The dihedral rigidity conjecture for n-prisms

2019/07/08 by Chao Li, Li, Chao · 3 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.1907.03855

Abstract

We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for n≤ 7, if an n-dimensional prism has nonnegative scalar curvature and weakly mean convex faces, then its dihedral angle cannot be everywhere not larger than its Euclidean model, unless it is isometric to an Euclidean prism. The proof relies on constructing certain free boundary minimal hypersurface in a Riemannian polyhedron, and extending a dimension descent idea of Schoen-Yau. Our result is a localization of the positive mass theorem.

Citations

• On the proof of the positive mass conjecture in general relativity
• Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass
• A new proof of the positive energy theorem
• Scalar curvature and harmonic one-forms on three-manifolds with boundary

Cited by

• On Gromov's dihedral extremality and rigidity conjectures
• Scalar Curvature on some Open 3-Manifolds
• Dihedral rigidity for cubic initial data sets

Related

• A polyhedron comparison theorem for 3-manifolds with positive scalar curvature
• Dihedral rigidity in hyperbolic 3-space
• On Gromov's dihedral extremality and rigidity conjectures
• Dihedral rigidity for cubic initial data sets
• Convex Polytopes, Dihedral Angles, Mean Curvature and Scalar Curvature