Yamabe Classification and Prescribed Scalar Curvature in the Asymptotically Euclidean Setting

TL;DR

This paper establishes a criterion involving the Yamabe invariant for when an asymptotically Euclidean manifold can be conformally transformed to have a prescribed nonpositive scalar curvature, linking geometric analysis and conformal geometry.

Contribution

It provides a necessary and sufficient condition based on the Yamabe invariant for prescribing scalar curvature in asymptotically Euclidean manifolds, and relates Yamabe classes to conformal compactifications.

Findings

Characterization of when a manifold can be conformally transformed to have a given scalar curvature.

Method to compute the Yamabe invariant of a set from generalized eigenvalues.

Equivalence of Yamabe class between a manifold and its conformal compactification.

Abstract

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also show that the Yamabe class of an asymptotically Euclidean manifold is the same as the Yamabe class of its conformal compactification.