2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119195In this paper, by modifying the arguments in \cite{WY}, we get some rigidity theorems on compact manifolds with nonempty boundary. The results in this paper are similar with those in \cite{ST} and \cite{WY}. Like \cite{ST} and \cite{WY}, we still use quasi-spherical metrics introduced by \cite{Ba} to get monotonicity of some quantities.13 pagesDifferential GeometryMathematical Physics53C20, 83C99text