Harmonic Functions On Manifolds Whose Large Sphere Are Small

TL;DR

This paper investigates the behavior of harmonic functions on complete Riemannian manifolds with sublinear growth of geodesic sphere diameters, extending previous results and providing gradient estimates.

Contribution

It generalizes Kazue's result to manifolds with sublinear sphere diameters and derives Cheng-Yau gradient estimates for harmonic functions.

Findings

Harmonic functions exhibit controlled growth on manifolds with sublinear sphere diameters

Established gradient bounds for harmonic functions in this geometric setting

Extended classical results to a broader class of Riemannian manifolds

Abstract

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.