Partial Differential Equations and Several Complex Variables

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9703678 Stanton The supported project lies in the area of several complex variables; in particular, it involves the study of real hypersurfaces in complex Euclidean space. Specifically, the investigator plans to find a complete normal form for the defining equation of a rigid hypersurface in complex Euclidean space; find a necessary and sufficient condition for analyticity of infinitesimal Cauchy-Riemann automorphisms; study the evolution of a strictly convex hypersurface by the Levi flow; study the off-diagonal small time behavior of the heat kernel on a CR-manifold; understand the singularities of kernels arising in the study of the Neumann problem and the corresponding heat kernel. Real hypersurfaces in complex Euclidean space generalize surfaces in space: they are higher dimensional analogs of surfaces sitting inside a space equipped with a complex structure. Complex structures - locally, arrays of complex numbers - were first introduced in the study of polynomial equations; they continue to find many uses in various parts of mathematics, often leading to conceptual as well as computational simplifications.