Mathematics Colloquium
Liam Mazurowski
Lehigh University
Title: Prescribed Mean Curvature Surfaces in Riemannian Manifolds
Date: Friday, January 16
Place and Time: Love 101, 3:05-3:55 pm
Abstract. Prescribed mean curvature surfaces arise as critical points of functionals involving the area of a surface and the volume it encloses. They model equilibrium interfaces in physical settings and appear naturally in general relativity, capillarity, materials science, and related areas. While stable examples can often be found via direct minimization, many geometrically interesting surfaces are inherently unstable and require different techniques to detect. In this talk, I will describe some new min-max approaches for finding prescribed mean curvature surfaces that go beyond the classical setting of smooth, compact ambient manifolds. These methods apply in situations where additional challenges arise such as the presence of constraints, singularities, or a lack of compactness. I will explain how these methods lead to new existence results for prescribed mean curvature surfaces, and discuss several open problems motivated by these developments.