Three circles theorems for harmonic functions

Abstract

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.

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

“…Finally by Arzela-Ascoli theorem we construct a harmonic function with the desired growth rate on the whole manifold. In [40], Xu proved a three circles theorem, which is different from the one given by Ding, see Theorem 3.2 in [40] and Lemma 1.1 in [19] respectively.…”

“…A key viewpoint in the proof of Theorem 1.3 is that, for a manifold (M n , g) with nonnegative Ricci curvature and maximal volume growth, its asymptotically conic property makes the harmonic functions with polynomial growth on it behave like harmonic functions on a cone. This viewpoint has influenced many previous works, see [14], [16], [19], [40], [24], [26] etc.…”

“…Note that the power n−1 in this result is sharp compared to the Euclidean case. In a recent paper, Xu obtained the existence of nontrivial harmonic functions with polynomial growth on some manifolds satisfying certain assumptions on its tangent cone at infinity, see Theorem 1.7 in [40]. Note that Xu's existence theorem does not require the manifold to have maximal volume growth, and an explicit example of a collapsed manifold satisfying the assumptions in Xu's existence theorem was constructed in [40].…”

“…In [19] and [40], the ideas to find nontrivial harmonic functions with polynomial growth are similar. Making use of the conic structure at infinity, we transplant suitable harmonic functions on the tangent cone at infinity back to obtain harmonic functions defined on larger and larger geodesic balls of the manifold.…”

Three Balls Theorem for Eigenfunctions of Dirac Operator in Clifford Analysis

“…From this aspect, Hadamard's three circles theorem has many interesting applications in partial differential equations and differential geometry (see e.g. [7,8,16,1,15,11,5] and the references therein).…”