##
**Harmonic maps into singular spaces and \(p\)-adic superrigidity for lattices in groups of rank one.**
*(English)*
Zbl 0896.58024

From the introduction: “In Part 1 of this paper we develop a theory of harmonic mappings into nonpositively curved metric spaces. The main application of the theory, which is presented in Part II, is to provide a new approach to the study of \(p\)-adic representations of lattices in noncompact semisimple Lie groups. We establish \(p\)-adic superrigidity and the consequent arithmeticity for lattices in the isometry groups of quaternionic hyperbolic space and the Cayley plane (the groups \(Sp(n,1)\), \(n\geq 2\) and \(F_4^{-20})\). We show here that representations of lattices in \(Sp(n,1)\) and \(F_4\) in almost simple \(p\)-adic algebraic groups have bounded image. This is accomplished by the construction of an equivariant harmonic map from the symmetric space into the Euclidean building of Bruhat-Tits associated to the \(p\)-adic group. We analyze the structure of such maps in detail, and show that their image is locally contained in an apartment at enough points so that differential geometric methods may be applied. In particular, we apply the Corlette vanishing theorem to show that the harmonic map is constant, and conclude that the representation has bounded image.

We also prove that equivariant harmonic maps of finite energy from a Kähler manifold into a class of Riemannian simplicial complexes (referred to as \(F\)-connected) are pluriharmonic. The class of \(F\)-connected complexes includes Euclidean buildings. This result generalizes work of Y. T. Siu which implies the same result for maps to manifolds with nonpositive curvature operator.

We now briefly outline the contents of this paper. We consider maps into locally finite Riemannian simplicial complexes, by which we mean simplicial complexes with a smooth Riemannian metric on each face.

In the first four sections of this paper we develop methods for constructing Lipschitz maps of least energy in homotopy classes or with the map specified on the boundary provided the receiving space (complex) has non-positive curvature in a suitable sense. This generalizes the theorems of J. Eells and J. H. Sampson and R. Hamilton who proved these results for maps to manifolds of nonpositive curvature. We also prove and use convexity properties of the energy functional along geodesic homotopies to prove uniqueness theorems generalizing those of P. Hartman. A key property of harmonic maps which we exploit to prove these results is a statement to the effect that harmonic maps can achieve their value at a point only to a bounded order, and near the point they can be approximated by homogeneous maps from the tangent space of the domain manifold to the tangent cone of the image complex at the image point. These homogeneous maps have degree at least one and, at most points, they must have degree equal to one. The homogeneous maps of degree one are compositions of an isometric totally geodesic embedding of an Euclidean space into the tangent complex with a linear map of Euclidean spaces. In particular, these maps identify flat totally geodesic submanifolds of the tangent complex.

In Section 5 we define an intrinsic notion of differentiability for harmonic maps based on how well approximated they are near a point by maps which are homogeneous of degree one in an intrinsic sense. We then prove a result which enables us to establish differentiability of a map based on the differentiability of maps into a totally geodesic subcomplex which approximately contains the local image of the map. This result is the main technical tool of the paper as it can be used to show that the local image of a harmonic map under appropriate conditions is actually in a subcomplex whose geometry is simpler than that of the ambient complex. We then apply this result to assert differentiability of harmonic maps into one-dimensional complexes.

In Section 6 we define a class of complexes which we refer to as \(F\)-connected. A \(k\)-dimensional complex is called \(F\)-connected if each of its simplices is isometric to a linear image of the standard simplex and any two adjacent simplices are contained in a \(k\)-flat, by which we mean a totally geodesic subcomplex isometric to a region in \(\mathbb{R}^k \). We then show that harmonic maps into \(F\)-connected complexes are differentiable, and we give a detailed discussion of the size of the set of nonsmooth points, by which we mean points for which the local image of the map is not contained in a \(k\)-flat.

In Section 7 we carry through the Bochner method (in particular the Corlette vanishing theorem) for maps into \(F\)-connected complexes. We establish pluriharmonic properties for maps of Kähler manifolds and show that finite energy equivariant maps are constant from the quaternionic hyperbolic space or the Cayley plane. We also extend the existence theory to include the construction of finite energy equivariant maps into buildings associated to an almost simple \(p\)-adic algebraic group \(H\). We show that either the harmonic map exists or the image of the representation lies in a parabolic subgroup of \(H\). In particular, if the image of the representation is Zariski dense in \(H\), then the harmonic map exists. The hypothesis on the domain manifold is very general here. One requires only that it be complete. In Section 8 we establish our \(p\)-adic superrigidity results and discuss the arithmetic of lattices.

Finally, in Section 9 we discuss the structure of harmonic maps of Kähler manifolds into trees and buildings. We describe an extension of our work to maps in \(\mathbb{Z}\)-trees and use it to show that the fundamental group of a Kähler manifold cannot be an amalgamated free product unless the manifold admits a surjective holomorphic map to a Riemann surface. Applications of harmonic maps into trees similar to those done in Section 9 were also obtained by C. Simpson.

The work in this paper was initiated by a suggestion of the first author that it might be possible to develop a harmonic map theory into nonpositively curved metric spaces and that, in interesting cases, the resulting maps might be regular enough so that the Bochner method could be applied. In particular, he had a conjecture on the singular structure of harmonic maps into trees. He also proposed a version of the heat equation method which might be used to produce such harmonic maps. The work in Part I of this paper comprises the second author’s solution to this problem. The approach taken is a variational approach rather than a heat flow method. The conjectured behavior of harmonic maps to trees is shown to be substantially correct (with a slightly worse blow-up of derivatives near singular points than conjectured). Note that the theory developed in this paper is largely independent of discrete group theory, and should be viewed as a part of the geometric calculus of variations”.

