  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.
