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

Abstract:

Let $(M,g)$ be a smooth compact $n$-dimensional Riemannian manifold with smooth boundary $\partial M$ for $n\ge 5$. We prove a trace inequality, that is \[ \|u\|^2_{L^q(\partial M)}\leq S\left (\int _{M}|\nabla _g u|^2 \mathrm {d} v_g+ \frac {n-2}{2}\int _{\partial M}h_g u^2 \mathrm {d} s_g\right )+A\|u\|^2_{L^r(\partial M)} \] for all $u\in H^1(M)$, where $S=\frac {2}{n-2}\omega _n^{-1/(n-1)}$ with $\omega _n$ the volume of the unit sphere in $\mathbb {R}^n$, $q=\frac {2(n-1)}{n-2}$, $r=\frac {2(n-1)}{n}$, $h_g$ is the mean curvature of $\partial M$, $\mathrm {d} v_g$ is the volume form of $(M,g)$, $\mathrm {d} s_g$ is the induced volume form on $\partial M$, and $A$ is a positive constant depending only on $(M, g)$. This inequality is sharp in the sense that $S$ cannot be replaced by any smaller constant, $h$ in general cannot be replaced by any smooth function which is smaller than $h$ at some point on $\partial M$, and $r$ cannot be replaced by any smaller number.