The local and global structure of variational solutions

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The projects in this proposal address classical questions in geometric analysis related to the existence, structure, and compactness of solutions to particular variational problems. In a previous work with N. Kapouleas, the PI developed a gluing construction to produce new examples of embedded constant mean curvature surfaces with finite topology in Euclidean three space. The PI expects that appropriate modifications to the linear problem will allow her to extend their techniques to produce infinitely many new embedded, constant mean curvature hypersurfaces. As a second project, the PI will consider compactness theory for extrinsic biharmonic maps. For sequences of harmonic maps, energy quantization and regularity theory play a key role in establishing the compactness result. The PI intends to show, using comparable results for the biharmonic setting, that the limiting biharmonic map and the maps arising from dilations at concentration points are connected in the image manifold without necks. Finally, the PI will attempt to combine and extend her study of the structure of complete, embedded minimal surfaces with finite topology and the structure of annular minimal surfaces with boundary to understand the structure of complete minimal surfaces with infinite topology. Solutions to variational problems are critical points for an energy functional and as such are deeply connected with physical phenomenon. As a particular example, minimal surfaces are critical points for area and on sufficiently small scales are area minimizers, with soap films occurring as natural models. The objects and questions considered in this project can be studied through the lens of partial differential equations, analysis, calculus of variations, and submanifold geometry and are of interest to pure and applied mathematicians as well as many physicists.