Differential Geometry and Minimal Surfaces

Minimal surfaces are shapes which are in an equilibrium position. They are physical objects which serve to model black holes, soap films, or polymers in material sciences. Minimal surfaces are studied from a purely mathematical point of view and play a central role in geometry. The overall goal of this project is to study how rigid or flexible they are in a given space. The analogous problem for geodesics has had a tremendous impact not only in mathematics but also in applied fields of science. Broader impacts of this project include work with students and postdoctoral researchers. These projects will deepen the relation between min-max methods and the existence theory of metrics with maximal families of minimal surfaces. Continuing efforts will study of the area-growth of minimal surfaces in negatively curved spaces. More precisely, we plan to study to which extent the volume spectrum of a Riemannian metric characterizes the metric and to relate the area-growth of minimal surfaces with classical quantities such as versions of entropy in dynamical systems. 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.