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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On compact $ 3$-manifolds with nonnegative scalar curvature with a CMC boundary component


Authors: Pengzi Miao and Naqing Xie
Journal: Trans. Amer. Math. Soc. 370 (2018), 5887-5906
MSC (2010): Primary 53C20; Secondary 83C99
DOI: https://doi.org/10.1090/tran/7500
Published electronically: April 17, 2018
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Abstract: We apply the Riemannian Penrose inequality and the Riemannian positive mass theorem to derive inequalities on the boundary of a class of compact Riemannian $ 3$-manifolds with nonnegative scalar curvature. The boundary of such a manifold has a CMC component, i.e., a $ 2$-sphere with positive constant mean curvature; and the rest of the boundary, if nonempty, consists of closed minimal surfaces. A key step in our proof is the construction of a collar extension that is inspired by the method of Mantoulidis-Schoen.


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Additional Information

Pengzi Miao
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33146
Email: pengzim@math.miami.edu

Naqing Xie
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: nqxie@fudan.edu.cn

DOI: https://doi.org/10.1090/tran/7500
Keywords: Scalar curvature, CMC surfaces, Riemannian Penrose inequality
Received by editor(s): January 30, 2017
Published electronically: April 17, 2018
Additional Notes: The first named author’s research was partially supported by Simons Foundation Collaboration Grant for Mathematicians #281105.
The second named author’s research was partially supported by the National Science Foundation of China #11671089, #11421061.
Article copyright: © Copyright 2018 American Mathematical Society

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