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A Generalization of the Sphere Covering Inequality

Changfeng Gui, Amir Moradifam

TL;DR

The paper generalizes the Sphere Covering Inequality to allow a boundary gap $c ≥ 0$ between two conformal factors, establishing a quantitative stability-type bound on the combined exponential mass. It develops auxiliary lemmas, a radial comparison principle, and a rearrangement argument to compare mass against model radial profiles, leading to the bound $∫ (e^{v_1}+e^{v_2}) ≥ 8(1 - c/M)π$ (and its $w = 2v$ form $∫ (e^{2v_1}+e^{2v_2}) ≥ 4(1 - c/M)π$). Equality characterizations link to spherical-cap geometry, highlighting when boundary matching is essential for sharpness. The results broaden analytic tools for elliptic PDEs with exponential nonlinearities and have implications in conformal geometry and mathematical physics by quantifying how boundary perturbations affect global area bounds.

Abstract

The Sphere Covering Inequality was introduced in \cite{GM} (\emph{Invent. Math.}, 2018) as a sharp geometric inequality that provides a lower bound for the total area of two distinct surfaces of Gaussian curvature 1. These surfaces are assumed to be conformal to the Euclidean unit disk and share the same conformal factor along the boundary. In this paper, we establish a quantitative generalization that relaxes the boundary matching condition by allowing the conformal factors to differ by a constant \( c \ge 0 \) on the boundary. This refinement reveals a new stability-type structure underlying the inequality. Our results show that the Sphere Covering Inequality is stable with respect to perturbations in the boundary data and provide a precise quantitative description of how the total-area bound varies under such perturbations. The generalized inequality provides new analytic and geometric tools for the study of elliptic equations with exponential nonlinearities, conformal geometry, and related problems in mathematical physics.

A Generalization of the Sphere Covering Inequality

TL;DR

The paper generalizes the Sphere Covering Inequality to allow a boundary gap between two conformal factors, establishing a quantitative stability-type bound on the combined exponential mass. It develops auxiliary lemmas, a radial comparison principle, and a rearrangement argument to compare mass against model radial profiles, leading to the bound (and its form ). Equality characterizations link to spherical-cap geometry, highlighting when boundary matching is essential for sharpness. The results broaden analytic tools for elliptic PDEs with exponential nonlinearities and have implications in conformal geometry and mathematical physics by quantifying how boundary perturbations affect global area bounds.

Abstract

The Sphere Covering Inequality was introduced in \cite{GM} (\emph{Invent. Math.}, 2018) as a sharp geometric inequality that provides a lower bound for the total area of two distinct surfaces of Gaussian curvature 1. These surfaces are assumed to be conformal to the Euclidean unit disk and share the same conformal factor along the boundary. In this paper, we establish a quantitative generalization that relaxes the boundary matching condition by allowing the conformal factors to differ by a constant on the boundary. This refinement reveals a new stability-type structure underlying the inequality. Our results show that the Sphere Covering Inequality is stable with respect to perturbations in the boundary data and provide a precise quantitative description of how the total-area bound varies under such perturbations. The generalized inequality provides new analytic and geometric tools for the study of elliptic equations with exponential nonlinearities, conformal geometry, and related problems in mathematical physics.
Paper Structure (3 sections, 7 theorems, 63 equations)

This paper contains 3 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Results and Preliminaries
  3. Generalized Sphere Covering Inequality

Key Result

Theorem 1.1

Let $\Omega$ be a simply-connected subset of $\R^2$ and assume $v_i \in C^2(\overline{\Omega})$, $i=1,2$ satisfy where $f_2 \geq f_1 \ge 0$ in $\Omega$. If $v_2 \ge v_1, v_2 \not \equiv v_1$ in $\omega$ and $v_2=v_1$ on $\partial \omega$ for some piecewise Liptschitz subdomain $\omega \subset \Omega$, then Moreover, the equality only holds when $f_2 \equiv f_1 \equiv 0$ in $\omega$, and $(\omeg

Theorems & Definitions (9)

  • Theorem 1.1: The Sphere Covering Inequality GM
  • Theorem 1.2: Generalized Sphere Covering Inequality
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 3.1
  • Theorem 3.1