Parabolic flows and canonical metrics in Kahler geometry.

卡勒几何中的抛物线流和规范度量。

• 批准号: 0504285
• 负责人: Benjamin Weinkove
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504285&HistoricalAwards=false
• 关键词: Parabolic; flows; canonical; metrics; Kahler

The problem of the existence of a constant scalar curvature Kahler metric in a given Kahler class is an important and difficult problem and has provided the motivation for much current research in Kahler geometry. For the special case of Kahler-Einstein metrics on Fano manifolds, existence was conjectured by Yau to be equivalent to the stability of the manifold in the sense of geometric invariant theory. The principal investigator proposes to study three parabolic flows of Kahler potentials which arise naturally in this context. The first is the J-flow, which is the gradient flow of a functional appearing in Chen's formula for the Mabuchi energy. The study of the J-flow has led to significant advances in understanding the lower boundedness and asymptotics of the Mabuchi energy. The second is the Kahler-Ricci flow. Its behavior in the Fano case is not yet well understood, and it is proposed that the method of multiplier ideal sheaves may capture the necessary information about its singularities to be able to provide a link with stability. The third is the Calabi flow. It is a fourth order parabolic PDE about which little is known in general. The principal investigator intends to study the problem of long time existence of this flow.An important problem in geometry and physics is whether a given space has a special notion of distance. Take, for example, the two dimensional sphere - the surface of a ball. With our usual sense of distance, this space is curved in the same way at every point. We say that the sphere admits a 'metric of constant curvature'. Not all spaces admit such metrics, and it is an interesting and deep problem to find conditions under which they do. A natural and beautiful approach to this problem is the parabolic, or 'heat flow' method. The idea is simple. The distribution of heat in an object (not subject to outside sources) will flow in time, becoming more even and finally constant - no matter what the initial distribution looked like. In a similar way, if we start with an arbitrary notion of distance on a space, then we can apply a natural 'heat flow' and hope to prove, under the right conditions, that we obtain convergence to a metric of constant curvature, or some other special metric, as time evolves. If no such metrics exist, then we expect the flow to go wrong - to develop singularities. The PI intends to study the question of convergence and singularities of three such parabolic flows corresponding to different types of special metrics.

给定Kahler类中恒定标量曲率Kahler度规的存在性问题是一个重要而困难的问题，它为当前Kahler几何的许多研究提供了动力。对于Fano流形上的Kahler-Einstein度量的特例，Yau在几何不变理论意义上推测了流形的存在性等价于流形的稳定性。首席研究员建议研究在这种情况下自然产生的卡勒势的三种抛物线流。第一个是j流，它是在Chen的Mabuchi能量公式中出现的泛函的梯度流。j流的研究在理解Mabuchi能量的下界性和渐近性方面取得了重大进展。第二个是Kahler-Ricci流。它在Fano情况下的行为尚未被很好地理解，并且提出乘子理想轴的方法可以捕获关于其奇点的必要信息，从而能够提供具有稳定性的链接。第三个是卡拉比流。它是一个四阶抛物型偏微分方程，一般对它知之甚少。本课题拟研究该流长期存在的问题。几何和物理中的一个重要问题是给定空间是否具有特殊的距离概念。以二维球面为例——一个球的表面。用我们通常的距离感，这个空间在每一点都以同样的方式弯曲。我们说球体允许一个“常曲率度规”。并不是所有的空间都允许这样的度量，找到它们存在的条件是一个有趣而深刻的问题。解决这个问题的一个自然而美丽的方法是抛物线法，或“热流”法。这个想法很简单。一个物体的热量分布（不受外界热源的影响）将随着时间的推移而流动，变得更加均匀，最终保持恒定——不管最初的分布是什么样子。类似地，如果我们从空间上的任意距离概念开始，那么我们可以应用自然的“热流”，并希望证明，在适当的条件下，随着时间的发展，我们可以收敛到一个恒定曲率的度规，或者其他一些特殊的度规。如果不存在这样的度量，那么我们预计流将出错——产生奇点。PI计划研究三种这样的抛物线流对应于不同类型的特殊度量的收敛性和奇异性问题。