CAREER: Rigidity and Boundary Phenomena in Geometric Variational Theory

Surfaces governed by their mean curvature model many physical phenomena, such as soap films, black hole horizons, capillary surfaces, and other interface behaviors. These topics, particularly minimal surfaces and mean curvature flow (MCF), are pivotal in geometric analysis and contribute to advances across mathematics, physics, and materials science. Under this project, the investigator will conduct a number of research projects concerning the theoretical construction and variational properties of minimal surfaces and mean curvature flow, including the emerging and physically relevant study of capillary action on the boundary. Additionally, the project will promote the development of a diverse and collaborative community through educational and outreach initiatives including undergraduate clubs, curriculum development, research workshops and mentorship programs to support early career researchers. This project includes three primary research directions. In the first subject, the investigator will study the singularity analysis of MCF near cylindrical and conical singularity models, in the presence of interior and boundary curvature. This subject revolves around uniqueness for blow-up analysis of MCF. In the second subjects, the investigator will continue their development of min-max constructions of hypersurfaces with prescribed mean curvature and boundary contact angle. The principle investigator will also discover the regularity and long-time behavior of MCF with capillary boundary action. In the third subject, the investigator will explore the complexity of submanifolds, quantified by area and entropy, and connections to the stability of minimal submanifolds. 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.