Mass, Conformal Capacity, and the Volumetric Penrose Inequality

2024/10/12 by Mazurowski, Liam, Yao, Xuan
#Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2410.09626

Abstract

Let Ω be a smooth, bounded subset of ℝ³ diffeomorphic to a ball. Consider M = ℝ³ ∖ Ω equipped with an asymptotically flat metric g = f⁴ g_euc, where f→ 1 at infinity. Assume that g has non-negative scalar curvature and that Σ= ∂ M is a minimal 2-sphere in the g metric. We prove a sharp inequality relating the ADM mass of M with the conformal capacity of Ω. As a corollary, we deduce a sharp lower bound for the ADM mass of M in terms of the Euclidean volume of Ω. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function.

Related

• Mass-capacity inequality modeled on conformally flat manifolds
• Riemannian Penrose inequality without horizon in dimension three
• The Penrose inequality in extrinsic geometry
• ADM mass and the capacity-volume deficit at infinity
• ADM mass, area and capacity in asymptotically flat 3-manifolds with nonnegative scalar curvature