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

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Abstract

In this article, we find a connection between Brown–York mass and the first Dirichlet eigenvalue of a Schrödinger 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).

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Acknowledgments

The author would like to express his deepest appreciation to Professor Jie Qing for his inspiring discussions and constant encouragement. Also, he would like to thank the referee for his or her patient work and valuable suggestions. This work is support by NSF Grant DMS-1005295 and DMS-1303543.

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  1. School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, Guangdong, China

    Wei Yuan

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  1. Wei Yuan
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Correspondence to Wei Yuan.

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Yuan, W. Brown–York Mass and Compactly Supported Conformal Deformations of Scalar Curvature. J Geom Anal 27, 797–816 (2017). https://doi.org/10.1007/s12220-016-9698-7

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  • DOI: https://doi.org/10.1007/s12220-016-9698-7

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