Liouville type theorems on manifolds with nonnegative curvature and strictly convex boundary

2019/12/11 by Qianqiao Guo, Guo, Qianqiao, Fengbo Hang +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1912.05574

openalex publication_date 2019/12/11 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.

Cited by

• Uniqueness results for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary
• A note on Wang's conjecture for harmonic functions with nonlinear boundary condition

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