Uniformly Area Expanding Flows in Spacetimes

TL;DR

This thesis proves the existence of infinitely many non-spherically symmetric, non-static spacetimes that admit smooth solutions to inverse mean curvature vector flow, advancing understanding of geometric flows in general relativity.

Contribution

It demonstrates the existence of smooth global solutions to inverse mean curvature vector flow in a broad class of spacetimes beyond prior symmetric cases, potentially impacting the Spacetime Penrose Conjecture.

Findings

Existence of infinitely many spacetimes with smooth solutions

Construction of solutions in non-spherically symmetric, dynamic spacetimes

Potential implications for the Spacetime Penrose Conjecture

Abstract

The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thesis might be important to prove the Spacetime Penrose Conjecture, which remains open today. Given a spacetime $(N^{4}, \overline{g})$ and a spacelike hypersurface $M$. For any closed surface $\Sigma$ embedded in $M$ satisfying some natural conditions, one can "steer" the spacetime metric $\overline{g}$ such that the mean curvature vector field of $\Sigma$ becomes tangential to $M$ while keeping the induced metric on $M$. This can be used to construct more examples of smooth solutions to inverse mean curvature vector flow from smooth solutions to inverse mean curvature flow in a spacelike hypersurface.