Rigidity results for complete manifolds with nonnegative scalar curvature

2020/08/16 by Jintian Zhu, Zhu, Jintian · 4 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Secondary 53C21 #math.DG #msc:53C21 #msc:53C24 #primary 53C24

paper · pdf · doi:10.48550/arxiv.2008.07028

18 pages

arxiv created 2020/08/16 · openalex publication_date 2020/08/16 · arxiv updated 2020/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to T^n-1× \mathbf R with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal 2-systole inequality for several classes of complete Riemannian manifolds with positive scalar curvature and further prove a rigidity result for the equality case.

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• Existence of hypersurfaces with prescribed mean curvature I - Generic min-max

Cited by

• Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature
• Area-charge inequalities and rigidity of time-symmetric initial data sets
• Rigidity of Complete Free Boundary Minimal Hypersurfaces in Convex NNSC Manifolds
• Band width estimates with lower spectral curvature bounds

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