Geometric Variational Theory and Applications

The main topics of this research program are mathematical objects that can be thought of as soap films spanning a closed piece of wire and soap bubbles. By the least action principle, the surface area of a soap film or a soap bubble will minimize among all films spanning the wire or among all bubbles including a fixed volume. Mathematically, a soap film spanning a wire is called a minimal surface, and a soap bubble is called a surface of constant mean curvature (abbreviated as CMC). Their study has driven progress in both mathematics and physics, with wide-reaching impact across scientific disciplines. This project aims to advance Geometric Variational Theory --- the primary framework for proving the existence of these surfaces --- and apply it to several open problems in the field. In addition to its theoretical contributions, the project will support educational and mentoring activities that foster the development of young researchers in geometric analysis and related areas. In this project, the PI will investigate a family of problems related to minimal surfaces, constant mean curvature (CMC) surfaces, and their applications. The research will focus on three main directions: investigating the existence of closed minimal surfaces with controlled genus in three-manifolds, with the goal of resolving Yau’s Four Minimal Spheres Conjecture; developing min-max theory for minimal hypersurfaces in non-generic settings; and studying the existence of CMC hypersurfaces across various geometric contexts. 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.