A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundary

TL;DR

This paper investigates the existence of conformal metrics with prescribed non-positive scalar and boundary mean curvatures on asymptotically Euclidean manifolds with inner boundary, providing a classification based on conformal invariants.

Contribution

It establishes a necessary and sufficient condition for the existence of such metrics and applies this to classify Yamabe types on these manifolds.

Findings

Derived a conformal invariant criterion for solution existence

Established the Yamabe classification for asymptotically Euclidean manifolds with boundary

Connected curvature prescription problems with conformal geometry invariants

Abstract

We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.