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。几何流理论不仅是数学中的一个基础学科，而且在计算机科学、材料科学和物理学等其他科学领域也有潜在的应用。几何流动研究中最重要的问题之一是理解流动的所有可能的奇异性，而这些奇异性又由流动的自相似解来模拟。对这种自相似解的模空间的拟议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，新的思想和工具将在从微分几何到分析的各种数学学科中发展。此外，PI将继续指导和组织本科生，研究生和年轻研究人员的研讨会和讲习班。PI还将积极参与促进妇女在数学方面的发展，以加强社会的多样性和性别平等。该拟议项目的主要目标是建立几何流自相似解空间的各种几何和分析性质。首先，PI与亚利桑那州立大学的Brett Kotschwar继续合作，将应用Carleman型技术来解决非紧梯度Ricci孤子的刚性问题。其次，呼吁工具的启发，部分理论的极小曲面，PI的目的是描述一个更详细的图片二维光滑非紧自收缩的有限属平均曲率流。为了实现这一点，PI将开始与调查的渐近结构在无穷远的这种自收缩体，这是假定为正常的圆锥或圆柱体。然后，PI打算解决的圆柱体刚度猜想的Ilmanen有关的唯一性的自收缩圆柱。最后，PI与约翰霍普金斯大学的Joel Spruck计划寻求自收缩方程渐近Dirichlet问题存在的充分和必要条件。第三，在更高的维度，PI与剑桥大学的Neshan Wickramasekera合作，将扩展稳定极小子流形的正则性理论，以获得熵稳定弱自收缩奇异集的大小估计。

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.