HARMONIC MORPHISMS AS A VARIATIONAL PROBLEM E. LOUBEAU Abstract. In this note, we establish a variational setting for harmonic morphisms for target space of any dimension. We then extend this result to horizontally weakly conformal p-harmonic maps, such maps being pharmonic morphisms.
1. Introduction Harmonic morphisms are a special class of harmonic maps. In this article, we develop ideas of Sanini and Uhlenbeck to describe harmonic morphisms as critical points of a variational problem. Though this fact has been known in the trade for a few years (cf. [13, 2]), the closest thing to a formal proof was the characterisation of weakly conformal harmonic maps from the sphere S2 which could be found in [10, 11], while important techniques for this proof were already in [1]. The author would like to thank P. Baird and J. C. Wood for help and comments. 2. Harmonic maps and harmonic morphisms Let (M; g ) and (N; h) be C 1 Riemannian manifolds. De nition 2.1. [4] A smooth map : M ! N is said to be harmonic if is a critical point of the functional Z 1 E ( ; K ) = 2 j d j2 dx; K for all compact sets K M , where j d j2 is the Hilbert-Schmidt norm of the linear map d (x). If (xi ) and (u ) are local coordinates around x and (x), we have j d j2= gij h ( ) i j : We can generalise this de nition for any real number p > 1: 1991 Mathematics Subject Classi cation. Primary:58E20 Secondary:58E11. Key words and phrases. harmonic morphisms, variational theory. To appear in Proc. Royal Soc. Edinburgh A. 1
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E. LOUBEAU
A smooth map : M ! N is said to be p-harmonic if is a critical point of the functional Z 1 Ep( ; K ) = p j d jp dx; K for all compact sets K M . The fundamental existence result for harmonic maps is due to Eells and Sampson. Theorem 2.1. [4] Let (M; g) and (N; h) be Riemannian manifolds with (M; g ) compact and (N; h) complete. If RiemN 0 then for any C 1-map : M ! N , there exists a harmonic representative in the homotopy class of . De nition 2.2. A map : M ! N is called a harmonic morphism if pulls back local harmonic functions on N to local harmonic functions on M , i.e. for any open subset U of N and function f : U ! R such that N f = 0, f : ?1 (U ) ! R is harmonic, i.e. M (f ) = 0. De nition 2.3. Let : (M; g) ! N , we de ne at each point p 2 M the vertical space by Vp = ker d p and its g -orthogonal complement the horizontal space by Hp = (ker d p)? . It is clear that Vp Hp = TpM: De nition 2.4. A map : (M; g) ! (N; h) is called horizontally weakly conformal if at each noncritical point x 2 M (that is such that d x has maximal rank) d x : Hx ! T (x)N is conformal of conformal factor (x) (called dilation). In local coordinates (xi ) around x 2 M and (y ) around (x) 2 N , this condition translates as
@ 2 gij (x) @ @xi @xj = (x)h ( (x)): If the function (x) is constant and non-zero then is called a horizontally
conformal submersion. The link between harmonic morphisms and harmonic maps is given by the Fuglede-Ishihara Characterisation; Theorem 2.2. [6, 8] Let (M; g) and (N; h) be Riemannian manifolds. A map : (M; g ) ! (N; h) is a harmonic morphism if and only if is a horizontally weakly conformal harmonic map. Proposition 2.1. [6] A harmonic morphism : (M; g) ! (N; h) satis es the following properties: If is non-constant then dim M dim N . For any function f : U N ! R, we have M (f ) 2 N f .
HARMONIC MORPHISMS AS A VARIATIONAL PROBLEM
3
The set of degenerate points, i.e. points at which d has not maximal rank, is of measure zero. Besides if p 2 M is a degenerate point then d p 0. The composition of with a harmonic morphism is a harmonic morphism.
Let (P; h0) be a Riemannian manifold and : (N; h) ! (P; h0) a harmonic map, then : M ! P is a harmonic map. If is non-constant then it is an open map.
