The Degree Theorem in Higher Rank

Abstract

Let N be any closed, Riemannian manifold. In this paper we prove that, for most locally symmetric, nonpositively curved Riemannian manifolds M, and for every continuous map f : N → M, the map f is homotopic to a smooth map with Jacobian bounded by a universal constant, depending (as it must) only on Ricci curvature bounds of N. From this we deduce an extension of Gromov's Volume Comparison Theorem for negatively curved manifolds to (most) nonpositively curved, locally symmetric manifolds.