Self-similar Solutions of Geometric Flows

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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.