Nonlinear Partial Differential Equations in Geometry and General Relativity

University Of Washington, Seattle WA

Investigators

Abstract

This proposal presents a broad array of projects within three distinct areas of geometric analysis. The first area concerns the application of nonlinear gluing techniques to the Cauchy problem in General Relativity. Nonlinear gluing techniques have played a central role in geometric analysis over the last 20 years. It is only recently that they have been introduced as a useful tool in General Relativity. Applications have included the existence of asymptotically flat vacuum spacetimes with arbitrary spacial topology and those with no maximal slices. The PI will extend and apply these powerful tools in a number of important ways. The second set of projects studies the existence and behavior of smooth Schr\"odinger maps. From a geometric point of view the Schr\"odinger flow is the most natural dispersive equation as it arises as the Hamiltonian flow of the Dirichlet energy for maps between manifolds. This work will establish a bridge between the well-developed theory of dispersive equations and important techniques from geometric analysis. The third area is a continuation of the PI's work on surfaces of constant mean curvature (CMC) in Euclidean 3-space. The local structure of the moduli space of all such surfaces with a fixed topology was previously worked out by the PI and his co-authors. The development and application of new gluing techniques for CMC surfaces has led to important advances in our ability to describe the global structure of these moduli spaces. The PI will carry this work significantly forward so that we may begin to obtain a more thorough understanding of these basic geometric objects. General Relativity is the physical theory which forms the cornerstone to our understanding of the large scale structure of the Universe, and has been intimately intertwined with differential geometry and partial differential equations since its inception at the beginning of the 20th century. As with any physical theory, its dynamical formulation (the Cauchy problem) is of central concern. The PI's research on the Cauchy problem will develop and apply new analytic techniques to General Relativity. The resolution of these problems will bring us further in the long term goal of understanding how to model physical phenomena via the mathematics of the initial data sets. Analytically, the Schroedinger flow is a generalization of the classical nonlinear Schroedinger equation, which has been intensively studied. The proposed program concerning the Schroedinger flow will foster the development of a new area of geometric dispersive systems. Surfaces of constant mean curvature arise naturally as the surfaces which locally minimize their surface area while maintaining a fixed enclosed volume. Soap bubbles form a familiar and important class of examples of surfaces of constant mean curvature. The significance of these research projects lies both in the importance of the particular results which the PI will obtain and also in the continual development of sophisticated techniques which enable one to approach and understand problems which exhibit increasingly complex phenomena. Much of this research is interdisciplinary both between distinct areas within Mathematics and between Mathematics and Physics. While important results have already been obtained, there is great potential for expansion in these areas.