Restricted mean value property on Riemannian manifolds

TL;DR

This paper extends the classical study of harmonic functions satisfying the restricted mean-value property from Euclidean spaces to general Riemannian manifolds, including negatively curved spaces, revealing new generalizations.

Contribution

It generalizes classical results of Fenton to Riemannian manifolds and provides new results for negatively curved spaces without radius restrictions.

Findings

Generalization of RMVP harmonicity results to Riemannian manifolds

New theorems for negatively curved, simply connected manifolds

Extension of classical Euclidean results to curved geometries

Abstract

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in $\mathbb{R}^n$, we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian manifolds of pinched negative curvature where there is no restriction on the radius function in the RMVP.