, respectively. Picking a different metric in the conformal class changes ` and µ by an n-conformal rescaling, almost by definition. If X “ Γ is a graph, we may pick a base metric and associated measure on Γ, and restrict our attention to cases when ‚ ` is a piecewise-constant multiple of the base metric; ‚ µ is a piecewise-constant multiple of the base Lebesgue measure; and ‚ the rescaling functions λ are piecewise-constant. Definition A.23 is then almost identical to Definition A.17, if we define the weight at a generic point x P Γ by µp∆xq wpxq “ `p∆xq

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where ∆x is a small interval centered on x. Definition A.24. For 1 ď p ď q ď 8 and φ : X1p Ñ X2q a suitable map from a p-conformal space to a q-conformal space, with Xip “ rpXi , `i , µi qsp , define Fillpconf pφq : Y q Ñ Rě0 Fillpconf pφq :“ φ˚ pLip``12 pφqp µ1 q{µ2 ` ˘1{p p Eq,conf pφq :“ kFillpconf pφqkq{pq´pq,X2 p To take the definition of Eq,conf step-by-step:

‚ The expression Lip``12 pφq : X1 Ñ R` is the best local Lipschitz constant of φ. ‚ Next, φ˚ pLip``12 pφqp µ1 q is the push-forward of the given measure on X1 to a measure on X2 . ‚ Fillpconf pφq “ φ˚ pLip``12 pφqp µ1 q{µ2 is the Radon-Nikodym derivative of the two measures. p ‚ Finally, Eq,conf pφq comes from the Lq{pq´pq -norm of Fillpconf . We do not attempt to define which maps φ are “suitable”, but it should include cases where φ is Lipschitz and the Radon-Nikodym derivative exists, i.e., φ˚ pLippφqp µ1 q is absolutely continuous with respect to µ2 . In the case of maps between graphs, PL maps are suitable in this sense. p For q “ 8 (so that X2 is just a length space), E8,conf can be rewritten ˆż ˙1{p

p p (A.25) E8,conf pφq “ Econf pφq “ Lip``12 pφqp µ1 “

Lip``12 pφq

. X1

p,X1

Note that we do not need the Radon-Nikodym derivative in Equation (A.25). One motivation for the specific exponents in Definition A.24 is that, up to an overall p power, Eq,conf is the unique expression constructed with this data and these operations that is invariant under both p-conformal rescaling on X1p and q-conformal rescaling on X2q . (For instance, the only allowed operations with measures are integration and Radon-Nikodym derivatives.) Proposition A.26. For f : Gp Ñ H q a PL map from a p-conformal graph to a p q-conformal graph, Eqp pf q “ Eq,conf pf q. Proof. Follows from expanding the definitions.



Definitions A.23 and A.24 point to a considerably more general setting. Note that there are likely to be substantial new difficulties: as mentioned in Section 1.1, much attention has been devoted to proving the existence of harmonic maps between various 2 types of spaces (presumably related to minimizing E8 ), and the general case is likely to be harder. Warning A.27. Definition A.24 does not agree with standard definitions of Dirichlet energy and harmonic maps. For instance, suppose p “ 2, q “ 8, X1 is a Riemann

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57

surface Σ (with its natural 2-conformal structure), and X2 is Rn with its Euclidean metric. Pick a base metric g on Σ in the given conformal class. Then, given a smooth map f : Σ Ñ Rn , we can consider the Jacobian dfx Ñ Tx Σ Ñ Rn , and specifically its singular values a λ1 , λ2 : Σ Ñ Rě0 , chosen so that λ1 pxq ě λ2 pxq. These are the eigenvalues of pdfx qT pdfx q. Alternatively, dfx maps the unit circle in Tx Σ to an ellipse, whose major and minor axis have length λ1 pxq and λ2 pxq. In particular, the local Lipschitz constant of f at x is λ1 pxq. Then, if µ is Lebesgue measure on Σ, the formulas above give ż ` 2 ˘2 E8,conf pf q “ λ1 pxq2 µpxq Σ

