The effect of curvature on convexity properties of harmonic functions and eigenfunctions
Analysis of PDEs
2014-01-14 v3 Differential Geometry
Spectral Theory
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.
Keywords
Cite
@article{arxiv.1112.4352,
title = {The effect of curvature on convexity properties of harmonic functions and eigenfunctions},
author = {Dan Mangoubi},
journal= {arXiv preprint arXiv:1112.4352},
year = {2014}
}
Comments
24 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