The growth rate of harmonic functions

TL;DR

This paper investigates the growth rates of harmonic functions on certain Riemannian manifolds, providing sharp gradient estimates and confirming a case of the Colding-Minicozzi frequency conjecture.

Contribution

It offers new sharp gradient estimates for positive harmonic functions and proves the existence of polynomial growth harmonic functions under specific geometric conditions.

Findings

Sharp gradient estimate for positive harmonic functions on surfaces

Existence of polynomial growth harmonic functions on manifolds with non-negative Ricci curvature

Confirmation of a special case of the Colding-Minicozzi frequency conjecture

Abstract

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than three, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding-Minicozzi conjecture on frequency.