Award: DMS 1307953, Principal Investigator: Jacob Bernstein
The project will study properties of classical minimal surfaces - that is, surfaces in Euclidean three-space which are critical points for area. The focus will be on understanding global properties of interesting classes of minimal surfaces, e.g., compactness properties. This will be done by studying the geometric structure of elements of the class. This approach is motivated by far-reaching work of Colding and Minicozzi who characterized the structure of embedded minimal disks. Their work has been the basis for much recent progress in the field. The PI will attack these questions primarily by using methods originating in the theory of integrable systems - especially the geometric theory of the Korteweg-de Vries (KdV) equation. Specifically, the PI will develop preliminary work which suggests that many spaces of minimal surfaces possess dynamics which are modeled on the KdV equation. The connection between minimal surfaces and the KdV equation is best understood using language and ideas coming from projective geometry. As such, one over-arching goal of the project is to formalize these observations in the language of projective geometry and to understand the interaction between the dynamics and the geometry of the minimal surfaces. A near-term goal is to apply the techniques developed to spaces of free-boundary minimal surfaces. These are particularly amenable to the methods and so act as good model spaces. Motivated by work of Meeks, Perez and Ros on properly embedded minimal surfaces of genus-zero, a longer-term goal is to blend this perspective with more established methods in order to answer questions about embedded minimal surfaces of finite genus. Chief among these is the question, raised by Colding and Minicozzi, whether a embedded minimal surface of finite genus which is complete is necessarily properly embedded.
A minimal surface mathematically models the shape of a soap film spanning a fixed wire frame. This is because, roughly speaking, the "energy" of such a film is given by its surface area and so stable configurations are those with least area - i.e., minimal surfaces. As such, the theory of minimal surfaces directly connects to problems arising in physics, chemistry, biology and materials science. More broadly, minimal surfaces are an important model for many geometric variational problems - that is problems where one seeks to find and study the properties of geometric objects which are optimal in some sense. In addition to being a fundamental principle in the physical sciences, such variational problems arise in diverse areas of pure and applied mathematics. A specific goal of the project is to understand the relationship between minimal surfaces and the KdV equation - an equation important in fluid dynamics. In contrast to minimal surfaces, which are static, the KdV equation models a dynamic physical phenomena, namely the motion of certain one-dimensional water waves. The KdV equation possesses many remarkable mathematical properties and is important in other areas of physics, notably in string theory. The relationship between the KdV equation and the minimal surface equation is poorly understood and any insight the project may shed on this connection should have broad importance both in mathematics and in physics.
