# Mini-Courses

Summer school schedule

Otis Chodosh (Princeton University)

Title: Geometric features of the Allen-Cahn equation

Abstract: The Allen-Cahn equation is a semilinear elliptic PDE with deep links with the theory of minimal surfaces. I will discuss several aspects of the theory including existence results and the convergence theory. I’ll assume basic facts about Riemannian geometry (e.g. the Laplacian on Riemannian manifolds, basic notions of curvature, etc) as well as some background in basic elliptic PDE (Sobolev spaces, Schauder theory), but the course should be understandable even without these prerequisites.

From there, you can read the following papers:

Ailana Fraser (University of British Columbia)

Title: Minimal surfaces and extremal eigenvalue problems

Abstract: When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this course we will discuss recent progress that has been made for surfaces with boundary, both on the eigenvalue problem and the minimal surfaces.

Exercises

Yng-Ing Lee (National Taiwan University)

Title: An Introduction to Lagrangian Mean Curvature Flow

Abstract: If the initial data of mean curvature flow in a Kahler-Einstein manifold is a Lagrangian submanifold, the Lagrangian condition will be preserved whenever the solution is smooth. It is thus called Lagrangian mean curvature flow and is a potential method to construct special Lagrangians that play an important role in string theory.

A Lagrangian submanifold is of middle dimension. Mean curvature flow in high codimension is more complicated because it involves system of nonlinear equations instead of one single scalar equation, and the concept of convexity is not available. We will discuss the study and results on Lagrangian mean curvature flow.

Lecture 1
Lecture 2
Lecture 3 (upcoming)

Two survey papers:

From there, you can read the following papers:

Richard Schoen (UC Irvine)

Title: Geometry and general relativity

Abstract: We will introduce the Einstein equations, the constraint equations, and the initial value problem. We will then discuss the gravitational mass and the positive mass theorem. We will focus on positivity proofs.