Selected Topics in Geometric Analysis

本研究的一个主要主题是平均曲率流，特别是拉格朗日曲面。沿流动方向，比表面积减小；因此，表面随着时间的推移而趋于平衡——即所谓的最小表面，前提是水流一直平稳存在并最终收敛。横跨金属线的肥皂膜是最小的表面。最小曲面在几何学和物理学中——在广义相对论、弦理论和镜像对称中——具有基本的重要性。拉格朗日几何植根于哈密顿力学。平均曲率流为构造和理解拉格朗日极小曲面提供了一种很有前途的方法。然而，像其他几何热流一样，平均曲率流通常在有限时间内变得不规则；换句话说，不断演化的曲面变得越来越单一。因此，研究这些奇异点附近的流动特性是至关重要的。为此，文献中已经引入了孤子解，研究这些特殊解是本文的重点之一。

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.