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

这个项目的第一个主要目标是最大限度地扩大胶合方法论在理解极小曲面理论中的重要存在问题方面的使用。另一个目标(在相反的方向)是理解相关的不存在、表征和唯一性问题。实现这些目标需要改进已知的方法，还需要开发全新的方法。在这个方案的第一个项目中，PI旨在与Mark Haskins合作研究重要的特殊拉格朗日和其他已定标子流形。特别地，他们打算继续研究由ODE系统控制的特殊拉格朗日圆锥，并以各种方式与Lawler颈项相关，其中一些在$So(P)乘以So(n-p)$的作用下不变；它们用作粘合新的特殊拉格朗日圆锥的构造块；以及这些对象的唯一性问题(对于这些对象，已知的理论似乎不够充分，因为高余维)。在其他项目中，PI单独或合作，打算继续他的工作，最大限度地推广他早期的去奇化和加倍三维流形中极小曲面的构造，并将这些构造应用于极小曲面理论中的基本问题，例如，Yau关于任何黎曼三维流形中存在无穷多个极小曲面的问题。PI还打算研究圆形三球面上极小曲面的存在性和分类问题，包括Lawson曲面的拓扑性的刻画。在与F.Martin和W.Meek的合作中，PI打算在有三个交点的情况下进行去奇异构造，以便它们可以用于理解嵌入情况下极小曲面的Calabi-Yau问题。在与Stephen Kleene和Niels Moller的合作下，PI打算研究平均曲率流的自缩者的存在性问题。在与克里斯汀·布赖纳的合作中，PI打算扩展他早期关于常平均曲率曲面的粘合构造的工作。最小和常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这里被提出，后来被应用到非线性偏微分方程组，广义相对论，爱因斯坦流形和其他领域。这并不令人惊讶，因为在某种意义上，该理论结合了所有这些领域的重要特征，同时也是最简单和最直观的。虽然已经取得了巨大的进展，但仍有许多根本问题是完全悬而未决的，主要是因为已知的方法不够充分。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。

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.