Extremal Kahler metrics, the Kahler Ricci flow and the Calabi flow

极值 Kahler 度量、Kahler Ricci 流和 Calabi 流

• 批准号: 1211652
• 负责人: Xiuxiong Chen
• 金额: $ 35.8万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211652&HistoricalAwards=false
• 关键词: Extremal; Kahler; metrics; Ricci; flow

AbstractAward: DMS 1211652, Principal Investigator: Xiuxiong ChenThe famous Calabi conjecture states that every Kaehler manifold whose first Chern class has a definite sign will always have a Kaehler Einstein metric with appropriate sign on its scalar curvature. This famous conjecture was proved by S. T. Yau in 1976 when the first Chern class vanishes, and independently by Yau and Aubin for the case when the first Chern class is negative. When the first Chern class is positive, the corresponding Calabi conjecture is only solved in dimension 2, by G. Tian in 1989. The higher dimensional case is largely open till this day. There are substantial and major progress made in this important area of mathematics in recent years. We propose to study a network of problems centered around the existence of extremal Kaehler metrics (a cousin of Kaehler Einstein metrics), stability of the underlying polarized Kaehler manifolds, and other related areas . The principal investigator believes that we will see more breakthrough in near future.The problem of the existence of extremal Kaehler metrics is the key problem in differential geometry. It has strong impact to other fields of sciences, in particular, physics. According to A. Einstein, the theory of gravity can be interpreted as the geometry of space-time. Thus the research in Kaehler geometry is crucially important in physics and cosmology. The research proposed also has impact in string theory, in algebraic geometry as well as nonlinear analysis. Progress in this proposed problems will be highly interesting to many different fields.

AbstractAward：著名的卡拉比猜想指出，任何第一个陈类有一个确定的符号的Kaehler流形总是有一个Kaehler Einstein度量，其标量曲率有适当的符号。这个著名的猜想是由S. T. Yau在1976年证明了第一个Chern类为零时的情形，而Yau和奥宾独立地证明了第一个Chern类为负时的情形。当第一类Chern类为正时，相应的Calabi猜想仅在二维空间中被G. 1989年的田。 更高维度的情况直到今天都是开放的。 近年来，数学的这一重要领域取得了重大进展。 我们建议研究一个网络的问题围绕极值凯勒度量（凯勒爱因斯坦度量的表弟）的存在，稳定性的基础极化凯勒流形，和其他相关领域。 本文的主要研究者相信在不久的将来我们会看到更多的突破。极值Kaehler度量的存在性问题是微分几何中的关键问题。 它对其他科学领域，特别是物理学有很大的影响。 根据A.爱因斯坦的引力理论可以解释为时空的几何学。因此，Kaehler几何的研究在物理学和宇宙学中具有重要意义。这项研究也对弦理论、代数几何和非线性分析产生了影响。 在这个问题上的进展将是非常有趣的许多不同的领域。