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An extension theorem for holomorphic mappings

Published online by Cambridge University Press:  24 October 2008

J. C. Wood
J. C. Wood
Affiliation:
University of Leeds
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Let X, Y be complex manifolds with smooth (C∞) boundaries ∂X, ∂Y. We give conditions which ensure that a smooth map Φ: ∂X → ∂Y has an extension to a holomorphic map X → Y. Let J denote the complex structure on X. We say that Φ satisfies the ‘tangential Cauchy-Riemann equation ∂¯bΦ = 0’if the differential dΦ restricted to the complex subspace Tp(∂X) ∩ JTp(∂X) of the tangent space Tp(∂X) is complex linear at all points p ∈ ∂X. Clearly this is a necessary condition for the existence of a holomorphic extension. A further necessary condition is that there exists no topological obstruction to extension, hence we assume that a smooth extension φ: X → Y is given and we shall look for a holomorphic map f: X → Y with the same boundary values.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 125 - 127
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

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