Harmonic maps, rigidity, and Hodge theory - International ...

Harmonie Maps, Rigidity, and Hodge Theory

KEVIN CORLETTE

Mathematics Department, University of Chicago Chicago, IL 60637, USA

Harmonic maps are nonlinear analogues of harmonic functions or, if one considers their differentials, harmonic 1-forms. As such, one can expect analogues of Hodgetheoretic results about harmonic 1-forms. Harmonic maps arise as critical points for the energy functional on maps between two Riemannian manifolds. If A4, N are Riemannian manifolds and / : A4 —> N is a smooth map between them, then the energy is defined by

E{f) = ! W\\ JM

where df is the differential of /. If / has finite energy, then we can ask whether / is a critical point for E; the corresponding Euler-Lagrange equation is D*df = 0, where D is the exterior derivative operator associated to the natural connection on f*TN and df is regarded as a 1-form on A4 with values in f*TN. The latter is the harmonic map equation. It is a nonlinear analogue of Laplace's equation. Existence theory for harmonic maps is well behaved when the target manifold N is nonpositively curved. The first important result is due to Eells and Sampson [ES]. 1. If A4,N are compact Riemannian manifolds, and N has nonpositive sectional curvature, then any homotopy class of maps from, A4 to N has a harmonic representative.

THEOREM

In some situations, it is necessary to consider more general classes of maps. Many of the applications to be discussed here are related to representations of the fundamental group of a manifold A4 in a semisimple Lie group G. In that case, it is natural to consider equivariant maps from the universal cover of A4 to the symmetric space X — G/K associated to G; here, K is a maximal compact subgroup of G. We shall refer to such a map as a twisted map from A4 to X. In this setting, the appropriate existence result in this case is proved in [C2]; related results were proved by Diederich-Ohsawa [DO], Donaldson [D], Labourie [L], and Jost-Yau [JY1]. Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994 © Birkhäuser Verlag, Basel, Switzerland 1995

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T H E O R E M 2. Suppose M is a compact Riemannian manifold, G a linear semisimple Lie group, and p : TVI(M) —> G a homomorphism. Then there exists a pequivariant harmonie map from M to X if and only if the Zariski closure of the image of p is a reductive group.

Harmonic maps are a natural tool to apply in trying to prove rigidity theorems for nonpositively curved manifolds. For example, Mostow's strong rigidity theorem for a locally symmetric space would follow if one had techniques for proving that harmonic maps are isometries. However, the first progress in this direction did not occur until the work of Siu [Si], 15 years after that of Eells and Sampson. His basic observation, later extended by Sampson [S], was that there was an analogue for harmonic maps whose domain is a Kahler manifold of the decomposition of a harmonic 1-form into a holomorphic (l,0)-form and an antiholomorphic (0,l)-form. The Siu-Sampson result depends on the notion of complex sectional curvature. N has nonpositive complex sectional curvature if, for any pair X, Y in the complexified tangent space of N at n, we have

(R(X,Y)Y,X)

1.) Another holonomy group that shares many features with the quaternionic Kahler case is that of the holonomy group Spin(9) in dimension 16. Here, the only examples are locally isometric to the elliptic Cayley plane jF4/Spin(9) and the hyperbolic Cayley plane HQ = F 4 _ 2 0 /Spin(9). In this case, there is a parallel 8-form with which one can work. For these examples, the result above implies the following. 6. Let A4 be a compact Riemannian manifold with holonomy Spin(9) or Sp(7i)Sp(l). If N and f are as in the previous result, then f is necessarily totally geodesic.

THEOREM

A fairly direct consequence of these ideas (extended slightly so as to allow A4 to be merely of finite volume) leads to an extension of Margulis' superrigidity results [M] to certain locally symmetric spaces of rank one. 7. Suppose Y is a lattice in Sp(7i, 1) or F^20, and p : Y —> G is a homomorphism into a semisimple real algebraic group with Zariski dense image. Either G is compact or p extends to a homomorphism from the ambient group into G.

