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Conformal metrics of the ball with constant $σ_k$-curvature and constant boundary mean curvature

Xuezhang Chen, Wei Wei

TL;DR

The paper extends Escobar's Obata-type ball rigidity from scalar curvature ($k=1$) to fully nonlinear conformal invariants $\sigma_k(A_g)$ under boundary curvature constraints. The authors develop an Obata-Escobar framework for the $\sigma_k$-cone $\Gamma_k^+$ and its negative, employing Newton transformations $T_k$ and the trace-free tensor $L_k(A_g)$ to produce integral inequalities that force the metric to be Einstein. Under three regimes -- zero $\sigma_k$, positive constant $\sigma_k$, or positive $\sigma_k(-A_g)$ in $\overline{\Gamma_k^+}$ -- they show the metric is conformally equivalent to a canonical Einstein metric $g_c$, with explicit formulas for $g_c$ in each case. The results unify Liouville-type rigidity on the hemisphere and provide explicit conformal models on the ball, contributing to fully nonlinear conformal geometry and boundary curvature problems.

Abstract

On the upper hemisphere, we use the Obata-Escobar argument to classify conformal metrics with constant $σ_k$ curvature and constant boundary mean curvature in all types of cones including positive and negative cones. This extends a result of Escobar in \cite{Es} for $k=1$.

Conformal metrics of the ball with constant $σ_k$-curvature and constant boundary mean curvature

TL;DR

The paper extends Escobar's Obata-type ball rigidity from scalar curvature () to fully nonlinear conformal invariants under boundary curvature constraints. The authors develop an Obata-Escobar framework for the -cone and its negative, employing Newton transformations and the trace-free tensor to produce integral inequalities that force the metric to be Einstein. Under three regimes -- zero , positive constant , or positive in -- they show the metric is conformally equivalent to a canonical Einstein metric , with explicit formulas for in each case. The results unify Liouville-type rigidity on the hemisphere and provide explicit conformal models on the ball, contributing to fully nonlinear conformal geometry and boundary curvature problems.

Abstract

On the upper hemisphere, we use the Obata-Escobar argument to classify conformal metrics with constant curvature and constant boundary mean curvature in all types of cones including positive and negative cones. This extends a result of Escobar in \cite{Es} for .

Paper Structure

This paper contains 2 sections, 9 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Proof of main theorem

Key Result

Theorem 1

Let $\tilde{g}$ be a conformal metric in the unit Euclidean ball $(\mathbb{B}^{n+1},|\mathrm{d} x|^2)$ for $n \geq 2$ and $\tilde{g}$ has constant boundary mean curvature $\tilde{H}$ on $\partial \mathbb{B}^{n+1}$. For $2\le k\le n$, suppose one of three conditions is true: Then $\tilde{g}$ is Einstein. Moreover, there exist a canonical conformal metric $g_{\mathrm{c}}$ and a conformal diffeomorp

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 2: LL03
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 6 more