Asymptotically harmonic manifolds of dimension 3 with minimal horospheres

TL;DR

This paper corrects a previous proof and completes the classification of 3-dimensional asymptotically harmonic manifolds with minimal horospheres, showing they are either flat or hyperbolic space.

Contribution

It provides the correct proof for the classification of 3D asymptotically harmonic manifolds with minimal horospheres, resolving a gap in prior work.

Findings

3D asymptotically harmonic manifolds are either flat or hyperbolic space

The paper completes the classification of such manifolds in dimension 3

It corrects and verifies previous results in the field

Abstract

In [14], it was shown that, if M is a 3-dimensional asymptotically harmonic with minimal horospheres, then M is flat. However, there is a gap in the proof of this paper. In this paper, we provide the correct proof of the result. Thus we complete the classification of asymptotically harmonic manifolds of dimension 3: An asymptotically harmonic manifold of dimension 3 is either a flat or real hyperbolic space.