权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Gluing, Rigidity and Uniqueness Questions in Geometric Analysis

几何分析中的粘合、刚性和唯一性问题

基本信息

批准号：
EP/J014206/1
负责人：
Jason Lotay
金额：
$ 1.23万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ014206%2F1
关键词：
Gluing Rigidity Uniqueness Questions Geometric

项目摘要

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.

我们想研究两种不同但相关的几何对象，称为特殊拉格朗日（SL）子流形和拉格朗日自膨胀。SL子流形具有吸引人的性质，它们是体积最小的，所以可以被认为是像肥皂膜。我们还希望考虑另一种相关的体积最小化对象类型，称为凯莱4-折叠。数学家们研究肥皂膜方程已经有两百多年的历史了，许多广泛适用的数学方法首先被开发出来研究肥皂膜方程（非线性椭圆方程）。这些技术现在被数学家、物理学家和工程师用于一系列与肥皂膜完全无关的问题。虽然现在对肥皂膜本身了解很多，但它们的研究仍然是一个活跃的研究领域，最近有几个重要的突破。对于广义的肥皂膜，如SL子流形，知之甚少;一些全新的现象发生，我们才刚刚开始了解。拉格朗日自膨胀器的研究是由一种独特而自然的方式来移动生活在更大空间中的几何物体，称为平均曲率流（MCF）。在MCF下，球体会简单地收缩，而拉格朗日自膨胀体会增长。MCF试图移动一个给定的表面，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被平滑掉，就像热量从热源扩散一样。由于这个原因，它已被许多工程师用作从经验数据中去除噪声的鲁棒方法，例如来自各种类型扫描仪的大脑图像。工程师通常依赖主要由数学家开发的工具。与MCF的一个困难是，在很长一段时间内，它的平滑效果可能会被非线性反馈压倒，因此奇点可能会在流动中发展。这个项目将有助于我们理解奇异性如何在一种特殊类型的MCF中形成，称为拉格朗日平均曲率流。我们的项目是研究SL子流形，拉格朗日自膨胀和Cayley 4-folds与“结束”。我们想证明，在某些情况下，只知道一个几何对象的“两端”完全决定了它的全局结构。这一点很重要，因为它表明，如果我们只了解一个物体在非常大的尺度下的样子，那么我们就可以推断出它在所有尺度下的样子。如果我们对正的x和y绘制曲线xy=1，我们看到它有两个“端点”：一个更接近x轴，另一个更接近y轴。总的来说，曲线xy=1接近一对直线（轴），这对直线仅相交于一点（原点）。如果我们考虑同样的方程xy=1，但是现在x和y是复数，我们得到一个有两个端点的曲面，每个端点都渐近于一个平面。而且，两个渐近平面只在一点相交，所以我们称它们为横截的。我们想研究具有相同性质的物体：它们有两端，每一端都接近一个平面，并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截平面，那么要么不存在两端渐近于它们的SL子流形或拉格朗日自扩张，要么只有一个（可能满足一些额外的条件使其唯一）。我们还希望探讨这些结构的结果的一些后果，希望这将最终导致解决的重要和困难的问题，找到SL子流形使用拉格朗日平均曲率流和定义不变量使用凯莱4倍。也有连接到更多样化的领域，包括广义肥皂膜和肥皂泡的研究，“胶合”问题，同调镜像对称（这是从弦理论和M理论的理论物理的想法启发），以及非线性偏微分方程的研究。