Mathematical Research Letters

Volume 28 (2021)

Number 5

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

Pages: 1419 – 1439

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n5.a6

Authors

Qianqiao Guo (Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, China)

Fengbo Hang (Courant Institute, New York, N.Y., U.S.A.)

Xiaodong Wang (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

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 for the first Steklov eigenvalue by Xia–Xiong and verifies partially a conjecture by the third named author. As a consequence, we derive several sharp Sobolev trace inequalities on such manifolds.

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The third-named author is partially supported by Simons Foundation Collaboration Grant for Mathematicians #312820.

Received 8 January 2020

Accepted 25 May 2020

Published 16 August 2022