Mean curvature flow: singularity formation beyond 2 convexity and applications

Geometric flows as a mean to attack problems in topology, geometry and physics, had 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.

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

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 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.

We also propose to study the formation of singularities more generally, and in particular, the structure of the singular set.

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