Minicozzi Mini-school Schedule
|Wedneday, April 12||
|Thursday, April 13||
William Minicozzi, Singularities in mean curvature flow:
Lecture 1: Geometric heat equations
The classical heat equation describes how a temperature distribution changes in time. Over time, the temperature spreads itself out more and more evenly and, as time goes to infinity, the temperature goes to a steady-state equilibrium. There are a number of geometric heat equations, where some geometric quantity evolves over time and – in the best case – approaches an equilibrium. A simple example is the curve shortening flow where a curve in the plane evolves to minimize its length, but other examples include the Ricci flow and the mean curvature flow. All of these flows behave like the classical heat equation for a short amount of time, but they are nonlinear and these nonlinearities dominate over longer time intervals leading to many new phenomena.
Lecture 2: Mean curvature flow
I will give an introduction to mean curvature flow (MCF) of hypersurfaces. MCF is a nonlinear heat equation where the hypersurface evolves to minimize its surface area and the major problem is to understand the possible singularities of the flow and the behavior of the flow near a singularity.
Lecture 3: Level set method for motion by mean curvature
Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to a degenerate elliptic nonlinear pde. A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. This is joint work with Toby Colding.
Paul Gallager, Asymptotics for Minimal Surfaces with Quadratic Area Growth
In 2005, Meeks and Wolf proved that a minimal surface in R^3 with the area growth of 2 planes and infinite symmetry group must either be a Scherk surface or a catenoid. They conjectured that the same should be true without the symmetry assumption. A first step towards this conjecture would be to prove that minimals surfaces in R^3 with quadratic area growth must have a unique tangent cone at infinity. We describe some recent progress towards this goal.
Jonathan Zhu, Moving-centre monotonicity for minimal submanifolds
We prove a new moving-centre monotonicity formula for minimal submanifolds of Euclidean space, a corollary of which is a sharp area bound for minimal submanifolds in a ball that pass through a prescribed point. This area bound had previously been proven by Brendle and Hung using a fixed-centre method and a carefully chosen vector field, resolving a conjecture of Alexander, Hoffman and Osserman. Our monotonicity formula provides a new perspective on this phenomenon.
Ao Sun, Carleman Estimate for Surface in Euclidean Space at Infinity
If a solution vanishes somewhere to high order, then it vanishes identically. This is called the unique continuation property. A classical approach to prove the unique continuation property of solutions to elliptic equation is Carleman estimate. In this talk, I will introduce some basic settings of unique continuation and Carleman estimate, and show how to generalize them to surface in Euclidean space. As a result, we will get some rigidity theorem in geometry.