Mean-value property on manifolds with minimal horospheres

TL;DR

This paper proves that on certain non-compact Riemannian manifolds with minimal horospheres, any bounded function satisfying the mean-value property must be constant, extending previous harmonic function results.

Contribution

It generalizes a known result for harmonic functions to all functions satisfying the mean-value property on manifolds with minimal horospheres.

Findings

Bounded mean-value functions are constant on the specified manifolds

Extends previous harmonic function results to broader class of functions

Manifolds considered have infinite injectivity radius and minimal horospheres

Abstract

Let (M,g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. We prove that bounded functions on (M,g) satisfying the mean-value property are constant. We extend thus a result of A. Ranjan and H. Shah who proved a similar result for bounded harmonic functions on harmonic manifolds with minimal horospheres.