A prescribed scalar and boundary mean curvature problem on compact manifolds with boundary

TL;DR

This paper investigates the problem of conformally deforming metrics on compact manifolds with boundary to achieve prescribed nonpositive scalar and boundary mean curvatures, establishing a key conformal invariant criterion.

Contribution

It introduces a necessary and sufficient condition based on a conformal invariant for the existence of such metrics with prescribed curvatures.

Findings

Established a conformal invariant criterion for existence

Characterized the zero set of the target curvatures

Provided conditions for conformal metric deformation

Abstract

We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures.