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<p>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. <b>31</b>, 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {B}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> case (Math. Ann. <b>391</b>, 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年6月3日

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  • Xiaohan Cai

摘要

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 \(\mathbb {B}^n\) 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.