while the standard Dirichlet energy is ż ` ˘ Dirpf q “ λ1 pxq2 ` λ2 pxq2 µpxq. Σ

These energies are both conformally invariant, but are not in general the same. They do agree if the target space is a graph, as in that case λ2 pxq “ 0. Appendix B. Relation to resistor networks The elastic graphs of this paper are closely related to the much better studied theory of resistor networks. Suppose we are given an elastic graph G with k marked vertices x1 , . . . , xk , called nodes; we will interpret the elastic constants αpeq as resistances. If we attach external voltage sources at voltages V1 , . . . , Vk to the nodes, then the remainder of the circuit will reach an electrical equilibrium, which has several pieces of data: ‚ an internal voltage V pvq for each vertex v of G (agreeing with V1 , . . . , Vk on x1 , . . . , x k ) ‚ an internal current Ip~eq flowing through each oriented edge ~e of G (with Ip´~eq “ ´Ip~eq; ‚ the total current I1 , . . . , IK flowing out of x1 , . . . , xk ; and ‚ the total energy E dissipated by the system per unit time. At equilibrium, these are related by Kirchhoff’s current laws. ‚ For each edge ~e of G, the current is related to the voltage difference. If ~e has source s and target t, let ∆V p~eq “ V ptq ´ V psq; then Ip~eq “

∆V p~eq . αpeq

‚ For each internal (non-marked) vertex v of G, the total current flowing in is 0 ÿ Ip~eq “ 0, ~e incident to v

while at the node xi , Ii “

ÿ ~e incident to xi

Ip~eq.

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‚ The energy is given by ÿ E“

αpeqIpeq2 “

ePEdgepGq

p∆V peqq2 . αpeq ePEdgepGq ÿ

Note the expression for the total energy is identical to Equation (1.7) for the Dirichlet energy of a map f , in the special case that the target of f is R with k marked points at V1 , . . . , Vk . In the case of electrical networks, the equations for the internal voltages and currents are linear, so V pvq, Ip~eq, and Ii are linear functions of the Vi , while E is a quadratic function of the Vi . (By contrast, in the more general case considered in the bulk of this paper, the energy as a function of lengths (the function Dirrf s from Definition 5.17) is only piecewise-quadratic.) The response matrix of the resistor network is the matrix that gives the external currents Ii as a function of the Vi . This turns out to be a symmetric matrix that also determines E as a quadratic function of the Vi . Much attention has been devoted to the question of when two resistor networks (with the same number of nodes) are electrically equivalent, in the sense that the response matrices are the same. Series and parallel reduction of resistors are one example of electrical equivalence. A more substantial example [Ken99] is the Y –∆ transform, that relates a 3-node network with the topology of a Y to one with the topology of a ∆: r1

(B.1)

”elec

R3

R2 R1

r2 r3 where

R2 R3 r2 r3 ` r1 r3 ` r1 r2 R1 “ R1 ` R2 ` R3 r1 r2 r3 ` r1 r3 ` r1 r2 R1 R3 R2 “ r2 “ R1 ` R2 ` R3 r2 R1 R2 r2 r3 ` r1 r3 ` r1 r2 r3 “ R3 “ R1 ` R2 ` R3 r3 It turns out that the Y –∆ transform and simpler reductions (like series and parallel reduction) are sufficient to relate any two electrically equivalent resistor networks if both are planar, with the nodes on the external face [CIM98, CdV94, CdVGV96]. By contrast, elastic networks with targets more general than R are almost never equivalent, so instead we can look for inequalities of the energy response function, as in Theorem 1. For instance, we have the following inequalities of energies: r1 “

