The spacetime Laplace equation on initial data sets for Einstein's equations

This talk will consider a 'spacetime Laplace operator' on an initial data set for the Einstein equations. This operator reflects the geometry of the underlying manifold in a manner similar to the Dirac operator appearing in Witten's proof of the Positive Mass Theorem. By analyzing linear-growth spacetime harmonic functions and their level sets, we obtain a rather approachable refinement of the Positive Mass Theorem for asymptotically flat 3-dimensional initial data sets. Applications to asymptotically hyperbolic initial data sets are also considered. The work I will discuss includes collaborations with Hugh Bray, Sven Hirsch, Marcus Khuri, Daniel Stern, and Yiyue Zhang.