Proof of the bounded conformal conjecture

2023/06/27 by Sameer Kumar, Kumar, Sameer
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2306.15322

openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given any asymptotically flat 3-manifold (M,g) with smooth, non-empty, compact boundary Σ, the conformal conjecture states that for every δ>0, there exists a metric g' = u⁴ g, with u a harmonic function, such that the area of outermost minimal area enclosure Σ_g' of Σ with respect to g' is less than δ. Recently, the conjecture was used to prove the Riemannian Penrose inequality for black holes with zero horizon area, and was proven to be true under the assumption of existence of only a finite number of minimal area enclosures of boundary Σ, and boundedness of harmonic function u. We prove the conjecture assuming only the boundedness of u.

Related

• Mass, Conformal Capacity, and the Volumetric Penrose Inequality
• Mass-capacity inequality modeled on conformally flat manifolds
• The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary
• Geometry of non-compact minimal and marginally outer-trapped surfaces in asymptotically flat manifolds
• A Conformal Positive Mass Theorem with Noncompact Boundary