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

TL;DR

This paper establishes Liouville type theorems on certain manifolds with nonnegative curvature and convex boundary, leading to new Sobolev trace inequalities and extending previous eigenvalue bounds.

Contribution

It provides a nonlinear generalization of eigenvalue bounds and verifies a conjecture, advancing understanding of geometric analysis on manifolds with boundary.

Findings

Proved Liouville type theorems for manifolds with nonnegative curvature and convex boundary.

Derived sharp Sobolev trace inequalities on these manifolds.

Partially verified a conjecture related to Steklov eigenvalues.

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.