Self-similar Solutions of Geometric Flows

几何流是与具有几何解释的流形上的泛函相关联的梯度流。几何流动理论不仅是数学的基础学科，而且在其他科学领域也有潜在的应用，包括计算机科学、材料科学和物理学。几何流动研究中最重要的问题之一是了解流动的所有可能的奇点，这些奇点反过来又由流动的自相似解来建模。对这种自相似解的模空间的研究有望拓宽和推进数学内外的知识和技术，如流形拓扑、图像处理、晶体生长和宇宙的大尺度结构。与此同时，从微分几何到分析，各种数学学科将发展出新的思想和工具。此外，PI将继续为本科生、研究生和年轻研究人员提供指导和组织研讨会和讲习班。PI还将积极参与促进女性在数学领域的发展，以促进社会的多样性和性别平等。本课题的主要目的是建立几何流的自相似解空间的各种几何和解析性质。首先，PI将继续与亚利桑那州立大学的Brett Kotschwar合作，应用Carleman型技术来解决非紧化梯度Ricci孤子的刚性问题。其次，利用部分受最小曲面理论启发的工具，PI旨在描述平均曲率有限属二维光滑非紧致自收缩体的详细图像。为了实现这一目标，PI将从研究这种自收缩器在无穷远处的渐近结构开始，这些自收缩器被推测为规则的锥体或圆柱体。在此基础上，提出了Ilmanen关于自缩圆柱的唯一性的圆柱刚度猜想。最后，PI计划与约翰霍普金斯大学的Joel Spruck一起，寻求自收缩方程的渐近Dirichlet问题存在的充分必要条件。第三，在更高的维度上，PI将与剑桥大学的Neshan Wickramasekera合作，扩展稳定最小子流形的正则性理论，以导出对熵稳定弱自收缩体的奇异集大小的估计。

A geometric flow is the gradient flow associated to a functional on a manifold with a geometric interpretation. Not only is the theory of geometric flows a fundamental subject in mathematics, but it also has potential applications to other scientific fields including computer sciences, material sciences and physics. One of the most important problems in the study of geometric flows is to understand all possible singularities of the flows, which are in turn modeled by self-similar solutions of the flows. The proposed research on the moduli space of such self-similar solutions is expected to broaden and advance the knowledge and techniques both within and outside of mathematics, such as the topology of manifolds, image processing, crystal growths and the large-scale structure of the universe. In the meanwhile, new ideas and tools will be developed in various mathematical disciplines ranging from differential geometry to analysis. In addition, the PI will continue mentoring and organizing seminars and workshops for undergraduates, graduate students and young researchers. The PI will also actively participate in the promotion of women in mathematics to enhance diversity and gender equity in the society.The main objective of this proposed project is to establish various geometric and analytic properties of the space of self-similar solutions of geometric flows. First, the PI, in the continuing collaboration with Brett Kotschwar at the Arizona State University, will apply the Carleman type technique to attack the rigidity problem for noncompact gradient Ricci solitons. Second, appealing to the tools inspired in part by the theory of minimal surfaces, the PI aims to describe a much detailed picture of two-dimensional smooth noncompact self-shrinkers of finite genus of mean curvature flow. To achieve this, the PI will begin with investigating the asymptotic structures at infinity of such self-shrinkers which are conjectured to be regular cones or cylinders. Then the PI intends to address the Cylinder Rigidity Conjecture of Ilmanen concerning the uniqueness of self-shrinking cylinders. At the end, the PI, with Joel Spruck at the Johns Hopkins University, plans to seek the sufficient and necessary conditions of the existence of the asymptotic Dirichlet problem for the self-shrinker equation. Third, in higher dimensions, the PI, joint with Neshan Wickramasekera at the University of Cambridge, will extend the regularity theory of stable minimal submanifolds to derive estimates on the size of singular sets of entropy stable weak self-shrinkers.