Stability of variational problems in differential geometry

微分几何中变分问题的稳定性

• 批准号: 1610202
• 负责人: Tamas Darvas
• 金额: $ 14.23万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1610202&HistoricalAwards=false
• 关键词: Stability; variational; problems; differential; geometry

The principle of least action says that the outcome of a physical model should always minimize a well-determined physical quantity, the action functional. This observation goes back to Fermat and Euler, and broadly speaking it says that, by the laws of nature, things are carried out in the most economical way. The phenomenon applies to many facets of physics, including Newtonian, Lagrangian and Hamiltonian mechanics, even general relativity. The abstract mathematical framework that studies such phenomena is the calculus of variations, and the principal investigator will apply this in the context of differential geometry. Namely, in differential geometry one often has a large collection of geometric objects (in this case Kahler metrics or Lagrangians) and is searching for special elements in this collection that have the nicest properties. These special elements often minimize a certain energy functional, and this is the starting point of the project. A thorough understanding of the problems at hand can lead to new insight into the shape of the universe, and it would help make exciting predictions in string theory and, more broadly, in theoretical physics.This project can be split into three main subjects: characterizing existence of constant scalar curvature metrics on Kahler manifolds; convexity and curvature properties of the L^p-Finsler geometry of the space of Kahler metrics, their finite dimensional approximations, and the structure of the associated space of geodesic rays; the metric structure of the space of positive Lagrangians. As a novelty, in the proposed study we will either specifically develop or use an adequate metric geometry, in hopes of understanding the underlying variational problems better. The metric spaces that the PI plans to use arise from the path length structure of infinite dimensional Finsler manifolds, and as such have a very rich geometry themselves. In the Kahler case it is hopeful that this will allow one to connect many notions of stability, including K-stability from the Yau-Tian-Donaldson conjecture, the energy properness of Tian, and geodesic stability, all conjectured to characterize existence of constant scalar curvature metrics. In the case of Lagrangian geometry much less is known. Following a recent program proposed by Solomon, the principal investigator intends to develop the underlying metric geometry further in order to formulate and prove stability conditions characterizing existence of special Lagrangians.

最少动作的原则表明，物理模型的结果应始终最大程度地减少确定的物理数量，即动作功能。这一观察结果可以追溯到Fermat和Euler，从广义上讲，它说，根据自然法则，事情是以最经济的方式进行的。该现象适用于许多物理学方面，包括牛顿，拉格朗日和哈密顿力学，甚至一般相对论。研究这种现象的抽象数学框架是变异的计算，主要研究者将在差异几何形状的背景下应用。也就是说，在差异几何形状中，一个几何对象（在这种情况下为Kahler指标或Lagrangians）通常都有很多，并且正在寻找该系列中具有最好属性的特殊元素。这些特殊要素通常最小化某种能量功能，这是项目的起点。对眼前问题的透彻理解可以导致对宇宙形状的新见解，这将有助于在弦理论中做出令人兴奋的预测，并且更广泛地在理论物理学中。该项目可以分为三个主要主题：表征卡勒流形上存在恒定标态曲率指标； Kahler指标空间的L^p-finsler几何形状的凸度和曲率特性，其有限的尺寸近似值以及相关射线相关空间的结构；积极的拉格朗日空间的度量结构。作为一种新颖性，在拟议的研究中，我们要么专门开发或使用足够的度量几何形状，希望更好地理解潜在的变异问题。 PI计划使用的度量空间是由无限尺寸鳍片歧管的路径长度结构产生的，因此本身具有非常丰富的几何形状。在卡勒（Kahler）的情况下，希望这将允许一个人连接许多稳定概念，包括Yau-tian-Donaldson的K稳定性，猜想，Tian的能量正常和地球稳定性曲率指标。在拉格朗日几何形状的情况下，还要鲜为人知。在所罗门提出的最新计划之后，首席研究人员打算进一步开发潜在的度量几何形状，以制定和证明特征特殊Lagrangians存在的稳定条件。