The Positive Mass Theorem is a fundamental result in differential geometry and general relativity. It states that if a space looks flat far away and has nonnegative scalar curvature (a measure of how the space bends), then its total mass must be nonnegative. Moreover, the only way the mass can be zero is if the space is completely flat.

A natural question is what happens to this total mass when we deform the geometry in a conformal way i.e, when we rescale distances by a smooth function. In joint work with Alex Freire, we study this question for asymptotically flat manifolds that have a noncompact boundary (such as a half-space). Using harmonic functions, we prove that a certain weighted combination of the masses of two conformally related metrics remains nonnegative under corresponding curvature conditions.

53 Differential geometry
83C General relativity