An important feature of harmonic morphisms is the geometry of their bres as shown by Baird and Eells: Theorem 2.3. [1] Let : (M m; g) ! (N n; h) (with m n 2) be a horizontally conformal submersion. Then 1. if n = 2, is harmonic (and therefore a harmonic morphism) if and only if the bres are minimal; 2. if n 6= 2, two of the following conditions imply the third: (a) is a harmonic map (b) the bres are minimal (c) re( ) is everywhere vertical, that is, is horizontally homothetic. 3. The space of Riemannian metrics Let M be a smooth compact manifold of nite dimension. Let S 2 T M be the vector bundle of symmetric 2-tensors on M and S+2 T M the open subset of positive de nite symmetric 2-tensors on M . The space M(M ) (or M if the context is unambiguous) of Riemannian metrics on M , is the space of smooth sections C (S+2 T M ) of the bre bundle S+2 T M . The space M = C (S+2 T M ) is a smooth manifold of in nite dimension ([3, 4]) and, since C (S+2 T M ) is open (for the Whitney C 1 -topology, cf. [9]) in the vector space C (S 2T M ) (denoted by D), the tangent bundle of M is T M = M D = C (S+2 T M ) C (S 2T M ): Let x 2 M , then g 2 M induces on S 2TxM the inner product (3.1) hh; kig = traceg (h k) = trace(g?1hg?1k) 8 h; k 2 S 2TxM; which in turn de nes a smooth Riemannian metric on M by the formula (3.2)
Gg (h; k) =
Z
M
traceg (h k) vg ;
where vg is the volume element associated to the metric g, i.e. vg = p det(g ) dx1 ^ ^ dxm for local coordinates (x1; : : :; xm) on M . The metric G is called the canonical Riemannian metric on M since it is invariant under the action of the di eomorphism group Di (M ) on M. The integral Gg (h; k) is well-de ned since M is compact. Besides the metric is positive de nite since Gg (h; h) > 0 and Gg (h; h) = 0 if and only if h = 0. The Riemannian metric G de nes an injective (though never surjective)
4
E. LOUBEAU dual (Tg M)0 .
linear map from Tg M into its More details on the geometry of the manifold of Riemannian metrics can be found in [5, 7]. 4. A variational formulation A starting point for a variational theory of harmonic morphisms is the following result of A. Sanini, which can be as well traced back to [1]. Proposition 4.1. [10] The energy of the map : (M m; g) ! (N; h) is a critical point with respect to variations of the metric g on M m if and only if a) either m = 2 and is a conformal map; b) or m 6= 2 and is a constant map. In a similar fashion, Proposition 4.2. [10] The energy of a non-constant map : (M m; g) ! (N; h) is critical with respect to isovolumetric variations of the metric g on M m if and only if a) m = 2 and is a conformal map; b) m 6= 2 and h is homothetic to g . Considering the m-energy, Sanini proved Proposition 4.3. [11] The m-energy functional of a map : (M m; g) ! (N; h) is critical with respect to variations of the metric g on M m if and only if is a weakly conformal immersion. Though the result of Proposition 4.1 does not seem to tell us anything about harmonic morphisms the proof can easily be adapted to include such maps. First we need the following de nition De nition 4.1. Let (M m; g) and (N n; h) be Riemannian manifolds such that m n and let : (M m ; g ) ! (N n ; h) a smooth map between them. As in Section 2, we decompose the tangent space of M at each point x 2 M in its horizontal (Hx) and vertical (Vx) parts. A variation of the metric g g() : ] ? ; +[ ! M(M ) t 7! g(t) g(0) = g such that g (t) = g on the vertical space V, will be called a horizontal variation of the metric g on M . The horizontal part of such a variation will be denoted by g~(t), so that g (t) can be described as g~(t) on H; g(t) = (4.1) g^(t) on V;
HARMONIC MORPHISMS AS A VARIATIONAL PROBLEM
with
5
g^(t) = g 8t 2] ? ; +[:
