2023/03/23 by Pengzi Miao, Miao, Pengzi, Annachiara Piubello +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2303.13633
openalex publication_date 2023/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a metric γ of nonnegative Gauss curvature and a positive function H on a 2-sphere Σ, we estimate the Bartnik quasi-local mass of (Σ, γ, H) in terms of the area, the total mean curvature, and a quantity depending only on γ, measuring the roundness of the metric. If γ has positive Gauss curvature, the roundness of γ in the estimate is controlled by the ratio κ between the maximum and the minimum of the Gauss curvature. As κ→ 1, the estimate approaches a sharp estimate for round spheres with arbitrary, positive mean curvature functions. Enroute we observe an estimate of the supremum of the total mean curvature among nonnegative scalar curvature fill-ins of a closed manifold with positive scalar curvature.