Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature

By Yuguang Shi, Luen-Fai Tam. 2002

314 citations recorded by the source corpus.

In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and with nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b73506ae3b55a602025267fd47032c99537e840

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In Applied Mathematics

1. M. Mars
2. A. Neves
3. Luen-Fai Tam
4. Yuguang Shi
5. Fengbo Hang
6. Pengzi Miao
7. Xiaodong Wang
8. F. C. Marques

In Mathematical Physics

1. M. Mars
2. Luen-Fai Tam
3. A. Neves
4. Yuguang Shi
5. Fengbo Hang
6. Pengzi Miao
7. Xiaodong Wang

Citations in the corpus

• On the volume functional of compact manifolds with boundary with constant scalar curvature (2008, background): “By the results in [19, 12], which are generalizations of the positive mass theorem in [17, 22, 18], we have H ≤ H0, and H = H0 if and only if (Ω, g) is isometric to a standard ball in R.”
• Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds (2007, methodology): “Finally, in Section 5 we adapt the work of Shi-Tam [11] and Mu-Tao-Yau [12] to prove long time existence for a flow inspired in [11].”
• Present status of the Penrose inequality (2009, methodology): “Using the inverse mean curvature flow, Shi and Tam prove [150] (among other things) that the inequality MBY (∂Ω) ≥ MG(∂Ω) holds with equality if only if Ω is a standard ball in R3.”
• Scalar curvature and singular metrics (2016, methodology): “Miao [21] and the authors [31] studied and proved positive mass theorems for metrics with corners.”
• Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature (2009, background): “Shi-Tam [ ST ] have also studied compact manifolds (Mn;g) whose boundaries isometrically embed in Rn as a convex hypersurface.”