Variational problems for Riemannian functionals and arithmetic groups

Abstract

In this paper we introduce a new approach to variational problems on the space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach often enables one to replace the considered variational problem on Riem(M^n) (or on some subset of Riem(M^n)) by the same problem but on spaces Riem(N^n) for every manifold N^n from a class of compact manifolds of the same dimension and with the same homology as M^n but with the following two useful properties: (1) If νis any Riemannian structure on any manifold N^n from this class such that Ric_(N^n,ν) >= -(n-1), then the volume of (N^n,ν) is greater than one; and (2) Manifolds from this class do not admit Riemannian metrics of non-negative scalar curvature. As a first application we prove a theorem which can be informally explained as follows: Let M be any compact connected smooth manifold of dimension greater than four, M et(M) be the space of isometry classes of compact metric spaces homeomorphic to M endowed with the Gromov-Hausdorff topology, Riem_1(M) in M et(M ) be the space of Riemannian structures on M such that the absolute values of sectional curvature do not exceed one, and R_1(M) denote the closure of Riem_1(M) in M et(M ). Then diameter regarded as a functional on R_1(M) has infinitely many "very deep" local minima.