Compact asymptotically harmonic manifolds

TL;DR

This paper classifies compact asymptotically harmonic manifolds, showing they are either flat or rank one locally symmetric spaces under certain conditions, and provides a new characterization of symmetric spaces among visibility manifolds.

Contribution

It proves that compact asymptotically harmonic manifolds are limited to flat or rank one symmetric spaces under various assumptions, and introduces a new characterization of symmetric spaces.

Findings

Compact asymptotically harmonic manifolds are either flat or rank one symmetric spaces.

Under nonpositive curvature or hyperbolic fundamental group, the classification is complete.

A new characterization of symmetric spaces among visibility manifolds is provided.

Abstract

A complete Riemannian manifold without conjugate points is called asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank one locally symmetric spaces of noncompact type. In this paper we show that this list exhausts the compact asymptotically harmonic manifolds under a variety of assumptions including nonpositive curvature or Gromov hyperbolic fundamental group. We then present a new characterization of symmetric spaces amongst the set of all visibility manifolds