Selected Topics in Geometric Analysis

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A major theme in this study is the mean curvature flow, especially for Lagrangian surfaces. Along the flow, surface area decreases; so the surface evolves with time toward an equilibrium - a so called minimal surface, provided the flow exists smoothly for all time and eventually converges. A soap film spanning a metal wire is a minimal surface. Minimal surfaces are of fundamental importance in geometry and in physics - in general relativity, string theory, and mirror symmetry. Lagrangian geometry has its root in Hamiltonian mechanics. The mean curvature flow provides a promising method to construct and understand Lagrangian minimal surfaces. However, like other geometric heat flows, mean curvature flows typically become irregular in a finite time; in other words, the evolving surfaces become more and more singular. Therefore it is crucial to study the behavior of the flow near these singularities. Soliton solutions have been introduced in the literature for this purpose, and studying these special solutions is one focus of the proposal. ***We also propose to explore Willmore surfaces. These are equilibriums of the Willmore functional, or the so called "bending energy", which measures how a surfaces bends in the ambient space. It is worth noting that these surfaces have been used in biology to model red blood cells. The differential equation governing these surfaces is complicated - it is of 4th order and nonlinear. Not much is known about this equation analytically. A basic question is to determine whether a solution as a graph over the whole 2-dimensional plane is always a plane. This is the so called Bernstein type property. We initiated a study on this problem and have obtained very meaningful results. It is proposed to settle the following conjecture: every entire graphic Willmore surface is flat. ***We also intend to continue our study on quaternionic maps between hyperkahler manifolds. This special class of spaces has been found very useful in gauge theory, for example, the space of instantons carries the hyperkahler structure. The quaternionic maps were first considered by physicists. They are harmonic maps - equilibriums of the (Dirichlet) energy, and share interesting properties with holomorphic maps, such as being defined by a system of first order differential equations. The regularity theory for these maps is not fully understood yet. Our long term goal is to use these maps to define smooth invariants that allow us to classify the hyperkahler manifolds. ***In the proposal, specific questions and possible approaches in the above topics are discussed and the problems that junior researchers may work on are highlighted. **