权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Complex Monge-Ampere Equations and the Calabi Flow

复杂的 Monge-Ampere 方程和卡拉比流

基本信息

批准号：
1914719
负责人：
Xiuxiong Chen
金额：
$ 39.28万
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：
2019
资助国家：
美国
起止时间：
2019-06-15 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1914719&HistoricalAwards=false
关键词：
Complex Monge Ampere Equations Calabi

项目摘要

The problem of the existence of constant scalar curvature metric is the key problem in complex differential geometry and it has close ties with mathematical physics. For instance, the work of Calabi-Yau directly provided mathematical foundation to mirror symmetry. According to A. Einstein, the theory of gravity can be interpreted as the geometry of space-time. Therefore, the research in complex geometry is crucially important in physics and cosmology. This project also has impact in algebraic geometry, physics as well as partial differential equations. Progress in these problems will be highly interesting to many different fields.The 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 Yau in 1976 with vanishing first Chern class, and independently by Yau and Aubin for the case of negative first Chern class. For general Fano manifolds, S. T. Yau first suggested that the existence of Kaehler Einstein metric is related to certain notion of stability of the underlying polarization. This is proved by PI, S. K. Donaldson and S. Sun in 2012 via a series of three papers. The PI believe that more exciting progress will follow after these work in this area which will impact not only on the rest of mathematics but also on physics. The PI shall study a network of problems centered around the existence of constant scalar curvature metrics and other related areas. Here constant scalar curvature Kaehler metrics includes the more famous Kaehler-Einstein metric as a special case. So it is a natural extension of the original Calabi conjecture. In 2018, the PI and Cheng Jingrui made major progress on this problem by establishing important a priori estimate for constant scalar Kaehler metrics. The PI shall push this further towards the existence of constant scalar curvature Kaehler metrics by studying a network of problems related algebraic stability to coerciveness of the appropriate variational energy functional.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

常数量曲率度量的存在性问题是复微分几何中的核心问题，与数学物理有着密切的联系。例如，卡-丘的工作直接为镜像对称提供了数学基础。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，复几何的研究在物理学和宇宙学中具有极其重要的意义。这个项目也有影响代数几何，物理学以及偏微分方程。著名的Calabi猜想指出，每个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，它的标量曲率有适当的符号。这个著名的猜想是由丘在1976年在第一陈氏类为零的情况下证明的，而在第一陈氏类为负的情况下，丘和奥宾独立地证明了这个猜想.对于一般的Fano流形，S. T.丘德威首先提出，凯勒爱因斯坦度规的存在与潜在极化的稳定性有关。这是由PI，S. K.唐纳森和S. Sun在2012年通过一系列的三篇论文。PI相信，在这一领域的这些工作之后，将会有更多令人兴奋的进展，这不仅会影响到数学的其他部分，还会影响到物理学。PI应研究围绕常数标量曲率度量和其他相关领域存在的问题网络。在这里，常数标量曲率Kaehler度量包括更著名的Kaehler-Einstein度量作为特例。因此它是原卡拉比猜想的自然推广。2018年，PI和程景瑞通过建立常数标量Kaehler度量的重要先验估计，在这个问题上取得了重大进展。 PI将进一步推动这对常数标量曲率Kaehler度量的存在，通过研究一个网络的问题相关的代数稳定性，以适当的变分能量functional.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。