THEOREM

This result has a geometric generalization that leads in particular to a metric rigidity result for manifolds that are locally quaternionic or Cayley hyperbolic. 8. If A4 is a finite volume quotient of H^ or HQ, then any complete Riemannian metric on A4 with nonpositive curvature operator is locally symmetric.

THEOREM

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Subsequently, Mok-Siu-Yeung [MSY] and Jost-Yau [JY2] (in less generality) found formulas that apply in greater generality and with weaker assumptions on curvature. The result of Mok-Siu-Yeung is the following. 9. Suppose M is a compact locally irreducible symmetric space either of rank at least two or locally isometric to the quaternionic or Cayley hyperbolic space. If N is a Riemannian manifold with, in the former case, nonpositive sectional curvature or, in the latter case, nonpositive complex sectional curvature, and f is a (possibly twisted) harmonic map from M to N, then f is totally geodesic.

THEOREM

The idea of the proof is related to Matsushima's technique for proving the vanishing of the first cohomology of certain locally symmetric spaces, although it is necessary to make somewhat more careful choices in the nonlinear setting. In the higher rank case, this result allows one to recover Margulis' superrigidity results over the reals for cocompact lattices in simple groups, as well as Gromov's metric rigidity theorem. Hernandez [H] and Yau-Zheng [YZ] have shown that any manifold with negative pointwise ^-pinched sectional curvature has nonpositive complex sectional curvature. Using this and the Siu-Sampson result, they proved that a Riemannian metric on a finite-volume complex hyperbolic manifold with pointwise ^-pinched sectional curvature is necessarily locally symmetric. The result of MokSiu-Yeung allows one to extend this to the quaternionic and Cayley hyperbolic cases. Gromov [G] indicated a different method for obtaining this extension based on a theory of harmonic maps from manifolds with foliations. Margulis' results apply to homomorphisms into p-adic Lie groups as well, and it is natural to ask whether there is an approach to this by means of harmonic maps. This requires one to study harmonic maps from Riemannian manifolds into the Bruhat-Tits building A associated to a semisimple p-adic group. The BruhatTits building is a metric simplicial complex whose every simplex is isometric to a simplex in Euclidean space. Furthermore, for each point of A, there is at least one subspace containing it that is isometric to a Euclidean space of the same dimension as A; these subspaces are called apartments. Gromov and Schoen developed a theory of harmonic maps into such spaces. As in the classical case, it is based on a notion of energy. Suppose / is a (possibly twisted) Lipschitz map from M to A. If we consider an isometric embedding of A in a Euclidean space MN (meaning that the lengths of curves in A are the same whether measured in A or M.N), then we can define the energy density e(f) to be the pointwise squared norm of the differential of the resulting Lipschitz map from M to RN. This function is independent of the choice of embedding. The energy of / is then the integral of e(f) over M, and / is said to be harmonic if it minimizes the energy among all nearby maps. In this situation, A is to be regarded as an analogue of a nonpositively curved manifold, so one does not expect to have to deal with the more general notion of a critical point for the energy. Gromov and Schoen [GS] have proved an analogue of Theorem 2 in this setting. To apply this, one needs to be able to apply the vanishing theorems of Siu-Sampson, the author, Jost-Yau, and Mok-Siu-Yeung. The first step toward this goal is to observe that there is a large subset of M on which / can be regarded as a map into a manifold. Define TTì G M to be a singular point for f il m has


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no neighborhood whose image under / is contained in an apartment. An informal examination suggests that the set of singular points has codimension two. Gromov and Schoen show the following. 10. Suppose f is a (possibly twisted) harmonic map from A4 to A. The singular set of f has Hausdorff codimension two.

