Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-20T22:32:46.492Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

Extract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bochner, S.Analytic and meromorphic continuation by means of Green's formula. Ann. of Math. 44 (1943), 652–673.Google Scholar
(2)Hamilton, R. S.Harmonic maps of manifolds with boundary (Springer Lecture Notes, no. 471, 1975).Google Scholar
(3)Harvey, F. R. and Lawson, H. B.Boundaries of complex analytic varieties. Bull. Amer. Math. Soc. 80 (1974), 180–183.CrossRefGoogle Scholar
(4)Harvey, F. R. and Lawson, H. B.On boundaries of complex analytic varieties. I. Ann. of Math. 102 (1975), 223–290.CrossRefGoogle Scholar
(5)Kohn, J. J. and Rossi, H.On the extension of holomorphic functions from the boundary of a complex manifold. Ann. of Math. 81 (1965), 451–472.Google Scholar
(6)Mostow, G. D. and Siu, Y.-T.A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math.Google Scholar
(7)Polking, J. C. and Wells, R. O.Boundary values of Dolbeault Cohomology classes and a generalized Bochner-Hartogs Theorem. Abhandlungen Math. Sem. Univ. Hamburg, 47 (1978), 3–24.CrossRefGoogle Scholar
(8)Siu, Y.-T.The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math.Google Scholar
(9)Schoen, R. M. Existence and regularity for some geometric variational problems (Thesis, Stanford University, 1977).Google Scholar
(10)Weinstock, B. M.An approximation theorem for ∂¯-closed forms of type (n, n − 1). Proc. Amer. Math. Soc. 26 (1970), 625–628.Google Scholar