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A priori estimates for negative constant scalar curvature conformal metrics with positive constant boundary mean curvature

Sergio Almaraz, Shaodong Wang

Abstract

On a compact Riemannian manifold with boundary, we study the set of conformal metrics of negative constant scalar curvature in the interior and positive constant mean curvature on the boundary. Working in the case of positive Yamabe conformal invariant, we prove that this set is a priori bounded in the three-dimensional case and in the locally conformally flat with umbilical boundary case in any dimension not less than three.

A priori estimates for negative constant scalar curvature conformal metrics with positive constant boundary mean curvature

Abstract

On a compact Riemannian manifold with boundary, we study the set of conformal metrics of negative constant scalar curvature in the interior and positive constant mean curvature on the boundary. Working in the case of positive Yamabe conformal invariant, we prove that this set is a priori bounded in the three-dimensional case and in the locally conformally flat with umbilical boundary case in any dimension not less than three.

Paper Structure

This paper contains 8 sections, 17 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The model solutions in $\mathbb R^n_+$
  4. The linearized equation
  5. Isolated and isolated simple blow-up points
  6. Blow-up estimates in dimension three
  7. The Pohozaev sign restriction in dimension three
  8. The proof of Theorem \ref{['compactness:thm']}

Key Result

Theorem 1.1

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with boundary $\partial M$. Suppose that $M$ is of positive type and it is not conformally equivalent to $\mathbb S^n_+$. Assume further that $n=3$ or $M$ is locally conformally flat with $\partial M$ umbilic and $n\geq 3$. For any $0<\kap for some $0<\alpha<1$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Lemma \ref{['classifLinear']}
  • Definition 3.1
  • ...and 28 more