Volume asymptotics and Margulis function in nonpositive curvature

TL;DR

This paper studies the asymptotic behavior of volume growth in nonpositive curvature manifolds, proving the smoothness of the Margulis function and establishing conditions for its constancy, with implications for geometric rigidity.

Contribution

It introduces new asymptotic estimates for volume growth, proves the Margulis function is C^1, and characterizes when it is constant in nonpositive curvature surfaces.

Findings

Established Margulis-type asymptotics for volume in nonpositive curvature.

Proved the Margulis function is continuously differentiable.

Characterized when the Margulis function is constant in certain surfaces.

Abstract

In this article, we consider a closed rank one $C^\infty$ 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 $x\in X$, and $h$ the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates \[\lim_{t\to \infty}b_t(x)/\frac{e^{ht}}{h}=c(x)\] for some continuous function $c: X\to \mathbb{R}$. We prove that the Margulis function $c(x)$ is in fact $C^1$. 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.