Complex Monge-Ampere Equations and the Calabi Flow

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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.