Assuming rst that n = 2, we can now show: Proposition 4.4. Let (M m; g) be a Riemannian manifold of dimension m 2 and (N 2; h) a Riemannian surface. A map : (M m; g) ! (N 2; h) is a critical point of the energy functional E (see Section 2) for variations of the map and horizontal variations of the metric (on the domain) if and only if is a harmonic morphism. Proof. As we have seen in Section 2, harmonic morphisms have been characterised by Fuglede [6] and Ishihara [8] as horizontally weakly conformal harmonic maps. Since harmonic maps are, by de nition, critical points of the energy functional E for variations of the mapping, Proposition 4.4 reduces to: Lemma 4.1. A map : (M m; g) ! (N 2; h) is a critical point of the energy for horizontal variations of the metric if and only if is horizontally weakly conformal. Proof. Let g () : ] ? ; +[ ! M(M ) be any variation of the metric g (= g (0)) 1 m m and denote dg dt (t) by g_ (t). Using local coordinates (x ; : : :; x ) on M g(t) = gij (t) dxi dxj : We rst establish the following relations: (4.2a) (4.2b) Since we have
d ?gij (t) = ?g ir (t)g_ (t)g sj (t); rs dt d ?v = 1 g_ (t)g rs (t) v : g(t) dt g(t) 2 rs g ij (t)gij (t) = Id;
d ?gij (t)g (t) = d ?g ij (t) g (t) + g ij (t) d (g (t)) ij ij dt dt dt ij = 0:
Therefore d ?gij (t) = d ?gir (t)g (t)g sj (t) rs dt dt
d ?gir (t) g (t)g sj (t) + g ir (t) d (g (t)) g sj (t) + g ir (t)g (t) d ?g sj (t) = dt rs dt rs dt rs
d (g (t)) g sj (t) + g ir (t) d (g (t)) g sj (t) ? g ir (t) d (g (t)) g sj (t) = ?g ir (t) dt rs dt rs dt rs
d (g (t)) g sj (t); = ?g ir (t) dt rs
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that is On the other hand and
d gij (t) = ?g ir (t)g_ (t)g sj (t): rs dt p
vg(t) = det g(t) dx1 ^ ^ dxm
d pdet g(t) = 1 d (det g(t)) (det g(t))? 12 dt 2 dt 1 1 = 2 traceg(t) (g_ (t)) det g (t)(det g (t))? 2 = 12 g_ rs (t)g rs(t) (det g (t)) 2 : 1
Therefore
d ?v = 1 g_ (t)g rs(t) v : g(t) dt g(t) 2 rs
Let : M m ! N 2 be a smooth map and consider the energy of when M is equipped with the metric g(t) and N 2 with h: Z 1 Et( ) = 2 j d j2 vg(t) M Z @ 1 = 2 g ij (t) @ @xi @xj h vg(t); M where (x1; : : :; xm ) are local coordinates on M and (y 1; y 2) are local coordinates on N 2. Using the Relations (4.2a) and (4.2b), we compute d E ( ) = 1 Z ?g ir (t)g_ (t)g sj (t) @ @ h v rs dt t 2 M @xi @xj g(t) Z @ 1 g_ (t)g rs(t) v 1 h + 2 g ij (t) @ g(t) i j @x @x 2 rs M Z @ ?gir(t)g_rs(t)gsj (t) + 21 gij (t)g_rs(t)grs(t) @ = 12 @xi @xj h vg(t): M At t = 0 we have d E ( ) = 1 Z ?g ir g_ (0)g sj + 1 g ij g_ (0)g rs @ @ h v : rs dt t t=0 2 M 2 rs @xi @xj g We conclude that
(4.3)
if and only if Z
M
d E ( ) = 0 dt t t=0
? traceg (g_ (0) h) + 21 traceg (g_ (0)g traceg ( h)) vg = 0
HARMONIC MORPHISMS AS A VARIATIONAL PROBLEM
so that (4.3) is equivalent to
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(4.4) Gg g_ (0); 21 g traceg ( h) ? h = 0: If we choose g (t) to be a horizontal variation of the metric g then g_ (t) has the form _ on H g_ (t) = g~(0t) on V; in particular for t = 0 _ on H g_ (0) = g~(0) 0 on V; and Equation (4.4) reads Z
traceg g~_ (0); 12 g traceg ( h)
? h = 0;
H for any choice of g~_ (0), therefore dtd Et( ) t=0 = 0 for any horizontal variation of the metric if and only if h = traceg2( h) g on the horizontal space H, i.e. is horizontally weakly conformal. M
This concludes the proof of Proposition 4.4. Proposition 4.4 can be generalised to higher dimensional target spaces following K. Uhlenbeck [12] and A. Sanini [11]; Proposition 4.5. Let (M m; g) and (N n; h) be Riemannian manifolds such that m n 2. A map : (M m ; g ) ! (N n ; h) is a critical point of the energy functional E for variations of the map and a critical point of the n-energy En for horizontal variations of the metric (of the domain) if and only if is a harmonic morphism. Proof. The proof runs along the same lines as the proof of Proposition 4.4. Because of the de nition of harmonic maps as critical points of the energy (for variations of the map) and the Fuglede-Ishihara Characterisation of harmonic morphisms, Proposition 4.5 reduces to Lemma 4.2. A map : (M m; g) ! (N n; h) (m n 2) is a critical point of En for horizontal variations of the metric on M m if and only if it is horizontally weakly conformal.
