CAREER:New Development in Geometric Variational Theory

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Minimal surfaces, Constant Mean Curvature (CMC) surfaces, and Prescribed Mean Curvature (PMC) surfaces are mathematical models of soap films, soap bubbles, and capillary surfaces. These types of surfaces have always been important topics in geometry and general relativity, and have also inspired advances of many other subjects in mathematics and science. Geometric Variational Theory is the major method for proving the existence of these types of surfaces. In this research program, the PI will conduct a number of research projects on the existence of minimal, CMC, and PMC surfaces by further advancing Geometric Variational Theory. This research program also includes support for educational activities. The PI will develop new curricula for graduate research topic courses, recruit and mentor Ph. D. students and postdocs, and direct advanced undergraduate students for honors theses. The PI will also organize a summer workshop for graduate students and junior postdocs on topics related to this research program. The goal is to promote early career researchers and encourage collaborations.In the first subject, the PI will explore important applications of the recent resolution of the Multiplicity One Conjecture. In particular, the PI will investigate new ergodic properties of minimal surfaces, and the Multiplicity One Conjecture in the Simon-Smith setting and the free boundary setting. In the second subject, the PI will continue the research on Geometric Variational Theory with Lagrange multipliers. The PI anticipates to prove topological bounds for the min-max CMC surfaces in three manifolds, as well as to prove the existence of multiple CMC surfaces. PI also intends to establish the general existence theory for capillary surfaces. In the last subject, the PI will investigate existence problems and applications for minimal surfaces in singular and noncompact spaces via approximations using the free boundary min-max theory.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.