Combinatorial scalar curvature and rigidity of ball ... - CiteSeerX

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COMBINATORIAL SCALAR CURVATURE

Combinatorial scalar curvature and rigidity of ball packings By Daryl Cooper and Igor Rivin*

Introduction

Let M 3 be a triangulated three-dimensional manifold. In this paper we de ne a combinatorial analogue of scalar curvature for M 3 ; and also a combinatorial analogue of conformal deformation of the metric. We further de ne a functional S on the combinatorial conformal deformation space, show that S is concave, and show that critical points of S correspond precisely to metrics of constant combinatorial scalar curvature on M 3 : These results are then applied to showing rigidity of ball packings with prescribed combinatorics (the concepts are quite similar to Colin de Verdiere's work on circle packing of surfaces [2]. See also [6] for a related variational argument). The plan of the paper is as follows. In section 1 we de ne the class of conformal simplices in E3 , and prove the necessary local versions of our results. In section 2 we extend these techniques to conformal simplices in H3 . In Section 3 we study the deformation space of conformal simplices. In section 4 we prove the results on scalar curvature alluded to above, and in section 5 we discuss the ball-packing results.

1. Conformal simplices Let T be a simplex in E3 : Denote the vertices of T by v1 ; : : : ; v4 ; and denote the edge joining v and v by e : The length of e shall be denoted by l ; while the dihedral angle of e shall be denoted by  : The solid angle at the vertex v of TPshall be denoted by S : From the Gauss-Bonnet formula we know that S = 6=  ? : We say that T is conformal, if there are r1 ; : : : ; r4 > 0 such that l(e ) = r + r : The collection of vectors (r1 ; : : : ; r4 ) 2 R4 which correspond to geometric simplices is an open set. Denote this set by C : Since the set of all three-dimensional Euclidean simplices is an open set in R6 (as parametrized i

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 The

authors would like to thank Oded Schramm for pointing out an oversight in a previous version of this paper.

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DARYL COOPER AND IGOR RIVIN

by edge-lengths, for example), it is clear that the property of conformality is rather special. In the sequel the simplex T and all of its geometric parameters will be viewed as functions of r = (r1 ; r2 ; r3 ; r4 ) 2 C : Let 4 X S (r) = S r : T

i

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=1

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The next two results form the foundation of what follows. Lemma 1.1. 4 X dS = S dr : i

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=1

i

Proof.PLet us rewrite the formula for S using the relations l = r + r and S = 6=  ? : A short computation shows that 4 X X S = l  ? r ; ij

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j

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j

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i

and thus dS =

X i

6=

ij

ij

i

j

 dl + ij

j

6=

X

ij

i

6=

=1

l d ij

ij

i

?

4 X i

j

=1

dr : i

However, by the Euclidean case of the Schla i dierential formula, the second sum above vanishes, and since l = r + r ; the claim of the lemma follows. Note. The Schla i dierential formula (for a good exposition and proof, see [1, Chapter 7, section 2.2.2]) states that as a simplex T (and thus any polyhedron) varies smoothly in one of the spaces of constant curvature H ; S or E ; its volume also varies smoothly, and furthermore X 1 (1.1) Kd vol T = vol F d ; n ? 1 dim = ?2 ij

i

j

n

n

n

F

F

n

where K is the curvature of the space, and  is the dihedral angle at the (n ? 2)-dimensional face F: Note that when K = 0; the formula simply says that the right-hand side vanishes. Lemma 1.2. The Hessian H of S is negative semi-de nite { its null-space is spanned by the vector r: Proof. It is clear that r is in the null-space of the Hessian, since it corresponds to a rescaling of the simplex. To show that all other eigenvalues of F

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COMBINATORIAL SCALAR CURVATURE

H are negative it is enough to show that the diagonal entries of H are negative, while the o-diagonal elements are positive. To see this, let (v1 ; : : : ; v4 ) be an eigenvector of H corresponding to a non-negative eigenvalue , and let w = v =r : We want to show that w = w ; for all i; j: If not, assume P (without loss of generality) that 0 < w1 = maxfw g: Then, ( ? H11 )v1 = 1 H1 v ; i

