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，主要研究人员：Nicolaos Kapouleas差异几何学是与数学和物理学其他领域密切相关的数学领域的重要领域，例如拓扑，非线性偏微分方程以及一般相对论。为了理解和发展差异几何理论，有必要有足够的考虑示例的示例。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使用奇异的扰动和已知的解决方案来构建解决方案解决方案的粘合方法。尽管已经在这个方向上取得了巨大进展，但该主题似乎还有很长的路要走，因为还有许多基本问题仍然完全开放。进一步的进步需要更多地发展新问题的方法和应用。首席研究人员计划在接下来讨论中继续进一步发展该主题。该议程中的第一个步骤将集中精力推进主要研究者的降级计划，以延长利曼尼亚歧管中最小表面的降低和加倍结构。各种合作者将在四个维度上与西蒙·布伦德尔（Simon Brendle）和弗雷德里克·冯（Frederick Fong）的一些项目一起在四个维度上进行其他合作者：爱因斯坦歧管和里奇·索蒂顿的构造。克里斯汀·布雷纳（Christine Breiner）的恒定平均曲率超曲面的结构。特殊拉格朗日锥的建筑与马克·哈斯金斯（Mark Haskins）。在与马丁·李（Martin Li）的单位球中最小表面的自由边界问题上。 与Jason Lotay共同交往的四个manifords的构造。 Stephen Kleene，Niels Moller和David Wiygul的平均曲率流量自我缩合器的构造。他还计划在与克里斯汀·布雷纳（Christine Breiner），雅各布·伯恩斯坦（Jacob Bernstein），斯蒂芬·克莱恩（Stephen Kleene），F。Martin，W。Meeks，Niels Moller和David Wiygul有关的项目上进行更多的合作，以与克里斯汀·布雷纳（Christine Breiner），雅各布·伯恩斯坦（Jacob Bernstein），斯蒂芬·克莱恩（Stephen Kleene）和戴维·维格尔（David Wiygul）进行更多合作。最后，首席调查员计划独自或与他的合作者一起工作，以与上述结构相关或动机的某些唯一性/特征/不存在问题。