On compact $3$-manifolds with nonnegative scalar curvature with a CMC boundary component

We apply the Riemannian Penrose inequality and the Riemannian positive mass theorem to derive inequalities on the boundary of a class of compact Riemannian $3$-manifolds with nonnegative scalar curvature. The boundary of such a manifold has a CMC component, i.e. a $2$-sphere with positive constant mean curvature; and the rest of the boundary, if nonempty, consists of closed minimal surfaces. A key step in our proof is the construction of a collar extension that is inspired by the method of Mantoulidis-Schoen \cite{M-S}.