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Geometric Analysis of the Bergman Kernel and Metric pp 71–85Cite as

The Bergman Metric

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Part of the book series: Graduate Texts in Mathematics ((GTM,volume 268))

Abstract

Poincaré’s theorem (see [KRA1, KRA9] for discussion) that the ball and polydisc are biholomorphically inequivalent shows that there is no Riemann mapping theorem (at least in the traditional sense) in several complex variables. More recent results of Burns, Shnider, and Wells [BSW] and of Greene and Krantz [GRK1, GRK2] confirm how truly dismal the situation is. First, we need a definition.

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Notes

  1. 1.

    For the readers’s convenience, we recall here that the Sobolev embedding theorem says that if a function on \({\mathbb{R}}^{N}\) has more than N ∕ 2 derivatives in L 2, then in fact it has a continuous derivative. See [STE1], for instance, for the details.

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    Steven G. Krantz

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Krantz, S.G. (2013). The Bergman Metric. In: Geometric Analysis of the Bergman Kernel and Metric. Graduate Texts in Mathematics, vol 268. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7924-6_2

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