Parabolic flows and canonical metrics in Kahler geometry.

在给定的Kahler类中，常数量曲率Kahler度量的存在性问题是一个重要而又困难的问题，也是当前许多Kahler几何研究的基础。对于Fano流形上的Kahler-Einstein度量的特殊情况，Yau猜想其存在等价于流形在几何不变量意义下的稳定性。首席研究员建议研究在这种情况下自然产生的三个Kahler势的抛物线流动。第一种是J流，它是出现在陈氏马布奇能量公式中的泛函的梯度流。对J流的研究在理解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情况下的行为还没有被很好地理解，而且有人提出乘子理想层方法可以捕捉到关于它的奇点的必要信息，从而能够提供与稳定性的联系。第三个是卡拉比流。它是一种四阶抛物型偏微分方程解，一般对它知之甚少。主要的研究人员打算研究这种流动的长期存在问题。几何和物理中的一个重要问题是，给定的空间是否有特殊的距离概念。以二维球体为例--球的表面。在我们通常的距离感下，这个空间在每一点都是以同样的方式弯曲的。我们说球体有一个“常曲率度规”。并不是所有的空间都承认这样的指标，找到它们在什么条件下承认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方法，或称“热流”方法。这个想法很简单。热在物体中的分布(不受外部来源的影响)将会及时流动，变得更加均匀，最终变得恒定--无论最初的分布是什么样子。以类似的方式，如果我们从一个空间上任意的距离概念开始，那么我们可以应用一个自然的‘热流’，并希望在正确的条件下证明，随着时间的演变，我们可以收敛到一个常曲率的度量，或者其他一些特殊的度量。如果不存在这样的指标，那么我们预计流程会出错--产生奇点。PI旨在研究与不同类型的特殊度量相对应的三种抛物线流的收敛和奇性问题。

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.