Intermediate curvature, spacetime harmonic functions and the monotonicity of the Hawking energy

Abstract

This dissertation is based on the manuscripts \cite{BHJ, Hirsch, HKK, HirschZhang}.The paper \cite{BHJ} is joint with Simon Brendle and Florian Johne, \cite{HKK} is joint with Demetre Kazaras and Marcus Khuri, \cite{HirschZhang} is joint with Yiyue Zhang, and \cite{Hirsch} is a solo-authored.First, we introduce \emph{$m$-intermediate curvature $\mathcal C_m$} which interpolates between Ricci ($m=1$) and scalar curvature ($m=n-1$) and prove in this context a generalized Geroch conjecture \cite{BHJ}.In particular, we show that $M^{n-m}\times \mathbb T^m$, $n\le7$, does not admit a metric with $\mathcal C_m>0$.Next we study initial data sets $(M,g,k)$ which are used in General Relativity to describe isolated gravitational systems.We introduce \emph{spacetime harmonic functions}, i.e. functions solving the PDE $\Delta u=-\tr_gk|\nabla u|$, to give a new lower bound for the mass of $(M,g,k)$. This lower bound in particular implies the spacetime positive mass theorem \cite{HKK} including the case of equality \cite{HirschZhang}.Finally, we discuss recent progress towards the spacetime Penrose conjecture \cite{Hirsch}.We demonstrate how the famous monotonicity formula for the Hawking energy under inverse mean curvature flow can be generalized to initial data sets. This leads to new notion of \emph{spacetime inverse mean curvature flow} which is based on double null foliations.Further papers I wrote during my time in graduate school \cite{BHHWZ, BHKKZ, BHKKZ2, HKKZ, HKKZ2, HirschLesourd, HirschLi, HirschMiao, HMT2, HMT, HirschZhu} will not be discussed.