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.

最小作用原理是说，一个物理模型的结果应该总是最小化一个确定的物理量，即作用函数。这个观察可以追溯到费马和欧拉，广义地说，根据自然规律，事物以最经济的方式进行。这种现象适用于物理学的许多方面，包括牛顿力学、拉格朗日力学和哈密顿力学，甚至广义相对论。研究这种现象的抽象数学框架是变分演算，主要研究者将在微分几何的背景下应用它。也就是说，在微分几何中，人们通常有大量的几何对象（在这种情况下是Kahler度规或拉格朗日量），并在这个集合中寻找具有最好性质的特殊元素。这些特殊的元素往往最小化了某种能量功能，这是项目的出发点。对手头问题的透彻理解可以导致对宇宙形状的新见解，它将有助于在弦理论中做出令人兴奋的预测，更广泛地说，在理论物理学中。本课题可分为三个主要课题：刻画Kahler流形上常数标量曲率度量的存在性；Kahler度量空间的L^p-Finsler几何的凸性和曲率性质，它们的有限维近似，以及测地线射线相关空间的结构；正拉格朗日空间的度规结构。作为一种创新，在提出的研究中，我们将专门开发或使用适当的度量几何，希望更好地理解潜在的变分问题。PI计划使用的度量空间源于无限维芬斯勒流形的路径长度结构，因此它们本身具有非常丰富的几何结构。在Kahler的例子中，我们希望这将允许人们将许多稳定性的概念联系起来，包括来自yu -Tian- donaldson猜想的k -稳定性，Tian的能量性和测地线稳定性，所有这些都被推测为表征常数标量曲率度量的存在性。对于拉格朗日几何，我们知道的就少得多。根据Solomon最近提出的一个计划，首席研究员打算进一步发展基本的度量几何，以制定和证明表征特殊拉格朗日存在的稳定性条件。