Poincar\'e inequality on complete Riemannian manifolds with Ricci curvature bounded below

TL;DR

This paper establishes uniform Poincaré inequalities on complete Riemannian manifolds with Ricci curvature bounded below and polynomial growth, with applications to horospheres in negatively curved manifolds.

Contribution

It proves the existence of uniform Poincaré inequalities under Ricci curvature bounds and polynomial growth, extending previous results to broader classes of manifolds.

Findings

Uniform Poincaré inequalities on manifolds with Ricci curvature bounds

Application to horospheres in negatively curved manifolds

Extension to manifolds with polynomial growth

Abstract

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincar\'e inequalities. A global, uniform Poincar\'e inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.