  In the pioneering work of Stern, level sets of harmonic functions have been
shown to be an effective tool in the study of scalar curvature in dimension 3.
Generalizations of this idea, utilizing level sets of so called spacetime
harmonic functions as well as other elliptic equations, are similarly effective
in treating geometric inequalities involving the ADM mass. In this paper, we
survey recent results in this context, focusing on applications of spacetime
harmonic functions to the asymptotically flat and asymptotically hyperbolic
versions of the spacetime positive mass theorem, and additionally introduce a
new concept of total mass valid in both settings which is encoded in
interpolation regions between generic initial data and model geometries.
Furthermore, a novel and elementary proof of the positive mass theorem with
charge is presented, and the level set approach to the Penrose inequality given
by Huisken and Ilmanen is related to the current developments. Lastly, we
discuss several open problems.
