Selected Topics in Geometric Analysis

在这项研究中的一个主要主题是平均曲率流，特别是拉格朗日曲面。沿着流动，表面积减小;因此，表面随着时间的推移朝向平衡--所谓的最小表面--演化，前提是流动一直平稳地存在并且最终收敛。横跨金属线的肥皂膜是最小的表面。极小曲面在几何学和物理学中具有根本的重要性--在广义相对论、弦理论和镜像对称中。 拉格朗日几何学起源于哈密顿力学。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，演化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，在文献中引入了孤子解，研究这些特殊的解决方案是该提案的重点之一。

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.