Invariants of the harmonic conformal class of an asymptotically flat manifold

TL;DR

This paper introduces and analyzes invariants within the harmonic conformal class of asymptotically flat manifolds, linking geometric optimization, scalar curvature, and applications in general relativity.

Contribution

It defines new invariants of the harmonic conformal class and explores their connections to geometric optimization problems and the positive mass theorem.

Findings

Invariants relate to hypersurface area-minimizers.

Connections established with ADM mass.

Implications for zero area singularities.

Abstract

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic conformal class. These invariants are closely related to constrained geometric optimization problems involving hypersurface area-minimizers and the ADM mass. In the final section, we discuss possible applications of the invariants and their relationship with zero area singularities and the positive mass theorem.