4

4 4

1

ĺelast 1

1

ĺelast

3

3 3

,

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s

2

59

4 t

1

5

7 6 Figure 5. A simple electric network and an associated tiling by rectangles. All resistances on the graph are 1 except for one edge, which has resistance 2. On the rectangle tiling, we have shown the widths of the rectangles, which is the total current flowing through the edge or |f 1 peq| in the associated harmonic map to an interval. where G1 ĺelast G2 means that, for any homotopy class rφ1 s of maps from G1 to a marked tree K and corresponding homotopy class rφ2 s : G2 Ñ K, Dirrφ1 s ď Dirrφ2 s. In fact, these inequalities hold more generally when K is any CAT(0) with three marked points. The second and third graphs are related by the Y –∆ relation of Equation (B.1), and so those two graphs have equal energies when K is Rn for any n. (The case n “ 1 is electrical equivalence, and for n ě 2 we can compute the Dirichlet energy by adding up the energies of the projections to the different coordinates.) We close by reminding the reader of the connection between electrical networks at equilibrium and rectangle tilings [BSST40]: loosely, if you assign each edge a rectangle of length equal to the voltage difference between the endpoints and width equal to the total current, then Kirchhoff’s laws say that the rectangles may be assembled into a single tiling, in which the aspect ratios are equal to the resistances. See Figure 5 for a simple example. In the more general setting of elastic graphs, the “weights” throughout this paper can also be reinterpreted as “widths”, giving similar tiling pictures. However, there are at least two additional complications. ‚ At vertices of the graph, the rectangle tiling necessarily has singularities, like zeroes of a quadratic differential. ‚ To get honest tilings, you need some additional structure, namely a ribbon structure on the graphs, and maps need to respect this ribbon structure. A non-obvious thing to check is that energy minimizers also respect the ribbon structure. References [Bes11] [BS15]

Mladen Bestvina, A Bers-like proof of the existence of train tracks for free group automorphisms, Fund. Math. 214 (2011), no. 1, 1–12, arXiv:1001.0325. Mark C. Bell and Saul Schleimer, Slow north-south dynamics on PML, Preprint, 2015, arXiv:1512.00829.

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[BSST40]

R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J. 7 (1940), 312–340. [CdV94] Yves Colin de Verdière, Réseaux électriques planaires. I, Comment. Math. Helv. 69 (1994), no. 3, 351–374. [CdVGV96] Yves Colin de Verdière, Isidoro Gitler, and Dirk Vertigan, Réseaux électriques planaires. II, Comment. Math. Helv. 71 (1996), no. 1, 144–167. [CIM98] E. B. Curtis, D. Ingerman, and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl. 283 (1998), no. 1-3, 115–150. Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization [DH93] of rational functions, Acta Math. 171 (1993), no. 2, 263–297. J. Eells and B. Fuglede, Harmonic maps between Riemannian polyhedra, Cambridge [EF01] Tracts in Mathematics, vol. 142, Cambridge University Press, Cambridge, 2001. Stefano Francaviglia and Armando Martino, Metric properties of outer space, Publ. Mat. [FM11] 55 (2011), no. 2, 433–473, arXiv:0803.0640. [GS92] Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. (1992), no. 76, 165–246. [Kah06] Jeremy Kahn, A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics, Preprint ims06-05, Stony Brook IMS, 2006, arXiv:math/0609045v2. [Ken99] A. E. Kennelly, Equivalence of triangles and stars in conducting networks, Electrical World and Engineer 34 (1899), 413–414. [KPT15] Jeremy Kahn, Kevin M. Pilgrim, and Dylan P. Thurston, Conformal surface embeddings and extremal length, Preprint, 2015, arXiv:1507.05294. [KS93] Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659. [Pic05] Jean Picard, Stochastic calculus and martingales on trees, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 631–683. [Thu16] Dylan P. Thurston, From rubber bands to rational maps: A research report, Res. Math. Sci. 3 (2016), Art. 15, arXiv:1502.02561. [Wol95] Michael Wolf, On the existence of Jenkins-Strebel differentials using harmonic maps from surfaces to graphs, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 269–278. Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA E-mail address: [email protected]