Singularity, Rigidity, and Extremality Phenomena in Minimal Hypersurfaces

Riemannian geometry is a modern version of geometry that studies shapes in any number of dimensions. Other than "lengths" and "angles," its key notions also include "minimal surfaces," which generalize the concept of a straight line, and "curvature," which measures how a shape is bent. The principal investigator (PI) will study problems involving minimal surfaces and their curvature that arise from physical theories including Einstein’s general theory of relativity and the van der Waals–Cahn–Hilliard theory for phase transitions in multicomponent alloy systems. In addition to the research, this project will also support the PI's continued efforts to promote student learning and training through seminar organization, conferences, expository articles, and notes. This project will specifically examine singularity, rigidity, and extremality phenomena in the theory of minimal surfaces. First, the PI will further investigate the structure of minimal surface singularities, meaning points of curvature blow-up, in area-minimization problems as well as their dynamic counterpart in mean curvature flow. Second, the PI will study enhanced rigidity properties of critical points in the van der Waals–Cahn–Hilliard phase transition theory, which can be thought of as diffuse variants of minimal surfaces. Third, the PI will study extremal behaviors of different quasi-local mass notions in general relativity, as seen through their interactions with scalar curvature and minimal surfaces, which correspond to energy density and boundaries of black hole regions. 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.