Extremal Kahler metrics, the Kahler Ricci flow and the Calabi flow

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AbstractAward: DMS 1211652, Principal Investigator: Xiuxiong ChenThe 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 S. T. Yau in 1976 when the first Chern class vanishes, and independently by Yau and Aubin for the case when the first Chern class is negative. When the first Chern class is positive, the corresponding Calabi conjecture is only solved in dimension 2, by G. Tian in 1989. The higher dimensional case is largely open till this day. There are substantial and major progress made in this important area of mathematics in recent years. We propose to study a network of problems centered around the existence of extremal Kaehler metrics (a cousin of Kaehler Einstein metrics), stability of the underlying polarized Kaehler manifolds, and other related areas . The principal investigator believes that we will see more breakthrough in near future.The problem of the existence of extremal Kaehler metrics is the key problem in differential geometry. It has strong impact to other fields of sciences, in particular, physics. According to A. Einstein, the theory of gravity can be interpreted as the geometry of space-time. Thus the research in Kaehler geometry is crucially important in physics and cosmology. The research proposed also has impact in string theory, in algebraic geometry as well as nonlinear analysis. Progress in this proposed problems will be highly interesting to many different fields.