Fully Nonlinear Equations and Minimal Submanifolds in Lagrangian Geometry

This project lies at the intersection of differential geometry and nonlinear partial differential equations. The differential equations under consideration have geometric motivation and content, and model natural problems such as the reconstruction of the shape of a geometric object based on knowledge of how that object curves in its ambient environment, or the motion of a geometric object which deforms in a manner determined by its extrinsic curvature. Other equations of interest describe the structure of geometric objects which locally minimize a suitable notion of area subject to predetermined constraints. Such objects arise classically in mathematical models for soap films. Prescribed curvature equations and the equations describing time-dependent curvature flow also find applications in mathematical physics, e.g., to the geometry of spacetime and to the deformation theory of elastic bodies. The project will generate research opportunities for graduate students and will facilitate the mentoring of graduate students and postdocs through interactive research seminars. In addition, the principal investigator will prepare publicly accessible educational materials through the writing of survey articles on the subject. Two important types of nonlinear geometric partial differential equations feature heavily in this project: Lagrangian mean curvature equations (and the associated flows), and the Hamiltonian stationary equation (along with other fourth order equations of a similar type). Lagrangian mean curvature equations feature in the existence theory for special Lagrangian submanifolds of Calabi-Yau manifolds, a central issue in mirror symmetry. The Hamiltonian stationary equation identifies critical points for the volume functional on Lagrangian submanifolds under Hamiltonian variations. A priori estimates are crucial for solving certain fully nonlinear equations and for determining fundamental properties of their solutions. Building on prior work in the complex Euclidean setting, the PI will investigate regularity and well-posedness of the variable Lagrangian phase function. In the Lagrangian context, variational problems for the volume functional lead to nonlinear equations of fourth order. A relevant challenge is to identify submanifolds that are minimal within a specific Hamiltonian isotopy class. In contrast with general minimal surfaces, the underlying constraints in this setting permit the existence of compact minimal submanifolds. From an analytic viewpoint, the maximum principle is no longer applicable. This project will develop new strategies for the existence theory for Hamiltonian stationary submanifolds. 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.