Gluing, Rigidity and Uniqueness Questions in Geometric Analysis

我们想研究两种不同但又相互关联的几何对象，即特殊拉格朗日(SL)子流形和拉格朗日自扩张器。SL子流形具有体积最小的吸引人的性质，因此可以被认为是类似于肥皂膜。我们还希望考虑另一种相关类型的体积最小化对象，称为Cayley 4-折叠体。数学家对肥皂膜方程的研究已有两百多年的历史，许多广泛应用的数学方法首先被用来研究肥皂膜方程(非线性椭圆型方程)。这些技术现在被数学家、物理学家和工程师用在一系列与肥皂片完全无关的问题上。虽然现在人们对肥皂膜本身有了很多了解，但他们的研究仍然是一个活跃的研究领域，最近取得了几项重要突破。对于像SL子流形这样的泛化肥皂片，我们知道的要少得多；一些全新的现象出现了，我们才刚刚开始理解。拉格朗日自扩张器的研究是由一种独特而自然的方式推动的，这种方式可以移动生活在更大空间中的几何对象，称为平均曲率流(MCF)。在MCF下，球体会简单地收缩，而拉格朗日自膨胀器会增长。MCF尝试移动给定的曲面，以使其面积尽可能快地缩小。MCF具有很强的平滑效果，其中局部不规则性往往会非常迅速地得到平滑，其方式与热量从热源传播的方式大致相同。出于这个原因，它被许多工程师用作从经验数据中去除噪声的稳健方法，例如来自各种类型扫描仪的大脑图像。工程师们经常依赖主要由数学家开发的工具。MCF的一个困难是，在很长的时间内，它的平滑效果可能被非线性反馈所淹没，因此可能在流动中产生奇异性。这个项目将有助于我们理解奇点是如何在一种特殊的称为拉格朗日平均曲率流的MCF中形成的。我们的项目是研究SL子流形、拉格朗日自扩张子流形和带“端”的Cayley 4-折叠。我们想要证明，在某些情况下，只知道一个几何对象的“末端”完全决定了它的全局结构。这一点很重要，因为它说，如果我们了解一个物体在非常大的尺度上是如何看的，那么我们就可以推断它在所有尺度上的样子。如果我们画正x和y的曲线xy=1，我们会看到它有两个“端”：一个更靠近x轴，另一个更靠近y轴。总的来说，曲线xy=1接近一对直线(轴)，这两条直线只在一点(原点)相交。如果我们考虑相同的方程xy=1，但现在x和y是复数，我们得到一个两端都渐近于一个平面的曲面。此外，这两个渐近平面只在一点相交，所以我们称它们为横向平面。我们想要研究具有相同性质的对象：它们的两端各接近一个平面，且这对渐近平面是横截的。我们的目的是证明，如果我们有一对横平面，那么要么没有SL子流形，要么没有两端渐近于它们的拉格朗日自扩张器，要么只有一个(可能满足一些额外的条件使其唯一)。我们还希望探索这些结构结果的一些结果；希望这将最终导致解决使用拉格朗日平均曲率流寻找SL子流形和使用Cayley 4-折叠定义不变量的重点和困难问题。它还与更多不同的领域有联系，包括研究广义肥皂膜和肥皂泡、“粘合”问题、同调镜面对称性(灵感来自弦理论和M理论中的理论物理思想)，以及研究非线性偏微分方程。

We want to study two different but related types of geometric object called special Lagrangian (SL) submanifolds and Lagrangian self-expanders. SL submanifolds have the attractive property that they are volume-minimizing, so can be thought of as like soap films. We also wish to consider another related type of volume-minimizing objects called Cayley 4-folds. Mathematicians have studied the equations governing soap films for over two hundred years and many widely applicable mathematical techniques were first developed to study the soap film equations (nonlinear elliptic equations). These techniques are now used by mathematicians, physicists and engineers in a whole range of problems completely unrelated to soap films. While much is now known about soap films themselves, their study is still an active area of research with several recent important breakthroughs. For generalised soap films like SL submanifolds, much less is known; some completely new phenomena occur which we are only just beginning to understand. The study of Lagrangian self-expanders is motivated by a distinguished and natural way to move geometric objects which live inside larger spaces called Mean Curvature Flow (MCF). Under MCF, a sphere will simply shrink, whereas Lagrangian self-expanders grow. MCF tries to move a given surface so that its area shrinks as rapidly as possible. MCF has a strong smoothing effect in which local irregularities tend to get smoothed out very rapidly, in much the same way that heat spreads out from a heat source. For this reason it has been used by many engineers as a robust way to remove noise from empirical data, e.g. images of brains from various types of scanners. The engineers often rely on tools developed primarily by mathematicians. One difficulty with MCF is that over long time periods its smoothing effects may be overwhelmed by nonlinear feedback and thus singularities may develop in the flow. This project will contribute to our understanding of how singularities can form in a special type of MCF called Lagrangian Mean Curvature Flow.Our project is to study SL submanifolds, Lagrangian self-expanders and Cayley 4-folds with "ends". We want to show that in certain circumstances knowing only the "ends" of a geometric object completely determines its global structure. This is important because it says if we understand how an object looks only at a very large-scale then we can infer how it looks at all scales. If we draw the curve xy=1 for positive x and y, we see that it has two "ends": one which gets closer to the x-axis and the other which gets closer to the y-axis. Overall the curve xy=1 approaches a pair of straight lines (the axes) which intersect at just one point (the origin). If we consider the same equation xy=1, but now where x and y are complex numbers, we get a surface with two ends each of which is asymptotic to a plane. Moreover, the two asymptotic planes meet at just one point, so we call them transverse. We want to study objects with the same property: they have two ends that each approach a plane and the pair of asymptotic planes are transverse.Our aim is to show that if we have a pair of transverse planes then either there is no SL submanifold or Lagrangian self-expander with two ends asymptotic to them, or there is just one (possibly satisfying some extra conditions to make it unique). We also hope to explore some of the consequences of these structural results; hopefully this will eventually lead to the solution of the important and difficult problems of finding SL submanifolds using Lagrangian Mean Curvature Flow and defining invariants using Cayley 4-folds. There are also connections to more diverse areas including the study of generalised soap films and soap bubbles, "gluing" problems, Homological Mirror Symmetry (which was inspired by ideas from theoretical physics in String Theory and M-Theory), and the study of nonlinear partial differential equations.