  In this article, we found a connection between Brown-York mass and the first
Dirichlet Eigenvalue of a Schr\"odingier type operator. In particular, we
proved a local positive mass type theorem for metrics conformal to the
background one with suitable presumptions. As applications, we investigated
compactly conformal deformations which either increase or decrease scalar
curvature. We found local conformal rigidity phenomena occur in both cases for
small domains and as for manifolds with nonpositive scalar curvature it is even
more rigid in particular. On the other hand, such deformations exist for closed
manifolds with positive scalar curvature. We also constructed such kind of
deformations on a type of product manifolds that either increase or decrease
their scalar curvature compactly and conformally. These results together
answered a natural question arises in \cite{Corvino, Lohkamp}.
