Sharp constants in weighted trace inequalities on Riemannian manifolds

We establish some sharp weighted trace inequalities $W^{1,2}({\rho }^{1-2\sigma },M)\hookrightarrow L^{\frac{2n}{n-2\sigma }}(\text{pa}M)$ on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of $M$ and $\sigma \in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.