The sharp σ₂-curvature inequality on the sphere in quantitative form

2024/12/17 by Rupert L. Frank, Jonas W. Peteranderl, Frank, Rupert L. +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2412.12819

openalex publication_date 2024/12/17 · openalex created_date 2024/12/19 · openalex updated_date 2026/08/01

Abstract

Among all metrics on \mathbb S^d with d>4 that are conformal to the standard metric and have positive scalar curvature, the total σ₂-curvature, normalized by the volume, is uniquely (up to Möbius transformations) minimized by the standard metric. We show that if a metric almost minimizes, then it is almost the standard metric (up to Möbius transformations). This closeness is measured in terms of Sobolev norms of the conformal factor, and we obtain the optimal stability exponents for two different notions of closeness. This is a stability result for an optimization problem whose Euler-Lagrange equation is fully nonlinear.

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