Abstract
We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi–Tam and Wang–Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown–York quasi-local mass without assuming that the boundary 2-sphere has positive Gauss curvature.
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Communicated by P. T. Chruściel
C. Mantoulidis research was partially supported by the Ric Weiland Graduate Fellowship at Stanford University. P. Miao research was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #281105.
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Mantoulidis, C., Miao, P. Total Mean Curvature, Scalar Curvature, and a Variational Analog of Brown–York Mass. Commun. Math. Phys. 352, 703–718 (2017). https://doi.org/10.1007/s00220-016-2767-8
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DOI: https://doi.org/10.1007/s00220-016-2767-8