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.

在给定的Kahler类中存在恒定标量曲率Kahler指标的问题是一个重要且困难的问题，并为Kahler几何学进行了许多当前研究的动机。 对于Fano歧管上的Kahler-Einstein指标的特殊情况，Yau的存在与几何不变理论意义上的歧管相同。 首席研究人员建议研究在这种情况下自然出现的卡勒电位的三个抛物面流。 第一个是J-flow，它是陈出的Mabuchi Energy中出现的功能性的梯度流。 对J-Flow的研究在理解Mabuchi能量的下界和渐近性方面取得了重大进展。 第二个是Kahler-Ricci流。 它在FANO案例中的行为尚不清楚，并提出乘数理想的系带方法可以捕获有关其奇异性的必要信息，以便能够提供与稳定性的链接。 第三是卡拉比流。 这是第四阶抛物线pde，涉及一般来说的几乎没有人知道。 主要研究者打算研究这种流程的长时间存在问题。几何和物理学中的重要问题是，给定的空间是否具有特殊的距离概念。 以二维球为例 - 球的表面。 有了我们通常的距离，这个空间在每个点都以相同的方式弯曲。 我们说球体承认“持续曲率的度量”。并非所有空间都承认这样的指标，找到他们所做的条件是一个有趣而深刻的问题。 抛物线代理或“热流”方法是一种自然而美丽的方法。 这个想法很简单。 物体中的热量分布（不受外部来源的约束）将随着时间的流逝而流动，变得更加均匀，最终 - 无论初始分布如何。 以类似的方式，如果我们从空间上的距离任意概念开始，那么我们可以应用自然的“热流”，并希望在正确的条件下证明，随着时间的发展，我们可以获得与恒定曲率或其他一些特殊度量的收敛性。 如果没有这样的指标，那么我们希望流动出错 - 发展奇异性。 PI旨在研究与不同类型的特殊指标相对应的三种此类抛物线流的收敛性和奇异性问题。