A sharp Sobolev trace inequality involving the mean curvature on Riemannian manifolds

In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $|u|{L^{q}(\pa M)}^2\leq S |\nabla_g u|{L^2(M)}^2 +A(M,g)|u|^2_{L^2(\pa M)}$onRiemannianmanifolds$(M,g)$withboundaries$\pa M$,where$q=\frac{2(n-1)}{n-2}$,$S$isthebestconstantand$A(M,g)$ is some positive constant depending only on $M$ and $g$. We obtain a sharp trace inequality involving the mean curvature in a remainder term, which would fail in general once the mean curvature is replaced by any smaller function.