Variational Problems In The Theory of Minimal Surfaces

A submanifold is called minimal if it is a critical point of the area functional. Minimal submanifolds are of central importance in differential geometry and arise naturally in mathematical physics, as soap films and black hole horizons, for example. Therefore, understanding their behavior is of great interest from the mathematical point of view but also for applications. The objective of this project is to take steps towards a full description of all minimal submanifolds in a given ambient manifold, inspired by the variational nature of these objects. The investigator will also conduct educational activities and practice community building, with particular attention to students and junior researchers. The project consists of three interwoven research lines. The first seeks new insights into the topological and analytical properties of minimal surfaces obtained via min-max constructions. The second line focuses on minimal surfaces with free boundary in the three-dimensional ball, with a focus on existence theorems and global properties. Finally, the project will investigate rigidity results for minimal submanifolds of higher codimension in ambient manifolds with positive curvature. 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.