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.
- A good survey The role of minimal surfaces in the study of the Allen-Cahn equation by Frank Pacard
From there, you can read the following papers:
- Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory by John Hutchinson and Tonegawa
- Min–max for phase transitions and the existence of embedded minimal hypersurfaces by Marco Guaraco
- Finite Morse index implies finite ends by Kelei Wang and Juncheng Wei
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.
- Section 1 and Section 4 of give an overview of some of what we will discuss: Minimal surfaces and eigenvalue problems, by Ailana Fraser and Richard Schoen
- Uniqueness theorems for free boundary minimal disks in space forms, by Ailana Fraser and Richard Schoen
From there you can read the following papers:
- Sharp eigenvalue bounds and minimal surfaces in the ball, by Ailana Fraser and Richard Schoen
Index characterization for free boundary minimal surfaces, by Hung Tran
- A characterization of critical catenoid, by Peter McGrath
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.
Two survey papers:
- Recent Progress on Singularities of Lagrangian Mean Curvature Flow by André Neves
- Mean Curvature Flow in Higher Codimension: Introduction and Survey by Knut Smoczyk
From there, you can read the following papers:
- Self-similar solutions and translating solitons for Lagragian mean curvature flow by Dominic Joyce, Yng-Ing Lee, and Mao-Pei Tsui
- Construction of Lagrangian self-similar solutions to the mean curvature flow in C^n by Henri Anciaux
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.
- A good survey paper Scalar Curvature and the Einstein Constraint Equations by Daniel Pollack and Justin Corvino
- The proof of the positive mass theorem in lower dimensions can be found in the lecture notes Variational Theory for the Total Scalar Curvature Functional for Riemannian Metrics and Related Topics by Richard Schoen
- The proof of the positive mass theorem in general dimensions is in this recent paper Positive Scalar Curvature and Minimal Hypersurface by Richard Schoen and Shing-Tung Yau. More details are presented in the lecture notes Topics in Scalar Curvature of Richard Schoen (taken by Chao Li)
- If you get really excited about positivity of mass, you can read a spacetime version in The spacetime positive mass theorem in dimensions less than eight by Michael Eichmair, Lan-Hsuan Huang, Dan Lee, and Richard Schoen
Exercises prepared by Justin Corvino
Lu Wang (University of Wisconsin)
Title: Curve Shortening Flow
Abstract: We will discuss a classical theorem due to Grayson and Gage-Hamilton that is any simple closed curve in plane will be evolved by its curvature to a round point in finite time. If time permitted, we will also discuss some applications of this result.
- The heat equation shrinking convex plane curves by M. Gage and R. S. Hamilton
- The heat equation shrinks embedded plane curves to a round point by Matthew Grayson