Geometry of extremal Kahler metrics and geometric flows in Kahler settings

近年来，卡勒几何学取得了令人瞩目的进展。著名的Calabi猜想指出，每个第一个Chern类C_1有确定符号的Kahler流形都有一个Kahler爱因斯坦度规，在它的标量曲率上有适当的符号。1976年，当C_1=0时，著名的猜想由Yau证明，而对于C_1=0，则由Yau和Aubin独立证明。在Fano曲面上，相应的Calabi猜想是由G.Tian在1989年解出的。直到今天，高维情况下的情况仍然很大程度上是开放的。这是一个令人兴奋的数学领域，在不久的将来，我们将看到更多重要的突破，对数学的其余部分和物理学都产生影响。例如，Calabi-Yau的工作直接为镜像所谓的Calabi-Yau流形中的对称性提供了数学基础。因此，我们建议研究一个围绕极值Kahler度量的存在性、基本极化Kahler流形的稳定性以及其他相关领域的问题网络。极端卡勒指标是著名的Calabi-Yau指标的自然延伸。这位首席研究员参与了威斯康星大学和其他地方许多研究生的教育和监督。建议积极促进美国、欧洲和亚洲(特别是中国、日本和韩国)数学家的交流。他在中国组织了几次国际会议，邀请了来自不同背景的数学家。因此，这项建议将增强我们培养美国下一代数学家的能力。更广泛地说，我们在这里提出的一组问题是微分几何中的关键问题，这对其他科学领域，如物理学具有强烈的影响。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。

In recent years, striking progress has been made in Kahler geometry. The famous Calabi conjecturestates that every Kahler manifold whose first Chern class C_1 has a definite sign will always have a Kahler Einstein metric with appropriate sign on its scalar curvature. This famous conjecture was proved by Yau in 1976 when C_1 =0, and independently by Yau and Aubin for the caseC_1 0. In Fano surface, the corresponding Calabi conjecture is solved by G. Tian in 1989. The high dimensional case is largely open till this day. This is an exciting area of mathematics in which we will see more important breakthroughs in the near future with impact both on the rest of mathematics and on physics. For instance, the work of Calabi-Yau directly provided mathematical foundation to mirror symmetry in the so called Calabi-Yau manifold. Therefore, we propose to study a network of problems centered around the existence of extremal Kahler metric, stability of the underlying polarized Kahler manifold, and other related areas. Extremal Kahler metrics is natural extension to the renown Calabi-Yau metrics.The principal investigator has been involved in the education and supervision of many graduate students at the University of Wisconsin and elsewhere. The proposer actively promotes the interaction of mathematicians from USA, Europe and Asian (particularly China, Japan and S.Korea). He has organized several international conferences in China where mathematicians from different background are invited. So this proposal will enhance our ability to train next generation of US based mathematicians.More broadly, The set of problems we propose here is key problems in differential geometry, which has strong impact to other fields of sciences like physics. According to Albert Einstein, 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 proposer's work, together with the works of other mathematicians and physicists, helps in understanding our Universe.