i

i

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so

r1 w1 ? H11 r1 w1 =

j

X i>

1

i

H1 r w i

i

i



X i>

1

i

i>

i

H1 r w1 ; i

i

and so   0; with equality if and only if all of the w are equal. Note now that H = @S =@r : If r1 ; r2 ; r3 are kept xed and r4 is increased, v4 is moving along a trajectory h such that the angles between the tangent vector to h and the edges e4 are equal (an easy consequence of the law of cosines). Thus, the simplex T (r1 ; r2 ; r3 ; r4 + ) (where  > 0) strictly contains T (r1 ; r2 ; r3 ; r4 ); while sharing its base. The desired inequalities are then clear. As a preview of the argument to come in the next section, we prove the following results. Theorem 1.3. Fix the scale of the simplices under consideration, by setting (1.2) r1 + r2 + r3 + r4 = 1: Then the regular simplex, for which r = 1=4 is a critical point of the function S , and furthermore, the regular simplex cannot be conformally deformed while keeping the solid angles xed. Proof. By Lemma 1.2, the function S is strictly concave on the slice of C determined by the equation 1.2. By the principle of Lagrange multipliers and Lemma 1.1, S has a critical point subject to equation 1.2, precisely when all of the solid angles are equal. By concavity, such a critical point is isolated, proving the rigidity statement. i

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A similar argument shows the following: Theorem 1.4. A conformal simplex cannot be deformed while keeping the solid angles xed; that is, the set C 1 2 3 4  C1 is discrete. Proof. We have seen above that this is true for the regular simplex. To showPthe theorem, we modify the function S as follows: Let S 1 2 3 4 = S + a r : Now S 1 2 3 4 is still strictly concave subject to the condition 1.2, but its gradient is (S1 + a1 ; : : : ; S4 + a4 ): Hence the critical points of S 1 4 occurs when S + a = S + a ; for all i; j  4: This shows the assertion of the theorem. S ;S ;S ;S

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DARYL COOPER AND IGOR RIVIN

2. Hyperbolic conformal simplices In this section we consider conformal simplices (de ned as in the previous section) in hyperbolic 3-space H3 ; rather than in E3 : Rather than repeating all of the de nitions, we will simply point out the dierences. The main one of these is in the de nition of the function S: We will now de ne S as X (2.1) S (r) = S r + 2 vol T: Then, Schla i's formula 1.1 immediately implies the following analogue of Lemma 1.1: Lemma 2.1. 4 X dS = S dr : h

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=1

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The following is the hyperbolic analogue of Lemma 1.2: Lemma 2.2. The Hessian H of S is negative de nite. Proof. The proof of the lemma is similar to the proof of Lemma 1.2. The rst observation is that as r1 ; r2 ; r3 are kept xed and r4 is increasing, v4 is moving along a trajectory h such that the angles between the tangent vector to h and the edges e4 are equal (an easy computation using the hyperbolic law of sines). Thus, the simplex T (r1 ; r2 ; r3 ; r4 + ) ( > 0) strictly contains T (r1 ; r2 ; r3 ; r4 ); while sharing its base. We see that S (r1 ; r2 ; r3 ; r4 + ) > S (r1 ; r2 ; r3 ; r4 ); for i < 4; so (2.2) @S =@r > 0; i 6= j: By containment, vol T (r1 ; r2 ; r3 ; r4 + ) > vol T (r1 ; r2 ; r3 ; r4 ); and so, by the Schla i formula 1.1, 4 @S X < 0; i = 1; : : : ; 4: r (2.3) @r h

h

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Equations 2.2 and 2.3 imply that H is negative de nite by Gershgorin's theorem (or the reader is welcome to modify the argument in the proof of Lemma 1.2). h

3. The space of conformal simplices In this section we will describe the space of Euclidean conformal simplices C introduced in section 1. The rst observation is the following: T

COMBINATORIAL SCALAR CURVATURE

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Lemma 3.1. A simplex T is conformal if and only if there exists a unique sphere s which is tangent to all of the edges of T: Furthermore, the point of tangency of s with the edge e is at distance r from v : T

