On 3-dimensional asymptotically harmonic manifolds with minimal horospheres

TL;DR

This paper proves that a 3-dimensional, simply connected, asymptotically harmonic manifold with minimal horospheres must be flat, contributing to the classification of such manifolds in differential geometry.

Contribution

It establishes that 3D asymptotically harmonic manifolds with zero mean curvature horospheres are necessarily flat, filling a gap in the understanding of these geometric structures.

Findings

Such manifolds are flat when asymptotically harmonic with minimal horospheres.

The result applies to complete, simply connected 3D manifolds without conjugate points.

It confirms the flatness under the specified conditions, extending previous classifications.

Abstract

Let $(M,g)$ be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that $M$ is a flat manifold, provided $M$ is asymptotically harmonic of constant $h = 0$.