Conformal metrics with prescribed scalar and mean curvature

TL;DR

This paper addresses the problem of prescribing scalar and boundary mean curvatures via conformal metric deformations in higher dimensions, including cases with negative curvatures, using variational methods and compactness techniques.

Contribution

It provides new existence results for conformal metrics with prescribed curvatures, especially in three dimensions, handling negative curvatures and boundary conditions.

Findings

Established existence results for prescribed curvatures with negative values.

Developed techniques to prevent bubbling and loss of compactness.

Applied variational methods and integral estimates to ensure solutions.

Abstract

We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur with profiles of hyperbolic balls or horospheres, and hence one may lose either pointwise estimates on the conformal factor or the total conformal volume. We can sometimes prevent them using integral estimates, Pohozaev identities and domain-variations of different types.