Complex Monge-Ampere Equations and the Calabi Flow

常标量曲率度量的存在性问题是复杂微分几何中的关键问题，它与数学物理有着密切的联系。例如，Calabi-Yau的工作直接为镜像对称提供了数学基础。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，复杂几何的研究在物理学和宇宙学中具有至关重要的意义。这个项目对代数几何、物理和偏微分方程也有影响。这些问题的进展将引起许多不同领域的极大兴趣。著名的卡拉比猜想指出，每一个第一个陈氏类具有确定符号的凯勒流形，其标量曲率上总会有一个带有适当符号的凯勒-爱因斯坦度规。这个著名的猜想由Yau在1976年用消失的第一陈氏类证明，并由Yau和Aubin独立地证明了负第一陈氏类的情况。对于一般的Fano流形，S. T. Yau首先提出了Kaehler - Einstein度规的存在与底层极化的一定稳定性概念有关。这是PI， S. K. Donaldson和S. Sun在2012年通过一系列三篇论文证明的。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，这不仅会影响数学的其他领域，也会影响物理学。PI应研究一个以常数标量曲率度量的存在性和其他相关领域为中心的问题网络。这里常数标量曲率卡勒度规包括了更著名的卡勒-爱因斯坦度规作为特例。所以它是原始卡拉比猜想的自然延伸。2018年，PI和程景瑞在这一问题上取得了重大进展，建立了常数标量Kaehler度量的重要先验估计。PI将通过研究代数稳定性与适当变分能量泛函的强制性相关的问题网络，进一步推动这一点，使其朝着常数标量曲率Kaehler度量的存在性发展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.