Existence and Uniqueness Questions in Differential Geometry

微分几何中的存在唯一性问题

• 批准号: 1405537
• 负责人: Nicolaos Kapouleas
• 金额: $ 22.79万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405537&HistoricalAwards=false
• 关键词: Existence; Uniqueness; Questions; Differential; Geometry

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.

AbstractAward：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，例如拓扑学，非线性偏微分方程和广义相对论。为了理解和发展微分几何的理论，有必要提供足够的几何对象的例子。找到这样的例子通常是一个困难的问题。由于其通用性和灵活性，一种非常成功的方法是胶合方法，用于构造偏微分方程的解，使用奇异摄动和已知的解的一部分问题的域。虽然在这方面已经取得了巨大的进展，但这个问题似乎还有很长的路要走，因为有许多根本问题仍然完全没有解决。要想取得进一步进展，就需要进一步发展方法并将其应用于新问题。首席研究员计划继续发展的主题进一步讨论下一个。在这个议程的第一步将集中在推进首席研究员的程序去奇异化和加倍的建设极小曲面在黎曼流形。其他胶合结构将与各种合作者进行：与Simon Brendle在四维中构建爱因斯坦流形和Ricci孤子，并在一些项目中与Frederick Fong合作。用克莉丝汀Breiner构造常平均曲率超曲面。特殊拉格朗日锥的构造与马克·哈斯金斯。单位球内极小曲面的自由边界问题。 用Jason Lotay构造余结合四维流形。与Stephen Kleene、Niels Moller和大卫Wiygul一起构造平均曲率流的自收缩器。他还计划与克莉丝汀布雷纳、雅各布伯恩斯坦、斯蒂芬克莱恩、F。Martin，W.米克斯、尼尔斯·穆勒和大卫·威古尔。最后，主要研究者计划单独或与他的合作者一起研究与上述结构相关或由其激发的一些独特性/特征化/不存在性问题。