Complex Analysis-1986 by S.G. Krantz

By S.G. Krantz

This convention accumulated jointly a small team of individuals with related pursuits within the geometric functionality concept of numerous advanced variables. whereas the speeches have been of a really expert nature, the papers within the lawsuits are principally of a survey and speculative nature. the amount is meant to serve either scholars and researchers as a call for participation to energetic new parts of analysis. the extent of the writing has been deliberately set in any such method that the papers should be obtainable to a vast viewers.

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This makes the complement of a compact into a product manifold, and defines a natural smooth structure on the boundary. The bounds on the curvature and the covariant derivatives of ¢ can be used to show that the metric ~)iT extends smoothly to a nondegenerate metric on the boundary, and the complex structure extends smoothly to a strongly pseudoconvex CR structure on the boundary. The resulting compactification is a complete finite strongly pseudoconvex manifold with a smooth K~thler metric ~)i~" The imbedding results of Rossi and Taylor apply to complete the proof.

1) ui(z) K2(fi(z),w) = Kl(Z,Fi(w)) Ui(w) . Let h(z) be any function in A°°(£22) and let s be a positive integer. Fact 1 together with the transformation formula for the Bergman kernels implies that there exist finitely many points {wj} in 02 with the following property. Given any point p E £21, there exist constants cj=cj(fi(p)) such that h(z) agrees to order s with the function H cj K2(z,w j) j=l at fi(p). The constants cj are uniformly bounded independent of the choice of p. 1) implies that the function order s with the function ui(z) h(fi(z)) agrees to 35 M ]"h(Z) = ~.

He said that a pseudoconvex domain D c c C n satisfied condition P if for every c > 0, there exists a smooth, pseudoconvex function ~ on D such that 0_<(~_< 1 on D and [~)ijq>_C[~ij] on ~D. In [14], Catlin showed that if D c c C n is weakly pseudoconvex of finite type (as in the sense of D'Angelo), then D satisfies condition P. With these preliminaries out of the way, we will now summarize what is known for weakly pseudoconvex domains. As in the strongly pseudoconvex case, we begin with the biholomorphic invariance of the boundary.

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