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Proof. Using the Relations (4.2a) and (4.2b), we compute dtd En;t ( ) for a variation g () of g : " # n d E ( ) = d 1 Z g ij (t) @ @ h 2 v n;t g(t) i j
dt
dt n
@x @x
M
2
Z @ 1 = n 4 n2 g ij (t) @ @xi @xj h M
+
Z
M Z
n2 @ @xi @xj h
g ij (t) @
n?2 2
@ @ ir sj ? @xi @xj h g (t)g_rs(t)g (t) vg(t)
1 g_ (t)g rs(t) v g(t) 2 rs
#
n?2
2 @ @ h gir (t)g_ (t)g sj (t) ij (t) @ @ h = 1 ? n g rs 2n M @xi @xj @xi @xj @ @ rs ij + g (t) @xi @xj h g_ rs (t)g (t) vg(t):
At t = 0 we have d E ( ) = 1 Z j d jn?2 ?n @ @ h gir g_ (0)g sj + trace ( h)g_ (0)g rs v g rs g dt n;t 2n @xi @xj rs t=0
(4.5)
= 21n
M
Z
M
j d jn?2 traceg (g_ (0)(?n h + g traceg ( h)) vg !
j n?2 (?n h + g trace ( h)) : = Gg g_ (0); j d g 2n Following the same reasoning as in Lemma 4.1, (4.5) shows that is a critical point of En for horizontal variation of the metric g if and only if h = tracegn( h) g on the horizontal space H, i.e. is horizontally weakly conformal.
This concludes the proof of Proposition 4.5. Combining Lemma 4.2 and the de nition of p-harmonic maps, we obtain: Proposition 4.6. Let (M m; g) and (N n; h) be Riemannian manifolds with m n 2 and let p 2. If a map : (M m ; g) ! (N n; h) is a critical point of the p-energy functional Ep for variations of the map and a critical point of the n-energy functional En for horizontal variations of the metric g on M m then is a p-harmonic morphism. Remark 4.1. As no generalisation of the Fuglede-Ishihara Characterisation exists for p-harmonic morphisms (p 6= 2), i.e. we cannot be sure that, for p 6= 2, any p-harmonic morphism is a horizontally weakly conformal pharmonic map, Proposition 4.6 does not reach an \if and only if" conclusion.
HARMONIC MORPHISMS AS A VARIATIONAL PROBLEM
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References [1] P. Baird and J. Eells. A conservation law for harmonic maps. In E. Looijenga, D. Siersma, and F. Takens, editors, Geometry Symposium. Utrecht 1980, volume 894 of Lecture Notes in Math., pages 1{25. Springer, Berlin, Heidelberg, New York, 1981. [2] P. Baird and J. C. Wood. Harmonic Morphisms between Riemannian Manifolds. [3] J. Eells. On the geometry of function spaces. In International Symposium in Algebraic Topology, pages 303{308. Universidad Nacional Aotonoma de Mexico and UNESCO, 1958. [4] J. Eells and J.H. Sampson. Energie et deformations en geometrie di erentielle. Ann. Inst. Fourier (Grenoble), 14:61{69, 1964. [5] D. S. Freed and D. Groisser. The basic geometry of the manifold of Riemannian metrics and its quotient by the di eomorphism group. Michigan Math. J., 36(3):323{ 344, 1989. [6] B. Fuglede. Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier (Grenoble), 28:107{144, 1978. [7] O. Gil-Medrano and P. W. Michor. The Riemannian manifold of all Riemannian metrics. Quart. J. Math. Oxford, 42(2):183{202, 1991. [8] T. Ishihara. A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ., 19:215{229, 1979. [9] P. W. Michor. Manifolds of Di erentiable Mappings, volume 3 of Shiva Mathematics Series. Shiva Publishing Limited, Orpington, Kent, U.K., 1980. [10] A. Sanini. Applicazioni tra varieta riemanniane con energia critica rispetto a deformazioni di metriche. Rend. Mat., 3:53{63, 1983. [11] A. Sanini. Conformal variational problems. Rend. Circ. Mat. Palermo, 41:165{184, 1992. [12] K. Uhlenbeck. Minimal spheres and other conformal variational problems. In E. Bombieri, editor, Seminar on Minimal Submanifolds, volume 103 of Ann. of Math. Stud., pages 169{176. Princeton Univ. Press, Princeton, N.J., 1983. [13] J.C. Wood. Harmonic morphisms between Riemannian manifolds. In T. Kotake, S. Nishikawa, and R. Schoen, editors, Geometry and Global Analysis, pages 413{422. T^ohoku Univ., Sendai, 1993. Universite de Bretagne Occidentale, UFR Sciences et Techniques, Departement de Mathematiques, 6, avenue Victor Le Gorgeu, BP 809, 29285 BREST Cedex, France
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