Extremal Kaehler Metrics and Geometric Flow Equations

DMS-0406346标题：极值Kaehler度量和几何流方程PI：陈秀雄，威斯康星大学(麦迪逊)摘要在90年代初，S.T.Yau猜想具有常标量曲率的Kaehler度量的存在与基础极化流形的稳定性有关。近年来，在这个方向上取得了令人瞩目的进展。1997年，G.Tian证明了任何具有正数量曲率的Kaehler-Einstein流形都是K-稳定的。2001年，S.K.Donaldson证明了具有离散自同构群且具有常数量曲率的Kaehler度量的代数流形是Chow稳定的，最近，在与G.Tian的合作中，证明了常数量曲率的Kaehler度量的存在蕴含着K能量关于给定上同调类的半K稳定性。受这些结果的启发，作者想要研究一个围绕极值Kaehler度量(包括作为特例的常数量曲率度量)的存在性、极化Kaehler流形的稳定性以及其他相关问题的网络问题。解决这些问题的主要思想是改进T.Mabuchi意义下的Kaehler度量空间中测地线的正则性，理解Calabflow的长期存在，以及Kaehler-Ricci流的收敛。这些想法涉及不同的数学领域，但作者认为它们都很有希望解决Yau的猜想(特别是解决Fano流形上Kaehler-Einstein度规的存在性问题)。常标量曲率的Kaehler度量的存在性问题是微分几何中的核心问题，它对物理学等其他科学领域有着重要的影响。根据阿尔伯特·爱因斯坦的说法，引力理论可以解释为时空的几何学。因此，微分几何的研究在物理学和宇宙学中都是至关重要的。这项研究的提出也对弦理论产生了影响，弦理论是将自然界的四种基本力量统一起来的理论。这位提出者的工作与其他数学家和物理学家的工作一起，有助于理解我们的宇宙。

DMS-0406346Title: Extremal Kaehler metrics and geometric flow equationsPI: Xiuxiong Chen, University of Wisconsin (Madison)ABSTRACTIn the early nineties, S. T. Yau conjectured that the existence of Kaehler metrics with constant scalar curvature is related to the stability of the underlying polarized manifold. In recent years, striking progress has been made in this direction. In 1997, G. Tian proved that any Kaehler-Einstein manifold with positive scalar curvature is K-Stable. In2001, S. K. Donaldson proved that an algebraic manifold with discrete automorphism group and Kaehler metric of constant scalar curvature is Chow stable.Very recently, in a joint work with G. Tian, the proposer proved that the existence of Kaehler metrics of constant scalar curvature implies Semi-KStability of the K energy with respect to the given cohomology class. Inspired by these results, the proposer wants to study a network of problems centered around the existence of extremal Kaehler metrics (which includes constant scalar curvature metric as a special case), stability of polarized Kaehler manifold, and other related problems. The main ideas of solving these problemsconsist of improving regularity for geodesics in the space of Kaehler metrics in the sense of T. Mabuchi, of understanding the long-time existence of the Calabiflow, and of the convergence of the Kaehler-Ricci flow. These ideas are related to different mathematical fields but the proposer believes that they are all very promising to solve the Conjecture of Yau (in particular to solve the problem of the existence of Kaehler-Einstein metrics on Fano manifold). The proposer is going to work on the problems through these ideas.The problem of the existence of Kaehler metrics of constantscalar curvature, being the key problem in differential geometry, has strong impact to other fields of sciences like physics. According to AlbertEinstein, the theory of gravity can be interpreted as the geometry of space-time. Thus the research in differential geometry is crucially important in physics and cosmology. The research proposed also has impact in string theory, which is the theory of unifying all four basic forces of the Nature. The proposer's work, together with the works of other mathematicians and physicists, helps in understanding our Universe.