THEOREM

On the complement of the singular set, one can proceed as before: the differential of the map / is well defined away from the singular set, and can be interpreted as a harmonic 1-form with values in a flat orthogonal vector bundle. To prove the analogous vanishing theorems, one needs to perform an integration by parts in order to show that the harmonic form df Au is co closed. This requires the fact above about the size of the singular set and information on the way in which the derivatives of / decay on approach to the singular set. Gromov and Schoen prove such a result, leading in particular to the following consequence. 11. Suppose Y is a lattice in Sp(n, 1) or F 4 ~ 20 . If Y acts on the BruhatTits building A by means of a homomorphism into the corresponding p-adic Lie group, then there is a fixed point for the action, either in A itself or at infinity. (In fa,ct, a more refined analysis shovis that there must be o,fixed,point in A itself.) THEOREM

This is an analogue for these lattices of Margulis' p-adic superrigidity results for higher rank lattices. It implies the following long-conjectured result. THEOREM

12. Any lattice in Sp(7i, 1) or F^20 is arithmetic.

Thus, the question of whether irreducible lattices in a semisimple Lie group are necessarily arithmetic is now open only for the group SU(n, 1), n > 3. Twisted harmonic maps from compact Kahler manifolds to trees have been studied by Gromov-Schoen and Simpson. Their basic observation is that such a map factors through a holomorphic map into a holomorphic curve with orbifold singularities (i.e. an orbicurve). Simpson exploited this fact to prove the following. 13. Suppose A4 is a smooth complex projective variety and p : TTI(A4) —> SX(2,C) is a homomorphism with Zariski dense image. If p is not locally rigid, then there is an orbicurve G and a holomorphic map f : A4 —> C such that p is induced by a homomorphism -K\(A4) —> SL(2,C). If p is locally rigid, then there is a Hilbert modular orbivariety V and a holomorphic map f : A4 —> V such that p is the pullback of one of the standard representations of 7V\(V) in SL(2,R). THEOREM

Simpson and the author have worked on extending this to quasiprojective varieties. Zimmer has been developing a program of using superrigidity and the ideas behind it to study questions about actions of lattices and semisimple groups on manifolds. One of the principal tools he has used is an extension of superrigidity to cocycles. In joint work, the author and Zimmer [CZ] have extended some of these results to the rank one case. The main technical tool is the theory of foliated harmonic maps first developed by Gromov [G]. As an example of the geometric consequences of these ideas, we mention the following.

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T H E O R E M 14. Suppose m > 2n. Then there is no discrete subgroup o / S p ( m , 2 ) that acts freely, properly dis continuously, and cocompactly on Sp(m, 2)/Sp(ra, 1). Korevaar and Schoen have also obtained a superrigidity result for cocycles in t h e rank one case, based on a generalization of Schoen's work with Gromov. T h e y have developed a theory of harmonic maps from Riemannian manifolds into length spaces of nonpositive curvature. This is a very general class of metric spaces, not requiring, for example, t h a t the target space be locally compact. A particular example would be t h e space of Riemannian metrics on a compact manifold compatible with a fixed volume form and endowed with an appropriate L2 metric. Application of t h e general theory to this example leads to t h e result on cocycles. Acknowledgment: To reflect on the way in which my understanding of this subject has developed is t o be reminded very forcefully of what is, in some circles, referred to as t h e dependent arising of phenomena. Many mathematicians have contributed to t h a t understanding in many ways; it would be a hopeless task to t r y to list t h e m all. All I can do is offer my gratitude. References [B]

[CI] [C2] [CZ] [DO] [D] [ES] [G] [GS] [H] [JY1] [JY2]

[L] [M]

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[MSY] N. Mok, Y.-T. Siu, and S. K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993), 57-83. [R] A. Reznikov, All regulators of flat bundles are torsion, Hebrew University preprint (1993). [S] J. Sampson, Applications of harmonic maps to Kahler geometry, Cont. Math. 49 (1986), 125-133. [Si] C. T. Simpson, Nonabelian Hodge Theory, Proc. Internat. Congr. Math., Kyoto 1990, V. I, Math. Soc. of Japan, Springer-Ver lag, 1991. [Su] Y.-T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. 112 (1980), 73-112. [YZ] S. T. Yau and F. Zheng, Negatively ^-pinched Riemannian metric on a compact Kahler manifold, Invent. Math. 103 (1991), 522-535.