Mean curvature flow: singularity formation beyond 2 convexity and applications

Introduction

Geometric flows as a means to attack problems in topology, geometry, and physics have been demonstrated to be an extremely powerful tool. The most successful such flow to date is the Ricci flow (RF), which was used in the proof of the geometrization conjecture.

Mean Curvature Flow

The mean curvature flow (MCF) - the most natural geometric flow for sub-manifolds in an ambient space - has also been successfully applied to address such problems. Nevertheless, the most striking potential applications of MCF are still out of reach.

Research Goals

The goal of the proposed research is to advance the understanding of the formation of singularities in MCF and to study a particular application of MCF for general relativity. More concretely, we propose to:

1. Continue the systematic study of the formation of bubble-sheet singularities in 4-space, initiated by Choi, Haslhofer, and the PI, with the goal of obtaining a mean convex neighbourhood theorem in this setting.
2. Study the formation of singularities more generally, and in particular, the structure of the singular set.

Second Objective

The second objective of the proposed research is to employ MCF in Lorentzian spacetime satisfying the Einstein equation with a positive cosmological constant to obtain versions of the cosmic no-hair conjecture, namely, geometric convergence results to de Sitter space.