Nonlinear Partial Differential Equations in Geometry and General Relativity

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This proposal presents a broad array of projects within three distinctareas of geometric analysis. The first area concerns the application ofnonlinear gluing techniques to the Cauchy problem in General Relativity.Nonlinear gluing techniques have played a central role in geometricanalysis over the last 20 years. It is only recently that they have beenintroduced as a useful tool in General Relativity. Applications haveincluded the existence of asymptotically flat vacuum spacetimes witharbitrary spacial topology and those with no maximal slices. The PI willextend and apply these powerful tools in a number of important ways. Thesecond set of projects studies the existence and behavior of smoothSchr\"odinger maps. From a geometric point of view the Schr\"odinger flowis the most natural dispersive equation as it arises as the Hamiltonianflow of the Dirichlet energy for maps between manifolds. This work willestablish a bridge between the well-developed theory of dispersiveequations and important techniques from geometric analysis. The third areais 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 ofall such surfaces with a fixed topology was previously worked out by thePI and his co-authors. The development and application of new gluingtechniques for CMC surfaces has led to important advances in our abilityto describe the global structure of these moduli spaces. The PI willcarry this work significantly forward so that we may begin to obtain amore thorough understanding of these basic geometric objects. General Relativity is the physical theory which forms the cornerstone toour understanding of the large scale structure of the Universe, and hasbeen intimately intertwined with differential geometry and partialdifferential equations since its inception at the beginning of the 20thcentury. As with any physical theory, its dynamical formulation (theCauchy problem) is of central concern. The PI's research on the Cauchyproblem will develop and apply new analytic techniques to GeneralRelativity. The resolution of these problems will bring us further in thelong term goal of understanding how to model physical phenomena via themathematics of the initial data sets. Analytically, the Schroedinger flowis a generalization of the classical nonlinear Schroedinger equation,which has been intensively studied. The proposed program concerning theSchroedinger flow will foster the development of a new area of geometricdispersive systems. Surfaces of constant mean curvature arise naturallyas the surfaces which locally minimize their surface area whilemaintaining a fixed enclosed volume. Soap bubbles form a familiar andimportant class of examples of surfaces of constant mean curvature. Thesignificance of these research projects lies both in the importance of theparticular results which the PI will obtain and also in the continualdevelopment of sophisticated techniques which enable one to approach andunderstand problems which exhibit increasingly complex phenomena. Much ofthis research is interdisciplinary both between distinct areas withinMathematics and between Mathematics and Physics. While important resultshave already been obtained, there is great potential for expansion inthese areas.