Existence and Uniqueness Questions in Differential Geometry

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AbstractAward: DMS 1405537, Principal Investigator: Nicolaos Kapouleas Differential Geometry is an important field in Mathematics closely related to other fields of Mathematics and Physics, for example Topology, nonlinear Partial Differential Equations, and General Relativity. In order to understand and develop the theory in Differential Geometry it is necessary to have a sufficient supply of examples of the geometric objects under consideration. Finding such examples is usually a difficult problem. A method which has been very successful because of its generality and flexibility is the gluing methodology for constructing solutions to partial differential equations, using singular perturbations and known solutions on part of the domain of a problem. Although enormous progress has been made in this direction already, the subject seems to have a long way to go because there are many fundamental questions which are still completely open. Further progress requires much more development of the methodology and application to new questions. The principal investigator plans to continue developing the subject further as discussed next.First steps in this agenda will concentrate on advancing the principal investigator's program for desingularization and doubling constructions for minimal surfaces in Riemannian manifolds. Other gluing constructions will be pursued with various collaborators: Constructions for Einstein manifolds and Ricci solitons in four dimensions with Simon Brendle and on some projects also with Frederick Fong. Constructions for Constant Mean Curvature hypersurfaces with Christine Breiner. Constructions of special Lagrangian cones with Mark Haskins. On free boundary problems for minimal surfaces in the unit ball with Martin Li. Constructions for coassociative four-manifolds with Jason Lotay. Constructions for self-shrinkers for the Mean Curvature flow with Stephen Kleene, Niels Moller, and David Wiygul. He also plans to pursue some more collaborations on projects related to doubling and desingularization constructions with Christine Breiner, Jacob Bernstein, Stephen Kleene, F. Martin, W. Meeks, Niels Moller, and David Wiygul. Finally, the principal investigator plans to work, alone or with his collaborators, on some uniqueness/characterization/nonexistence questions related to or motivated by the above constructions.