Special Langrangian submanifolds in C^n and minimal surfaces in 3-manifolds

C^n 中的特殊朗格朗日子流形和 3 流形中的最小曲面

• 批准号: 1105371
• 负责人: Nicolaos Kapouleas
• 金额: $ 22.05万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105371&HistoricalAwards=false
• 关键词: Special; Langrangian; submanifolds; minimal; surfaces

The first main objective of this project is to expand the use of gluing methodology to the greatest possible extent in understanding important existence questions in the theory of minimal surfaces. Another objective (in the opposite direction) is to understand related non-existence, characterization, and uniqueness questions. Achieving these objectives requires the refinement of the known methodology and also the development of entirely new methods. In the first project of this proposal, the PI aims to study in collaboration with Mark Haskins important special Lagrangian and other calibrated submanifolds. In particular they intend to continue to study special Lagrangian cones in $\mathbb{C}^n$ controlled by ODE systems and related in various ways to the Lawlor necks, some of them invariant under the action of $SO(p)\times SO(n-p)$; their use as building blocks for gluing constructions for new special Lagrangian cones; and uniqueness questions for these objects (for which the known theory seems inadequate because of the high codimension). In other projects, the PI, alone or in collaboration, intends to continue his work on generalizing his earlier desingularization and doubling constructions for minimal surfaces in three-manifolds to the greatest possible extent, and apply these constructions to fundamental questions in the theory of minimal surfaces, for example to a question of Yau about the existence of infinitely many minimal surfaces in any Riemannian three-manifold. The PI also intends to study existence and classification questions for minimal surfaces in the round three-sphere, including characterizations of a topological nature for the Lawson surfaces. In a collaboration with F. Martin and W. Meeks, the PI intends to work on desingularization constructions where there are triple points of intersection so that they can be used to understand the Calabi-Yau problem for minimal surfaces in the embedded case. In collaboration with Stephen Kleene and Niels Moller, the PI intends to work on existence questions for self-shrinkers of the mean curvature flow. In collaboration with Christine Breiner, the PI intends to expand his earlier work on gluing constructions for constant mean curvature surfaces.Minimal and constant mean curvature surfaces have historically been an important field where many important ideas were first developed, and later applied to nonlinear Partial Differential Equations, General Relativity, Einstein manifolds, and other fields. This is not surprising because in some sense the theory combines important features of all these fields while it is at the same time the simplest and most intuitive. Although enormous progress has been made, there are many fundamental questions which are completely open, mostly because the known methodologies are inadequate. Successful completion of these pending projects would answer many important such questions and would be the basis for progress in the related fields as well.

该项目的第一个主要目标是最大程度地扩展粘合方法的使用，以理解最小曲面理论中的重要存在问题。 另一个目标（相反的方向）是理解相关的不存在性、特征性和独特性问题。 实现这些目标需要改进已知的方法并开发全新的方法。 在本提案的第一个项目中，PI 旨在与 Mark Haskins 合作研究重要的特殊拉格朗日和其他校准子流形。特别是，他们打算继续研究由 ODE 系统控制的 $\mathbb{C}^n$ 中的特殊拉格朗日锥体，并以各种方式与劳勒颈相关，其中一些锥体在 $SO(p)\times 的作用下保持不变SO(n-p)$；它们用作新型特殊拉格朗日锥体粘合结构的构建块；以及这些物体的唯一性问题（由于余维数高，已知的理论似乎不足以解决这些问题）。 在其他项目中，PI 打算单独或合作继续他的工作，以最大可能的程度推广他早期的去奇异化和三流形极小曲面的双重构造，并将这些构造应用于极小理论中的基本问题。曲面，例如丘关于任何黎曼三流形中是否存在无限多个最小曲面的问题。 PI 还打算研究圆形三球体中最小曲面的存在性和分类问题，包括劳森曲面的拓扑性质的表征。在与 F. Martin 和 W. Meeks 的合作中，PI 打算研究存在三相交叉点的去奇异化结构，以便它们可以用于理解嵌入式案例中最小曲面的 Calabi-Yau 问题。 PI 打算与 Stephen Kleene 和 Niels Moller 合作，研究平均曲率流自收缩器的存在性问题。 PI 打算与 Christine Breiner 合作，扩展他早期在恒定平均曲率曲面粘合结构方面的工作。最小和恒定平均曲率曲面在历史上一直是一个重要的领域，许多重要的思想首先在该领域得到发展，后来应用于非线性偏微分方程、广义相对论、爱因斯坦流形和其他领域。 这并不奇怪，因为从某种意义上来说，该理论结合了所有这些领域的重要特征，同时又是最简单、最直观的。 尽管已经取得了巨大进展，但仍有许多完全开放的基本问题，主要是因为已知的方法还不够充分。这些悬而未决的项目的成功完成将回答许多重要的此类问题，也将成为相关领域取得进展的基础。