On 3-dimensional Asymptotically Harmonic Manifolds

TL;DR

This paper proves that a 3-dimensional complete, simply connected Riemannian manifold without conjugate points is hyperbolic with constant sectional curvature if it is asymptotically harmonic with positive constant h.

Contribution

It establishes a rigidity result linking asymptotic harmonicity to constant negative curvature in 3D manifolds without conjugate points.

Findings

M is hyperbolic with constant sectional curvature

Asymptotic harmonicity implies hyperbolicity in this setting

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

Abstract

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