Spacetime Harmonic Functions and Applications to Mass

In the pioneering work of Stern (2019) “Scalar curvature and harmonic maps to S¹”, J. Diff. Geom., (2019), arXiv:1908.09754, 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 chapter, 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.