As a branch of Mathematics, Differential Geometry studies the shapes of spaces through distances and angles. The key concept involved is that of a so-called curvature, the simplest kind being the scalar curvature function on a space. The extremal cases are often the most interesting to study. In particular, the existence of a constant scalar curvature metric is of a major importance in Differential Geometry. It has a strong impact on other fields of sciences such as physics. For instance, the work of Calabi-Yau directly provided a mathematical foundation in mirror symmetry. According to A. Einstein, the theory of gravity can be interpreted as the geometry of the space-time. Therefore, the research in complex geometry is crucially important in physics and cosmology. The research proposed here also has impacts on algebraic geometry and partial differential equations. The project has an integrated education component which will enable the PI to continue supporting graduate students financially to pursue their research.
In recent years, striking progress has been made in Kaehler geometry, particularly on the existence of the Kaehler-Einstein metrics and the limiting behavior of the Kaehler-Ricci flow solution, both in Fano manifolds. More exciting progress will follow after these works in this and adjacent area. In Kaehler geometry, the focus of the field is now on the existence of constant scalar curvature Kaehler metrics which is more general and harder than the existence of the Kaehler-Einstein metrics. This program on constant scalar curvature Kaehler metrics, which was first proposed by E. Calabi in 1950s, amounts to solving a 4th-order partial differential equation which naturally interacts with metric geometry as well as algebraic geometry. In this project, the PI will study a network of problems centering around the existence of constant scalar curvature metrics and other related areas. These include some fundamental problems in complex Monger-Ampere equations, a priori estimates for constant scalar curvature metrics, metric geometry as well as geometric flow (the Calabi flow and the Kaehler-Ricci flow).