Uniqueness results for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary

TL;DR

This paper proves Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and convex boundary, confirming parts of Wang's conjecture and offering new proofs and partial verifications.

Contribution

It introduces a P-function method leveraging conformal vector fields to establish uniqueness results and verifies aspects of Wang's conjecture on warped product manifolds.

Findings

Confirmed some cases of Wang's conjecture for compact manifolds.

Provided an alternative proof of Gu-Li's result in the Euclidean ball case.

Partially verified Wang's conjecture on warped product manifolds.

Abstract

We prove some Liouville-type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary, thereby confirming some cases of Wang's conjecture (J. Geom. Anal. 31, 2021). We further investigate Wang's conjecture on warped product manifolds and provide a partial verification of this conjecture, which also yields an alternative proof of Gu-Li's resolution of the conjecture in the $\mathbb{B}^n$ case (Math. Ann. 391, 2025). Our approach is based on a general principle of employing the P-function method to such Liouville-type results, with particular emphasis on the role of a closed conformal vector field inherent to such manifolds.