Complex Monge-Ampere Equations and the Calabi Flow

常标量曲率度量的存在性问题是复微分几何中的核心问题，它与数学物理有着密切的联系。例如，卡拉比-尤的工作直接为镜像对称性提供了数学基础。根据爱因斯坦的观点，引力理论可以解释为时空的几何学。因此，复杂几何的研究在物理学和宇宙学中都是至关重要的。这个项目对代数几何、物理以及偏微分方程式也有影响。这些问题的进展对许多不同的领域来说都是非常有趣的。著名的Calabi猜想指出，每个第一个Chern类具有确定符号的Kaehler流形总是有一个在其标量曲率上具有适当符号的Kaehler-Einstein度量。这个著名的猜想在1976年由Yau用消失的第一类证明，并由Yau和Aubin在负第一类的情况下独立证明。对于一般的Fano流形，S.T.Yau首先提出Kaehler爱因斯坦度规的存在与基本极化稳定性的某些概念有关。2012年，Pi、S.K.Donaldson和S.Sun通过一系列三篇论文证明了这一点。国际数学联合会相信，在这一领域的这些工作之后，将取得更令人振奋的进展，这不仅将对数学的其余部分产生影响，而且将对物理产生影响。PI应研究以恒定标量曲率度量和其他相关领域的存在为中心的问题网络。这里，恒定标量曲率的Kaehler度量包括更著名的Kaehler-Einstein度量作为特例。因此，这是最初的卡拉比猜想的自然延伸。2018年，PI和程景瑞通过建立常量Kaehler度量的重要先验估计，在这个问题上取得了重大进展。PI应通过研究与适当变分能量函数的强制性有关的代数稳定性的问题网络，进一步推动恒定标量曲率Kaehler度量的存在。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.