NSF Award
1609198
This project concerns optimal objects for their respective energy functionals, and as such existence and structural results are of interest in engineering, physics, and chemistry. The most classically studied of these are minimal surfaces, which locally minimize area subject to a fixed boundary. Of particular interest in this project are so-called constant mean curvature (CMC) and minimal surfaces as well as harmonic maps. CMC surfaces are also critical for the area functional, but with constraint now given by enclosed volume. Delaunay determined a family of CMC examples in 1841, but it was another 150 years before any new examples were known, at which time Kapouleas produced infinitely many new examples via gluing techniques. The variational solutions studied in this project have characterizations in many areas of mathematics and the proposed questions and desired results are of broad interest in mathematics and beyond.<br/><br/>The PI will continue her study of classical questions in geometric analysis related to the existence, regularity, and compactness of solutions to variational problems. The project will use and refine the gluing techniques pioneered by Kapouleas to produce new examples of minimal and CMC surfaces. The understanding of singularity development for a sequence of complete, properly embedded minimal disks, developed by Colding-Minicozzi, was of critical importance for the resolution of the uniqueness of the helicoid. In contrast to the picture developed when the disks are complete and proper, the structure of the singular set for sequences of embedded minimal disks with boundary in a ball can be pathological. These pathological examples are helpful in the resolution of uniqueness and regularity results. Gluing techniques will be used to produce even wilder singularities in settings where problems are intractable via former techniques. For CMC gluing, the project aims to extend the generalized gluing techniques developed in Euclidean space to more general manifolds. In the setting of harmonic maps, the aim of the project is to establish the existence of conformal harmonic maps into metric spaces with upper curvature bounds. This work generalizes a classical result of Sacks and Uhlenbeck on the existence of minimal 2-spheres. Existence of the established maps could help answer the unresolved portions of Thurston's Hyperbolization Conjecture.