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Proof. We will show this result in all dimensions greater than 1: The \if" direction is trivial. We will show the \only if" direction by induction on dimension. In dimension 2; where all triangles are conformal, the result is obvious. Now let T be a conformal simplex with radii r : First note that there is a unique sphere s tangent to e1 ; j 6= 1: such that the points of tangency are r1 away from v1 : Consider the intersection of s with one of the faces of T containing v1 { by induction we know that that the lower dimensional sphere is tangent to all of the edges of that face. Since all edges of T belong to a face containing v1 ; the lemma follows. Now, rst normalize so that the radius of s is equal to 1; and consider the pattern of circles formed on s by the spheres of radii r centered at the vertices of T: These circles are mutually tangent. Conversely, any such pattern of circles C determines a conformal simplex. Since for any two circle packings C1 and C2 ofthe sphere S2 whose nerve is the simplex graph, there is a conformal transformation sending C1 to C2 (this is elementary geometry, not requiring the Andreev-Thurston theorem, in particular it also works in any dimension), the set of such packings can be naturally identi ed with the group of conformal transformations of the sphere { P SL(2; C): Since we are only interested in the congruence class of the conformal simplex T; and are no interested in which way it is pointing, we must further normalize by factoring out the rotation group SO(3): What we have just determined is: Theorem 3.2. The space of similarity classes of Euclidean conformal simplices C can be naturally identi ed with the hyperbolic space H3 : In particular, C is homeomorphic to the ball B 3 : Note. The discussion above applies just as well to hyperbolic conformal simplices, except that since we can no longer normalize by xing the radius of the mid-sphere s ; and so C ' H3  R+ ; where C is the deformation space of hyperbolic conformal simplices. It will be useful to know exactly how C is parametrized by our coordinates, in particular, what the boundary of C (as a subset of R4 ) looks like. Note that while for any three positive real numbers r1 ; r2 ; and r3 ; the sums r1 + r2 ; r1 + r3 ; and r2 + r3 satisfy the triangle inequality, not every positive quadruple r = (r1 ; r2 ; r3 ; r4 ) de nes a geometric conformal simplex. Degeneracy occurs when r4 is large enough so that the sphere of radius r4 is large enough to be tangent to the other three spheres, yet small enough that its ceni

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DARYL COOPER AND IGOR RIVIN

ter v4 lies in the plane de ned by v1 , v2 ; and v3 : This condition is de ned by Soddy's theorem { below we state a version for arbitrary dimension, recently obtained by M. Kovalev ([3]). Theorem 3.3. Let S ; i = 1; : : : ; n +2; be n +2 pairwise tangent distinct spheres in the n-dimensional Euclidean space E ; such that S has radius R : Then i

n

(3.1)

n

+2 X

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=1

K2 =

+2 !2 X

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=1

K

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where K = 1=R ; if S does not contain the other spheres, otherwise K = ?1=R : i

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While the above theorem is generally quite useful, it is not necessary to obtain the following observation:

 R4 is not convex. Proof. Consider the section C 1 1 of C de ned by r1 = r2 = 1: We can think of C 1 1 as a subset of a plane. As described above, one component @1 of the boundary of C 1 1 is de ned by the condition that there are four mutuallytangent circles in the plane, of radii 1; 1; r3 ; r4 { C 1 1 itself lies above @1 : Notice Theorem 3.4. The space C

T

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that as r4 approaches in nity, r3 also grows, but to a nite limit, and so @1 is not convex.

Note. Using Theorem 3.3, it can actually be shown that @1 ; as de ned in the proof above is strictly concave.

4. Conformal deformation space of singular Euclidean metrics Consider a closed three-dimensional manifold M 3 , together with a topological triangulation T : If we equip all of the simplices of T with metrics, so that they are isometric to geodesic simplices in E3 ; and, furthermore, these metrics are compatible (which is to say that all of the gluing maps are Euclidean isometries), M 3 acquires a metric, which is at on the complement of the 1-skeleton of T : We de ne the solid angle S of (M 3 ; T ) at a vertex v to be the sum of the solid angles of the simplices of T incident to v at the vertex identi ed to v: The quantity 4 ? S is a combinatorial analogue of scalar curvature, in that it measures the dierence in the growth rates of the volumes of a small ball centered at v in M 3 one one hand, versus the growth rate of the volume of a Euclidean ball of the same radius. v

v

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COMBINATORIAL SCALAR CURVATURE

