Three circles theorems for harmonic functions

TL;DR

This paper establishes new three circles theorems for harmonic functions on manifolds, leading to existence results for polynomial growth harmonic functions and confirming conjectures about frequency bounds.

Contribution

It introduces two three circles theorems for harmonic functions in integral form on manifolds, generalizing previous results and solving conjectures about harmonic function growth and frequency.

Findings

Existence of nonconstant harmonic functions with polynomial growth on certain manifolds.

Confirmation of L. Ni's conjecture regarding harmonic functions.

Bounded frequency of harmonic functions with polynomial growth.

Abstract

We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.