Mean Curvature Flow and Geometric Analysis

Continuing investigations on mean curvature flow and related areas of geometric analysis, including projects on the uniqueness of tangent cones for Einstein manifolds. The first broad area of the proposal centers on the study of singularities in MCF, including estimates for the size of the singular sets, a compactness theorem for the possible types of singularities, a classification of generic singularities of the flow, and a canonical neighborhoods theorem that describes the flow in a neighborhood of the generic singularities. These are the most important questions about singularities and the PI and his collaborator have already obtained significant results in this direction. The second main area of the proposal concerns the structure of Einstein manifolds and, in particular, the question of when an Einstein manifold has a unique asymptotic structure (or tangent cone at infinity). The main prior result in this direction is due to Cheeger and Tian in 1994, where they showed uniqueness under an integrability assumption that the PI would like to remove. These sort of uniqueness questions have played an important role in a number of areas of geometric analysis and are extremely important in understanding the structure of the singular set for limits of Einstein manifolds.

This project focuses on several geometric variational problems. The problems are mathematical, but many of them arose first in science and engineering. Perhaps the most natural geometrically are minimal surfaces that locally minimize their surface area and, thus, model soap films in perfect equilibrium (so the film does not change other time). These have been studied at least since Lagrange's 1762 memoir, but recent years have seen breakthroughs on many long-standing problems in the theory of minimal surfaces, with important contributions from many mathematicians. There is a time-varying analog of this where a surface (which is not in equilibrium) evolves to minimize its surface area as quickly as possible; this is called mean curvature flow or MCF. Mathematically, this leads to a nonlinear partial differential equation which is formally similar to the equation that governs the flow of heat in physics. Clearly, minimal surfaces remain static under the MCF. MCF and other geometric flows were developed for their intrinsic beauty as well as their potential applications to other fields to model, for instance, option pricing, motion of grains in annealing metals, and crystal growth. While key foundational results have been obtained, several of the most basic questions remain unanswered. In contrast, the Einstein equation is a nonlinear differential equation for the curvature of a space (or a space-time in general relativity). Hilbert realized a century ago that this comes up variationally as the Euler-Lagrange equation for the Einstein-Hilbert functional.