Hypersurfaces with nonnegative scalar curvature

2011/02/28 by Lan-Hsuan Huang, Huang, Lan-Hsuan, Damin Wu +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1102.5749

A point in the proof of Theorem 2 that was overlooked in the previous versions is fixed. The appendix of some topological results is added. To appear in J. Differential. Geom

openalex publication_date 2011/02/28 · arxiv created 2013/05/02 · arxiv updated 2013/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete non-compact hypersurfaces of finitely many regular ends with nonnegative scalar curvature are weakly mean convex, and prove a positive mass theorem for such hypersurfaces.

Citations

• The Higher Dimensional Positive Mass Theorem II
• Rigidity phenomena involving scalar curvature
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• On the proof of the positive mass conjecture in general relativity
• A new proof of the positive energy theorem
• On the Positive Mass, Penrose, an ZAS Inequalities in General Dimension
• The equality case of the Penrose inequality for asymptotically flat graphs

Related

• A note on scalar curvature and the convexity of boundaries
• Mean Curvature in the Light of Scalar Curvature
• Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature
• Topological obstructions to nonnegative scalar curvature and mean convex boundary
• On the scalar curvature of constant mean curvature hypersurfaces in space forms