Scalar Curvature Splittings I: Minimal Factors

2020/12/22 by Joachim Lohkamp, Lohkamp, Joachim
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Curvature #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Isoperimetric inequality #Mathematical analysis #Mathematics #Pure mathematics #Scalar (mathematics) #Scalar curvature #Tangent #math.DG

paper · pdf · doi:10.48550/arxiv.2012.12223

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2020/12/22 · arxiv created 2022/03/29 · arxiv updated 2022/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Scalar curvature constraints can be studied by means of splitting procedures. The success of this strategy depends on the control we can get on its splitting factors. We introduce canonical so-called minimal splitting factors. They have positive scalar curvature while other properties strongly resemble those of area minimizing hypersurfaces. This includes the presence of Poincare, Sobolev and isoperimetric inequalities and the fact that singular points admit tangent cones but now with positive scalar curvature.

Citations

• Lectures on Analysis on Metric Spaces
• Positive scalar curvature and the Dirac operator on complete riemannian manifolds
• Metric Spaces of Non-Positive Curvature
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Related

• Scalar Curvature Splittings II: Removal of Singularities
• On Area Comparison and Rigidity Involving the Scalar Curvature
• Contracting Maps and Scalar Curvature
• Singular Minimal Hypersurfaces and Scalar Curvature
• Minimal Smoothings of Area Minimizing Cones