Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields

2023/07/11 by S. Cecchini, Martin Lesourd, Cecchini, Simone +3
Mathematics · #53C21 (Primary) 53C24 #53C27 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2307.05277

openalex publication_date 2023/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a positive mass theorem for spin initial data sets (M,g,k) that contain an asymptotically flat end and a shield of dominant energy (a subset of M on which the dominant energy scalar μ-|J| has a positive lower bound). In a similar vein, we show that for an asymptotically flat end E that violates the positive mass theorem (i.e. E < |P|), there exists a constant R>0, depending only on E, such that any initial data set containing E must violate the hypotheses of Witten's proof of the positive mass theorem in an R-neighborhood of E. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

Citations

• Boundary value problems for Dirac--type equations, with applications
• Conformally flat manifolds, Kleinian groups and scalar curvature
• The mass of an asymptotically flat manifold
• A new proof of the positive energy theorem
• Dihedral rigidity for cubic initial data sets

Related

• The Positive Mass Theorem for Creased Initial Data
• Positive Energy Theorems for Spin Initial Data with Charge
• Tilted spacetime positive mass theorem with arbitrary ends
• A tilted spacetime positive mass theorem
• Equality in the spacetime positive mass theorem II