  In 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.
