CAREER: Existence and Regularity of Solutions to Variational Problems in Geometric Analysis

Brown University, Providence RI

Investigators

Abstract

This project studies optimization questions in geometric analysis, namely constructs that optimize energy or area subject to a constraint. Existence and structural results are of interest in areas such as engineering, physics, and chemistry. Classical examples are minimal surfaces, which locally minimize area subject to fixed boundary conditions, such as soap films supported by wires of various shapes. This project studies constant mean curvature (CMC) and minimal surfaces as well as harmonic maps. CMC surfaces optimize area, but with constraint given by enclosed volume -- CMC surfaces appear in nature as soap bubbles. Harmonic maps optimize energy rather than area and are closely related to minimal surfaces. The objects studied in this project have characterizations in many branches of mathematics; the questions and desired results are of broad interest in mathematics and beyond. This research project primarily studies CMC surfaces immersed in smooth manifolds and harmonic maps into metric spaces. In the work on harmonic maps, the project aims to provide a new direction for resolution of Cannon's conjecture. It is planned to establish the existence of a harmonic homeomorphism from the round unit sphere into a sphere with a metric possessing upper curvature bounds. In a second direction, the project aims to refine techniques that produced a compactness theory for harmonic maps into metric spaces with upper curvature bounds. While the techniques for proving compactness in this setting are necessarily geometric and variational (rather than analytic), the results are analogous to those that establish compactness in the smooth setting. Using the refined techniques, the investigator plans to establish a harmonic replacement argument using energy rather than modulus of continuity methods. Other research directions relate to the study of CMC surfaces. The investigator plans to extend and refine a gluing construction that produced CMC hypersurfaces in Euclidean space. The new construction is expected to produce non-rotational, toroidal drops in Euclidean space and will serve as a model for a subsequent construction to produce CMC tori in three-manifolds. 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.