Volume asymptotics and Margulis function in nonpositive curvature

2021/06/14 by Weisheng Wu, Wu, Weisheng
Mathematics · #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2106.07493

Abstract

In this article, we consider a closed rank one C^∞ Riemannian manifold M of nonpositive curvature and its universal cover X. Let b_t(x) be the Riemannian volume of the ball of radius t>0 around x∈ X, and h the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates lim_{t→ ∞}b_t(x)/\frace^hth=c(x) for some continuous function c: X→ ℝ. We prove that the Margulis function c(x) is in fact C¹. If M is a surface of nonpositive curvature without flat strips, we show that c(x) is constant if and only if M has constant negative curvature. We also obtain a rigidity result related to the flip invariance of the Patterson-Sullivan measure.

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