Geometric Flows and Applications

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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.