2 problems

Let $(\Sigma,g_\Sigma)$ be a closed Riemannian $n$-manifold associated with a smooth function $h:\Sigma\to \mathbf R$. A fill-in is a compact Riemannian $(n+1)$-manifold…

Let $(X,g)$ be a compact Riemannian manifold with scalar curvature $R\geq\sigma$. Let $H$ denote the mean curvature of the boundary $\partial X$ in $(X,g)$ with respect to the outw…