Smooth Volume Rigidity for Manifolds with Negatively Curved Targets

TL;DR

This paper proves that under certain volume or entropy-volume conditions, a continuous map of nonzero degree between a closed manifold and a negatively curved manifold of dimension greater than four can be homotoped to a smooth cover, often a diffeomorphism.

Contribution

It establishes sharp volume and entropy-volume conditions ensuring homotopy to smooth covers or diffeomorphisms for maps between manifolds with negative curvature.

Findings

Maps with degree one are homotopic to diffeomorphisms under volume conditions.

Volume and entropy-volume differences are necessary for the rigidity results.

Results apply specifically to manifolds of dimension greater than four.

Abstract

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normalization of the metrics. The results are qualitatively sharp in the sense that the dependencies are necessary. We give a number of corollaries.