Extremal Kaehler Metrics and Geometric Flow Equations

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DMS-0406346Title: Extremal Kaehler metrics and geometric flow equationsPI: Xiuxiong Chen, University of Wisconsin (Madison)ABSTRACTIn the early nineties, S. T. Yau conjectured that the existence of Kaehler metrics with constant scalar curvature is related to the stability of the underlying polarized manifold. In recent years, striking progress has been made in this direction. In 1997, G. Tian proved that any Kaehler-Einstein manifold with positive scalar curvature is K-Stable. In2001, S. K. Donaldson proved that an algebraic manifold with discrete automorphism group and Kaehler metric of constant scalar curvature is Chow stable.Very recently, in a joint work with G. Tian, the proposer proved that the existence of Kaehler metrics of constant scalar curvature implies Semi-KStability of the K energy with respect to the given cohomology class. Inspired by these results, the proposer wants to study a network of problems centered around the existence of extremal Kaehler metrics (which includes constant scalar curvature metric as a special case), stability of polarized Kaehler manifold, and other related problems. The main ideas of solving these problemsconsist of improving regularity for geodesics in the space of Kaehler metrics in the sense of T. Mabuchi, of understanding the long-time existence of the Calabiflow, and of the convergence of the Kaehler-Ricci flow. These ideas are related to different mathematical fields but the proposer believes that they are all very promising to solve the Conjecture of Yau (in particular to solve the problem of the existence of Kaehler-Einstein metrics on Fano manifold). The proposer is going to work on the problems through these ideas.The problem of the existence of Kaehler metrics of constantscalar curvature, being the key problem in differential geometry, has strong impact to other fields of sciences like physics. According to AlbertEinstein, the theory of gravity can be interpreted as the geometry of space-time. Thus the research in differential geometry is crucially important in physics and cosmology. The research proposed also has impact in string theory, which is the theory of unifying all four basic forces of the Nature. The proposer's work, together with the works of other mathematicians and physicists, helps in understanding our Universe.