Mathematical Sciences: Research in Several Complex Variables

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The principal investigator will study three projects in the area of several complex variables. The first concerns imbedding strictly pseudoconvex CR manifolds in higher-dimensional spheres. In the second project he will generalize certain theorems about the Kobayashi metric and related notions from convex domains and CR manifolds. The third project involves the study of functions satisfying homogeneous complex Monge-Ampere equations which have a prescribed singular behavior along a smooth manifold. How surfaces and their higher-dimensional analogues, called manifolds, may lie in multi-dimensional space has been a subject of great interest to mathematicians for many decades. Certain surfaces, like the Klein bottle, cannot be placed in three- dimensional space without forcing self intersections. But this surface can be placed into four-dimensional space without such intersections. The principal investigator will study deep related questions for manifolds which are based, not on real multi-dimensional space, but on complex spaces.