Dihedral rigidity for cubic initial data sets

2021/08/19 by Tin-Yau Tsang, Tsang, Tin-Yau
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2108.08942

openalex publication_date 2021/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we pose and prove a spacetime version of Gromov's dihedral rigidity theorem (Gromov, Li) for cubes when the dimension is 3 by studying the level sets of spacetime harmonic functions (Stern, Bray-Stern, Hirsch-Kazaras-Khuri), extending the work of Chai-Kim. As a corollary, we also obtain an alternative proof of dihedral rigidity for prisms in hyperbolic space (Li). We then discuss the relation between polyhedra and the spacetime positive mass theorem. This generalises the work of Miao-Piubello and Li. Finally, we show dihedral rigidity of charged Riemannian cubes by charged harmonic functions (Bray-Hirsch-Kazaras-Khuri-Zhang).

Citations

• The mass of an asymptotically flat manifold
• A new proof of the positive energy theorem
• Scalar curvature and harmonic one-forms on three-manifolds with boundary
• The dihedral rigidity conjecture for n-prisms

Related

• Initial data set rigidity results for polyhedra
• Dihedral rigidity in hyperbolic 3-space
• The dihedral rigidity conjecture for n-prisms
• On Gromov's dihedral extremality and rigidity conjectures
• Rigidity of discrete conformal structures on surfaces