Lu Wang - US grants

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.

Geometric flows have many real-world applications including material sciences, biology and image processing. Mathematically they are parabolic partial differential equations that deform geometric objects to their optimal shapes. In addition to their importance in geometric analysis, they also have potential applications to other mathematical disciplines, such as mathematical physics and low-dimensional topology. This award supports the investigation of two fundamental examples of geometric flows, mean curvature flow and Ricci flow. The PI will develop new ideas and robust techniques that will benefit the study of other geometric partial differential equations and related applications. In addition, the PI will place a strong emphasis on education in differential geometry and related topics through teaching, supervising undergraduate, graduate students and young scholars, and organizing seminars and conferences. The PI will also play an important role in the promotion of women and other underrepresented groups in STEM to enhance diversity and equity in the society.

The first part of the project is on the properties of closed hypersurfaces with low entropy. It involves an exploration of global features of the moduli space of asymptotically conical self-expanders of mean curvature flow. An overarching goal is to verify the smooth four-dimensional Schoenflies conjecture for hypersurfaces with low entropy. The second part concerns the variational construction of new examples of asymptotically conical self-expanders. The third part probes the asymptotic structure of soliton solutions to mean curvature flow as well as Ricci flow. The PI aims to show the geometry of these soliton solutions under mild topological restrictions is bounded in various senses.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Mean curvature flow is a process that evolves hypersurfaces in an ambient space so that the area of the hypersurfaces decreases in the steepest direction. A minimal hypersurface is a hypersurface that locally minimizes the area, and it is a stationary solution to mean curvature flow. In addition to being beautiful subjects in themselves, mean curvature flow and minimal hypersurfaces arise as simplified models for various physical processes that involve surface tension, and they could be applied to questions in other scientific fields, such as materials science and computer vision. The project aims to exploit suitable notions of entropy to study global features of mean curvature flow and minimal hypersurfaces. Significant educational activities that are integrated into the project include: mentoring undergraduate and graduate students and postdocs on some questions in the project; recruiting women and other underrepresented groups; teaching mini-courses to attract young students to the field; and organizing seminars, workshops and research programs promoting young scholars.

The project has three parts. The first concerns quantitative understanding of resolutions of conical singularities of mean curvature flow and its application to the higher homotopy group of the space of closed hypersurfaces in Euclidean space of low entropy. The second is to develop a Morse theory for self-expanding solutions to mean curvature flow. The last is to study geometric and topological properties of minimal hypersurfaces in sphere and hyperbolic space.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.