The first main objective of this project is to expand the use of gluing methodology to the greatest possible extent in understanding important existence questions in the theory of minimal surfaces. Another objective (in the opposite direction) is to understand related non-existence, characterization, and uniqueness questions. Achieving these objectives requires the refinement of the known methodology and also the development of entirely new methods. In the first project of this proposal, the PI aims to study in collaboration with Mark Haskins important special Lagrangian and other calibrated submanifolds. In particular they intend to continue to study special Lagrangian cones in $mathbb{C}^n$ controlled by ODE systems and related in various ways to the Lawlor necks, some of them invariant under the action of $SO(p) imes SO(n-p)$; their use as building blocks for gluing constructions for new special Lagrangian cones; and uniqueness questions for these objects (for which the known theory seems inadequate because of the high codimension). In other projects, the PI, alone or in collaboration, intends to continue his work on generalizing his earlier desingularization and doubling constructions for minimal surfaces in three-manifolds to the greatest possible extent, and apply these constructions to fundamental questions in the theory of minimal surfaces, for example to a question of Yau about the existence of infinitely many minimal surfaces in any Riemannian three-manifold. The PI also intends to study existence and classification questions for minimal surfaces in the round three-sphere, including characterizations of a topological nature for the Lawson surfaces. In a collaboration with F. Martin and W. Meeks, the PI intends to work on desingularization constructions where there are triple points of intersection so that they can be used to understand the Calabi-Yau problem for minimal surfaces in the embedded case. In collaboration with Stephen Kleene and Niels Moller, the PI intends to work on existence questions for self-shrinkers of the mean curvature flow. In collaboration with Christine Breiner, the PI intends to expand his earlier work on gluing constructions for constant mean curvature surfaces.
Minimal and constant mean curvature surfaces have historically been an important field where many important ideas were first developed, and later applied to nonlinear Partial Differential Equations, General Relativity, Einstein manifolds, and other fields. This is not surprising because in some sense the theory combines important features of all these fields while it is at the same time the simplest and most intuitive. Although enormous progress has been made, there are many fundamental questions which are completely open, mostly because the known methodologies are inadequate. Successful completion of these pending projects would answer many important such questions and would be the basis for progress in the related fields as well.
