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的梯度流。几何流动理论不仅是数学中的一门基础学科，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。几何流研究中最重要的问题之一是理解流的所有可能的奇点，而这些奇点又由流的自相似解来建模。对这种自相似解的模空间的研究有望拓宽和发展数学内外的知识和技术，如流形的拓扑、图像处理、晶体生长和宇宙的大尺度结构。与此同时，从微分几何到分析的各种数学学科都将开发新的思想和工具。此外，国际和平研究所将继续为本科生、研究生和青年研究人员提供指导并组织研讨会和讲习班。国际数学联合会还将积极参与促进妇女在数学领域的工作，以增强社会中的多样性和性别平等。这一拟议项目的主要目标是建立几何流动的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类型的技术来解决非紧致梯度Ricci孤子的刚性问题。其次，PI部分地受到极小曲面理论的启发，旨在更详细地描述平均曲率流的有限亏格的二维光滑非紧自缩子的图景。为了实现这一点，PI将从研究这种自缩器在无穷远处的渐近结构开始，这些自缩器被猜想为正则圆锥或柱面。然后，PI打算解决Ilmanen关于自缩圆柱体唯一性的圆柱体刚性猜想。最后，与约翰·霍普金斯大学的Joel Spruck一起，PI计划寻求自收缩方程渐近Dirichlet问题存在的充要条件。第三，在高维上，PI与剑桥大学的Neshan Wickramasekera合作，将稳定极小子流形的正则性理论推广到对熵稳定的弱自缩子流形奇异集的大小的估计。

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.