Brown–York Mass and Compactly Supported Conformal Deformations of Scalar Curvature

Abstract

In this article, we find a connection between Brown–York mass and the first Dirichlet eigenvalue of a Schrodinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly supported conformal deformations which either increase or decrease scalar curvature. We find 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 or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question that arises naturally in (Corvino in Commun Math Phys 214:137–189, 2000; Lohkamp in Math Ann 313:385–407, 1999).