The effect of curvature on convexity properties of harmonic functions and eigenfunctions

TL;DR

This paper provides a simplified proof of the Donnelly-Fefferman growth bound for Laplace-Beltrami eigenfunctions, introducing new geometric estimates related to curvature that enhance previous results.

Contribution

It offers an elementary proof of the growth bound and extends convexity properties of harmonic functions to curved manifolds with improved curvature estimates.

Findings

Simplified proof of Donnelly-Fefferman growth bound

New quantitative geometric estimates involving curvature

Generalization of Agmon's convexity theorem to curved manifolds

Abstract

We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.