Analysis and Mathematical Physics

An interesting feature of General Relativity is the presence of singularities which can occur in even the simplest examples such as the Schwarzschild spacetime. However, in this case the singularity is cloaked behind the event horizon of the black hole which has been conjectured to be generically the case. To analyze this so-called Cosmic Censorship Conjecture, Roger Penrose proposed in 1973 a test which involves Hawking's area theorem, the final state conjecture and a geometric inequality on initial data sets (M,g,k). For k=0 this so-called Penrose inequality has been proven by Gerhard Huisken and Tom Ilmanen via inverse mean curvature flow and by Hubert Bray using the conformal flow, but in general the question is wide open. I will present a new approach to spacetime inverse mean curvature flow via double null foliations which is the first instance where there are both monotonicity and exsitence results. This is based on spacetime harmonic functions which have been introduced together with Demetre Kazaras and Marcus Khuri in the context of the spacetime positive mass theorem.