Selected Topics in Geometric Analysis

这项研究的一个主要主题是平均曲率流，特别是对于拉格朗日曲面。沿着流动，表面积减小；因此，表面随着时间的推移而演变为一种平衡--即所谓的极小表面，前提是流动始终平稳存在并最终收敛。横跨金属线的肥皂膜是一个最小的表面。极小曲面在几何学和物理学中--在广义相对论、弦理论和镜像对称性中--都是非常重要的。拉格朗日几何起源于哈密顿力学。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，演变的表面变得越来越奇异。因此，研究这些奇点附近的流动行为是至关重要的。为此目的，文献中已经引入了孤子解，而研究这些特殊解是该提议的重点之一。 我们还建议探索Willmore表面。这些是Willmore泛函的平衡，或所谓的“弯曲能”，它测量表面在环境空间中的弯曲程度。值得注意的是，这些表面已经在生物学中被用来模拟红细胞。控制这些曲面的微分方程很复杂--它是四阶的、非线性的。从分析上讲，人们对这个方程式知之甚少。一个基本的问题是确定在整个2维平面上作为图形的解是否总是平面。这就是所谓的Bernstein型属性。我们开始了对这一问题的研究，并取得了非常有意义的结果。本文提出了如下猜想：每个完整的图域Willmore曲面都是平坦的。 我们还打算继续研究Hyperkahler流形之间的四元数映射。这类特殊的空间已经被发现在规范理论中非常有用，例如，瞬子空间带有超卡勒结构。四元数地图最初是由物理学家研究的。它们是调和映射-(Dirichlet)能量的平衡点，并且与全纯映射共享有趣的性质，例如由一阶微分方程组定义。这些映射的正则性理论还没有完全被理解。我们的长期目标是使用这些映射来定义光滑不变量，从而允许我们对Hyperkahler流形进行分类。 在提案中，讨论了上述专题中的具体问题和可能的方法，并强调了初级研究人员可能研究的问题。

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.