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

TL;DR

This paper establishes a priori bounds for conformal metrics with negative scalar curvature and positive boundary mean curvature on certain compact manifolds, advancing understanding in geometric analysis.

Contribution

It provides new a priori estimates for such metrics in three dimensions and in higher dimensions under specific geometric conditions.

Findings

Set of conformal metrics is a priori bounded in 3D.

Boundedness also holds in higher dimensions with conformally flat, umbilical boundary.

Results depend on positive Yamabe invariant.

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.