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在理解最小曲面理论中重要的存在性问题时，最大程度地扩展粘接方法的使用。另一个目标（相反方向）是理解相关的不存在性、特征化和唯一性问题。实现这些目标需要改进已知的方法，也需要开发全新的方法。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流形和其他校准子流形。特别地，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，并以各种方式与Lawlor颈相关，其中一些在$SO(p)\乘以SO(n-p)$的作用下是不变的；它们被用作粘接新的特殊拉格朗日锥结构的积木；以及这些物体的唯一性问题（由于高余维数，已知的理论似乎不够充分）。在其他项目中，PI，单独或合作，打算继续他的工作，在最大可能的范围内推广他早期的三流形最小曲面的去象素化和加倍结构，并将这些结构应用于最小曲面理论中的基本问题，例如用于Yau关于任何黎曼三流形中存在无限多个最小曲面的问题。PI还打算研究圆形三球中最小曲面的存在性和分类问题，包括Lawson曲面的拓扑性质的表征。在与F. Martin和W. Meeks的合作中，PI打算在有三个交叉点的非物化结构上工作，这样它们就可以用来理解嵌入情况下最小表面的Calabi-Yau问题。在与Stephen Kleene和Niels Moller的合作中，PI打算研究平均曲率流的自收缩的存在性问题。在与Christine Breiner的合作中，PI打算扩展他早期关于恒定平均曲率表面的粘合结构的工作。最小和常数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被发展出来，后来被应用到非线性偏微分方程、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，同时又是最简单和最直观的。虽然已经取得了巨大的进展，但仍有许多基本问题是完全开放的，主要是因为已知的方法是不够的。这些未决项目的顺利完成将回答许多重要的此类问题，并将成为有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.