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

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.

arXiv comment24 pages. This is a major revision. The main theorem now treats the case of pinched curvature between any two real values. The proof is simplified in several points. Ricci curvature is replaced by sectional curvature. Referee's remarks incorporated

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