Extremal Kaehler Metrics and Geometric Flow Equations

陈秀雄，威斯康星州大学（麦迪逊）摘要九十年代初，S. T.丘成桐指出，具有常数量曲率的Kaehler度量的存在性与底层极化流形的稳定性有关。近年来，在这方面取得了显著进展。1997年，G. Tian证明了任何具有正数量曲率的Kaehler-Einstein流形是K-稳定的。2001年，S. K.唐纳森证明了具有离散自同构群和常数量曲率的Kaehler度量的代数流形是Chow稳定的。Tian等人证明了常数量曲率Kaehler度量的存在性蕴涵着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，提出者希望研究一个网络的问题，围绕极值Kaehler度量的存在性（其中包括常数标量曲率度量作为特例），极化Kaehler流形的稳定性，以及其他相关问题。解决这些问题的主要思想是在T. Mabuchi，理解Calabiflow的长期存在，以及Kaehler-Ricci流的收敛。这些想法与不同的数学领域有关，但提出者认为它们都很有希望解决丘氏猜想（特别是解决Fano流形上Kaehler-Einstein度量的存在性问题）。常数量曲率Kaehler度量的存在性问题是微分几何中的一个关键问题，对物理学等其他科学领域有着重要的影响。爱因斯坦认为，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具有极其重要的意义。这项研究也对弦理论产生了影响，弦理论是统一自然界所有四种基本力的理论。这位提议者的工作，以及其他数学家和物理学家的工作，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.