Existence and Uniqueness Questions in Differential Geometry

摘要奖：DMS 1405537，首席研究员：Nicolaos Kapouleas微分几何是数学中的一个重要领域，与数学和物理的其他领域密切相关，如拓扑学、非线性偏微分方程组和广义相对论。为了理解和发展微分几何中的理论，有必要提供足够的所考虑的几何对象的例子。寻找这样的例子通常是一个困难的问题。一种因其通用性和灵活性而非常成功的方法是粘合方法，用于构造偏微分方程解，使用问题区域的一部分的奇异摄动和已知解。虽然在这方面已经取得了巨大的进展，但这一主题似乎还有很长的路要走，因为还有许多根本问题仍然完全悬而未决。要取得进一步的进展，需要更多地发展方法论并将其应用于新的问题。主要研究人员计划继续下一步讨论的主题。这一议程中的第一步将集中在推进主要研究人员的去奇化程序和加倍黎曼流形中极小曲面的构造。其他胶合结构将与不同的合作者一起进行：与西蒙·布伦德尔合作构建爱因斯坦流形和利玛窦四维孤子，以及与弗雷德里克·方合作进行一些项目。用Christine Bliner构造常平均曲率超曲面。用Mark Haskins构造特殊的拉格朗日锥。关于单位球中极小曲面的自由边界问题。与Jason Lotay共结合的四维流形的构造。与Stephen Kleene，Niels Moller和David Wiygul一起构造平均曲率流的自缩子。他还计划与克里斯汀·布赖纳、雅各布·伯恩斯坦、斯蒂芬·克莱恩、F·马丁、W·米克斯、尼尔斯·穆勒和大卫·维格尔在加倍和去单元化建设相关项目上进行更多合作。最后，首席研究人员计划单独或与他的合作者合作，解决与上述结构相关或受其激励的一些独特性/表征/不存在的问题。

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.