Variational problems for Riemannian functionals and arithmetic groups

TL;DR

This paper introduces a novel approach to variational problems on Riemannian structures, enabling analysis across different manifolds with similar properties, and demonstrates complex behavior of the diameter functional on certain Riemannian spaces.

Contribution

It develops a new method to study variational problems on Riemannian structures by transferring them to manifolds with specific geometric constraints, revealing intricate properties of the diameter functional.

Findings

The diameter functional on R_1(M) has infinitely many deep local minima.

Manifolds in the considered class do not admit non-negative scalar curvature metrics.

The approach links variational problems across manifolds with shared homology and geometric bounds.

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 \nu is any Riemannian structure on any manifold N^n from this class such that Ric_(N^n,\nu) >= -(n-1), then the volume of (N^n,\nu) 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.