Uniqueness results for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary
Xiaohan Cai
TL;DR
This work establishes Liouville-type rigidity for positive harmonic functions on compact manifolds with nonnegative Ricci curvature and strictly convex boundary, addressing aspects of Wang's conjecture. It develops a P-function method anchored by a closed conformal vector field to derive divergence identities and integral estimates, enabling partial confirmation of Wang's conjecture on general manifolds and a complete rigidity result on warped products. The results encompass non-sharp but dimension-free ranges for the boundary parameter $\lambda$, connections to Gu-Li and Guo-Hang-Wang results in the model ball, and a precise classification of Sobolev trace minimizers on warped products. Overall, the paper links curvature, boundary convexity, and conformal structure to Liouville-type rigidity and sharp functional-inequality consequences.
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.