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

著名的Calabi猜想指出，每一个第一类具有确定符号的Kaehler流形，其标量曲率上总是有一个带有适当符号的Kaehler爱因斯坦度量。这个著名的猜想是由S. T. Yau于1976年在第一个陈氏类消失的情况下证明的，并由Yau和Aubin分别在第一个陈氏类为负的情况下证明。当第一个Chern类为正时，对应的Calabi猜想仅在2维中由G. Tian（1989）求解。高维的情况在很大程度上是开放的，直到今天。近年来，这一重要的数学领域取得了实质性的重大进展。我们建议研究一个以极值Kaehler度量（Kaehler - Einstein度量的表亲）的存在性、底层极化Kaehler流形的稳定性以及其他相关领域为中心的问题网络。首席研究员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。根据爱因斯坦的理论，引力理论可以解释为时空的几何学。因此，对凯勒几何的研究在物理学和宇宙学中是至关重要的。所提出的研究在弦理论、代数几何和非线性分析中也有影响。这一问题的进展将引起许多不同领域的极大兴趣。

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.