Just as in the previous section we use the notation e for the edge joining adjacent vertices v and v of T ; l for the length of that edge, and  for the total cone angle at that edge. We say that a singular Euclidean structure on (M 3 ; T ) is conformal, if there exists a function r : V (T ) ! R+ ; such that (denoting r(v ) by r ), l = r + r : Note that a conformal structure always exists, since we can make all simplices regular and of the same size. The term \conformal" can be justi ed { we think of a general conformal structure as obtained by rescaling the \metric" of this regular one. Let C  Rj (T )j be the set of those r which correspond to all the simplices of T being geometric and non-degenerate { it's clear that C is an open set in Rj (T )j: As before, we will think of geometric invariants of (M 3; T ) as functions on C : P P Let S = 2 (T ) S r : It is clear that S = 2T S ; where S is the function de ned in section 1. It is then immediate that S is weakly concave on C ; and if in addition we restrict our attention to the slice C1 of C given by the condition X (4.1) r = 1; then S is strictly concave. Now, the proof of Theorem 1.4 goes through unchanged to show Theorem 4.1. A conformal structure on (M 3 ; T ) is cannot be deformed (except by rescaling) while keeping the solid angles at the vertices of T xed, that is the set of conformal structures with prescribed solid angles is a discrete subset of C1 : Remark 4.2. By the results in section 2 everything in this section translate essentially verbatim to the case of hyperbolic conformal structures on M 3 (i.e., those where the simplices of T are hyperbolic confromal simplices. ij

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5. Applications to ball packing Consider the special case of the situation in the previous section where the metric on (M; T ) is in fact non-singular Euclidean or hyperbolic in the interior (though the boundary may be polyhedral). The conformality of the structure means that there are Euclidean (resp. hyperbolic) balls centered at the vertices of T ; such that the radius of the ball B centered at v is equal to r : Two of such balls B and B are tangent whenever v and w share an edge, and Theorem 4.1 tells us: Theorem 5.1. Let M 3 be a Euclidean (hyperbolic) 3-manifold with polyhedral boundary. A ball packing whose nerve is a triangulation T of M is v

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DARYL COOPER AND IGOR RIVIN

cannot be deformed (except by rescaling in the Euclidean case) while keeping the solid angles at the boundary of M xed. Note. If M is a closed manifold the clause following \while' is vacuous. The theorem says that the sets of structures corresponding to ball-packings are discrete subsets of C1 and C respectively. H

A particularly simple example of a Euclidean manifold with boundary is a polyhedron in R : Since one can always stereographically project the sphere S onto R in such a way that n + 1 mutually tangent spheres go to spheres all of radius one (as observed in [2]), the above theorem implies n

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Theorem 5.2. The geometry of ball-packing of the sphere S 3 whose nerve

is a triangulation T is rigid up to Mobius transformations.

The results above can be viewed as generalizing of the work of Colin de Verdiere [2] to higher dimensions. A rather dicult problem (perhaps intractable in dimension greater than 3) is obtaining any sort of existence results. It is known that no analogue of the results in two dimensions (that any triangulation of a surface is the nerve a circle packing) exists (see [4]), but no characterization is known.

6. Remarks he hypothesis in sections 4 and 5 requiring M 3 to be a manifold is clearly too strong { almost any semi-simplicial complex works just as well, though there are some degenerate examples, like two simplices joined at the vertices only. A necessary and sucient condition is that if, for each connected component of M we form a graph whose vertices are top-dimensional faces of T and such that two vertices are joined by an edge whenever the corresponding simplices share an edge, then that graph must be connected. Finally, the non-convexity of C ; as shown in section 3 shows that the current methods do not suce to give global uniqueness in Theorems 4.1 and 5.1. For Theorem 4.1 this might be an indication of the true state of aairs (as opposed to an indication of the weakness of the methods), since in the smooth category it is known that there are usually several metrics of constant scalar curvature in a xed conformal class (as shown by D. Pollack, [5]). University of California, Santa Barbara Melbourne University, Victoria

COMBINATORIAL SCALAR CURVATURE [1] [2] [3] [4] [5] [6]

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References D. V. Alekseevskij, E. Vinberg and A. S. Solodovnikov. Geometry of spaces of constant curvature. Geometry II , EMS 29, Springer Verlag, Berlin 1993. Yves Colin de Verdiere. Un principe variationnel pour les empilements des cercles , Inventiones Mathematicae 104, 1991. M. D. Kovalev. On curvatures of tangent spheres , (in Russian), Trudy Mat. Inst. Steklov. 196 (1991), 90{92. Greg Kuperberg and Oded Schramm. Average kissing numbers for non-congruent sphere packings , Math. Res. Lett. 1, no. 3 (1994). Daniel Pollack. Nonuniqueness and high energy solutions for a conformally invariant scalar equation , Comm. Anal. Geom. 1 (1993), 347-414. Igor Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Annals of mathematics 139(2), 1994.