Strong convexity for harmonic functions on compact symmetric spaces

TL;DR

This paper proves that certain derivatives of the squared magnitude of harmonic functions are nonnegative on compact symmetric spaces, extending known Euclidean results and highlighting strong convexity properties of harmonic functions on these spaces.

Contribution

It generalizes a nonnegativity property of derivatives of harmonic functions from Euclidean space to compact symmetric spaces, revealing new convexity features.

Findings

$ riangle^k |h|^2$ is nonnegative for harmonic functions on spherical disks

Extension of Euclidean harmonic properties to symmetric spaces of compact type

Reveals strong convexity of the $L^2$-growth function for harmonic functions

Abstract

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on $\mathbb{R}^n$ discovered by the first two authors and is related to strong convexity of the $L^2$-growth function of harmonic functions.