Global Analysis on Riemannian Manifolds

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9703656 Shi This project lies in the area of Riemannian and Kahler geometry. More specifically, the investigator is to use the Ricci flow to study various problems in Riemannian and Kahler geometry. For example, the investigator wishes to show that a complete noncompact Kahler manifold with positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space, a well-known problem in the area. Rimannian manifolds are higher dimensional generalizations of curved surfaces; Kahler manifolds are a complex analog of Riemannian manifolds. Physicists often attempt to describe the universe as a Riemannian or Kahler manifold possessing certain additional properties. The idea behind the Ricci flow is to vary the given metric - a metric defines the notion of distance on the underlying manifold - over time in a special way and to see how the resulting family of metrics interact with the topology or shape properties of the manifold.