We also prove that equivariant harmonic maps of finite energy from a Kähler manifold into a class of Riemannian simplicial complexes (referred to as \(F\)-connected) are pluriharmonic. The class of \(F\)-connected complexes includes Euclidean buildings. This result generalizes work of Y. T. Siu which implies the same result for maps to manifolds with nonpositive curvature operator.

We now briefly outline the contents of this paper. We consider maps into locally finite Riemannian simplicial complexes, by which we mean simplicial complexes with a smooth Riemannian metric on each face.

In the first four sections of this paper we develop methods for constructing Lipschitz maps of least energy in homotopy classes or with the map specified on the boundary provided the receiving space (complex) has non-positive curvature in a suitable sense. This generalizes the theorems of J. Eells and J. H. Sampson and R. Hamilton who proved these results for maps to manifolds of nonpositive curvature. We also prove and use convexity properties of the energy functional along geodesic homotopies to prove uniqueness theorems generalizing those of P. Hartman. A key property of harmonic maps which we exploit to prove these results is a statement to the effect that harmonic maps can achieve their value at a point only to a bounded order, and near the point they can be approximated by homogeneous maps from the tangent space of the domain manifold to the tangent cone of the image complex at the image point. These homogeneous maps have degree at least one and, at most points, they must have degree equal to one. The homogeneous maps of degree one are compositions of an isometric totally geodesic embedding of an Euclidean space into the tangent complex with a linear map of Euclidean spaces. In particular, these maps identify flat totally geodesic submanifolds of the tangent complex.

In Section 5 we define an intrinsic notion of differentiability for harmonic maps based on how well approximated they are near a point by maps which are homogeneous of degree one in an intrinsic sense. We then prove a result which enables us to establish differentiability of a map based on the differentiability of maps into a totally geodesic subcomplex which approximately contains the local image of the map. This result is the main technical tool of the paper as it can be used to show that the local image of a harmonic map under appropriate conditions is actually in a subcomplex whose geometry is simpler than that of the ambient complex. We then apply this result to assert differentiability of harmonic maps into one-dimensional complexes.

In Section 6 we define a class of complexes which we refer to as \(F\)-connected. A \(k\)-dimensional complex is called \(F\)-connected if each of its simplices is isometric to a linear image of the standard simplex and any two adjacent simplices are contained in a \(k\)-flat, by which we mean a totally geodesic subcomplex isometric to a region in \(\mathbb{R}^k \). We then show that harmonic maps into \(F\)-connected complexes are differentiable, and we give a detailed discussion of the size of the set of nonsmooth points, by which we mean points for which the local image of the map is not contained in a \(k\)-flat.

In Section 7 we carry through the Bochner method (in particular the Corlette vanishing theorem) for maps into \(F\)-connected complexes. We establish pluriharmonic properties for maps of Kähler manifolds and show that finite energy equivariant maps are constant from the quaternionic hyperbolic space or the Cayley plane. We also extend the existence theory to include the construction of finite energy equivariant maps into buildings associated to an almost simple \(p\)-adic algebraic group \(H\). We show that either the harmonic map exists or the image of the representation lies in a parabolic subgroup of \(H\). In particular, if the image of the representation is Zariski dense in \(H\), then the harmonic map exists. The hypothesis on the domain manifold is very general here. One requires only that it be complete. In Section 8 we establish our \(p\)-adic superrigidity results and discuss the arithmetic of lattices.

Finally, in Section 9 we discuss the structure of harmonic maps of Kähler manifolds into trees and buildings. We describe an extension of our work to maps in \(\mathbb{Z}\)-trees and use it to show that the fundamental group of a Kähler manifold cannot be an amalgamated free product unless the manifold admits a surjective holomorphic map to a Riemann surface. Applications of harmonic maps into trees similar to those done in Section 9 were also obtained by C. Simpson.

The work in this paper was initiated by a suggestion of the first author that it might be possible to develop a harmonic map theory into nonpositively curved metric spaces and that, in interesting cases, the resulting maps might be regular enough so that the Bochner method could be applied. In particular, he had a conjecture on the singular structure of harmonic maps into trees. He also proposed a version of the heat equation method which might be used to produce such harmonic maps. The work in Part I of this paper comprises the second author’s solution to this problem. The approach taken is a variational approach rather than a heat flow method. The conjectured behavior of harmonic maps to trees is shown to be substantially correct (with a slightly worse blow-up of derivatives near singular points than conjectured). Note that the theory developed in this paper is largely independent of discrete group theory, and should be viewed as a part of the geometric calculus of variations”.

### MSC:

58E20 | Harmonic maps, etc. |

22E40 | Discrete subgroups of Lie groups |

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |

58E40 | Variational aspects of group actions in infinite-dimensional spaces |

### Keywords:

\(p\)-adic representations; \(p\)-adic superrigidity; harmonic mappings; nonpositively curved metric spaces; lattices in noncompact semisimple Lie groups; arithmeticity for lattices; isometry groups; quaternionic hyperbolic space; Cayley plane
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\textit{M. Gromov} and \textit{R. Schoen}, Publ. Math., Inst. Hautes Étud. Sci. 76, 165--246 (1992; Zbl 0896.58024)

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