  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.
