Geometric Variational Theory and Application

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The main topics of this research program are mathematical models for soap films spanning a closed 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. The physical characterization of such surfaces says that the soap film and the soap bubble are respectively critical points of area or area subtracting enclosed 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). These two types of surfaces already caught interests by physicists and mathematicians in 1760s, and have been extensively-studied topics which also inspired the advances of many other subjects in mathematics and science. One general theory for proving the existence of such geometric objects, called the "min-max theory", has had striking recent successes, and will be the major object of study in this research program. More specifically, the PI will conduct research on the min-max theory and its applications for CMC surfaces and minimal surfaces with free boundary. In the first subject, the PI intends to prove topological bounds for the min-max CMC surfaces by Heegaard genus in a given three manifold, as well as to prove the existence of multiple CMC surfaces for a given mean curvature. The PI also plans to establish the min-max theory for constructing surfaces with mean curvature prescribed by an arbitrary smooth function, generalizing that of the CMC surfaces where the curvature functions are constants. In the second subject, the PI will investigate the compactness property, properness property, and Morse index upper bounds of the free boundary min-max minimal surfaces obtained by the PI with collaborators before; as applications, the PI plans to study minimal surfaces in singular or non-compact spaces by approximations using the free boundary solutions. The PI will finish a program on constructing min-max minimal disks with free boundary using the theory of harmonic maps. 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.