Stability of variational problems in differential geometry

最小作用量原理说，物理模型的结果应该总是最小化一个确定的物理量，即作用泛函。这一观察可以追溯到费马和欧拉，广义地说，它说，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法，首席研究员将在微分几何的背景下应用这一点。也就是说，在微分几何中，人们通常有大量的几何对象（在这种情况下是卡勒度量或拉格朗日），并在这个集合中搜索具有最好属性的特殊元素。这些特殊的元素往往最小化了某个能量泛函，而这就是工程的出发点。深入理解当前的问题可以使我们对宇宙的形状有一个新的认识，它将有助于在弦理论和更广泛的理论物理学中做出令人兴奋的预测。这个项目可以分为三个主要课题：描述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，以及测地线的相关空间的结构;正拉格朗日空间的度量结构。作为一种新奇，在拟议的研究中，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来自无限维芬斯勒流形的路径长度结构，因此本身具有非常丰富的几何。在卡勒的情况下，这是有希望的，这将允许一个连接的许多概念的稳定性，包括K-稳定性从姚田-唐纳森猜想，能量适当的田，和测地稳定性，所有的特征存在的常数标量曲率度量。在拉格朗日几何的情况下，已知的要少得多。根据最近的计划提出的所罗门，首席研究员打算进一步发展的基本度量几何，以制定和证明稳定性条件的存在特征的特殊拉格朗日。

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.