Minimal submanifolds in Arbitrary Geometries as Nodal sEts: Towards hIgher Codimension

Introduction

The goal of this research proposal is to advance the calculus of variations of area in codimension higher than one, specifically existence and regularity of its critical points (minimal submanifolds) and properties of its gradient flow (mean curvature flow).

Importance of the Topic

These are central objects in mathematics since three centuries and contributed to the birth of geometric analysis, geometric measure theory, and calculus of variations. Their (non-)existence often reveals deep links between small-scale geometry (curvature) and large-scale structure (topology).

Current Understanding

While the hypersurface case is by now well understood, with several deep results in the last two decades, very little is known in codimension at least two, especially for unstable submanifolds not minimizing area.

Proposed Projects

Several projects will focus on the intimate link between area and some well-known physical energies:

1. Phase transitions are understood to give diffuse approximations of hypersurfaces.
2. Vortices in models of superconductivity relate to codimension two submanifolds.

An energy proposed by me and D. Stern in this context is the abelian Higgs model, which I plan to use to extend the Lagrangian mean curvature flow past singularities and to relate stability and regularity of minimal submanifolds. These are two long-standing questions in geometric analysis (among other projects), by exploiting the much richer structure given by the PDEs solved by critical points of this energy.

Further Exploration

I will also look at candidates in codimension three and higher, inspired by energies from gauge theory and others of Ginzburg–Landau type, relating stability and minimality in critical dimension and attacking other basic open questions.

Additional Research Directions

Finally, I will also work on another set of projects exploiting parametrized varifolds, a variational object pioneered by me and T. Rivière. This approach combines advantages of the parametrized and intrinsic viewpoints to study Lagrangian surfaces and minimal submanifolds of higher dimension.