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

We prove some Liouville 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.