Differential Geometry and Minimal Surfaces

A classical problem in mathematics is to encode a complicated structure by a collection of numerical measurements (the data) and then reconstruct the unknown structure from those measurements. Such problems are important because they provide mathematical models for applications such as medical imaging and geophysical exploration. One well-understood case is when the unknown structure is the geometry of a surface, and the data are the lengths of its closed curves of minimal length, called geodesics. This project aims to study analogous questions in the more general setting where the unknown structure is the geometry of more general spaces called manifolds, and the data are the areas of surfaces in equilibrium within it. In quantum information science, there is interest in reconstructing a bulk metric from measurements of entanglement entropy, a problem closely related to part of this project. The project includes an active mentoring component, involving graduate students and postdoctoral researchers. In addition, the PI will continue organizing conferences, serving on editorial boards, and contributing to the advancement of the field in the broader mathematical community. The project aims to make progress on three problems. The first is to determine the ambient metric on a 4-manifold from its area spectrum. The second is to find infinitely many unstable minimal surfaces in 4-manifolds, and the third is to investigate the existence of unstable minimal laminations in gaps between stable minimal foliations. The approach used will combine variational methods with tools from integral geometry and the theory of harmonic maps. In addition, techniques from Aubry–Mather theory will be adapted to the setting of minimal hypersurfaces. 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.