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Continuous Boundary Values of Holomorphic Functions on Kähler Domains

Published online by Cambridge University Press:  20 November 2018

Barnet M. Weinstock
Barnet M. Weinstock*
Affiliation:
University of Kentucky, Lexington, Kentucky
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Let M be a complex manifold of dimension n which admits a Kähler metric, and let D be a relatively compact domain on M whose boundary B is a C ∞ submanifold of M of real codimension one. The object of this paper is to use the potential theory associated with the Laplace-Beltrami operator on M to characterize the continuous functions on B which have holomorphic extensions to D.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 513 - 522
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Andreotti, A. and HillDenson, C., E. E. Levi convexity and the Hans Lewy problem, Part I: Reduction to vanishing theorems, Ann. Scuola Norm. Sup. Pisa 26 (1972), 325–363.Google Scholar
2. Aronszajn, N., Krzywicki, A. and Szarski, J., A unique continuation theorem for exterior differential forms on Riemannian manifolds, Arkiv Mat. 4 (1962), 417–453.Google Scholar
3. Bidal, P. and Georges de Rham, Les formes différentielles harmoniques, Comm. Mat. Helv. 19 (1946), 1–49.Google Scholar
4. Bochner, S., Analytic and meromorphic continuation by means of Green's formula, Ann. Math. U (1943), 652–673.Google Scholar
5. Dolbeault, P., Formes différentielles et cohomologie sur une variété analytique complexe, I, Ann. Math. 64 (1956), 83–130.Google Scholar
6. Duff, G. F. D., Boundary value problems associated with the tensor Laplace equation, Can. J. Math. 5 (1953), 196–210.Google Scholar
7. Duff, G. F. D. and Spencer, D. C., Harmonic tensors on Riemannian manifolds with boundary, Ann. Math. 56 (1952), 128–156.Google Scholar
8. Fichera, G., Caratterizzazione delta traccia sulla frontier a di un campo di unafunzione analitica di pitl variabili complesse, Atti Accad. Nat. Lincei Rend. (8) 22 (1957), 706–715.Google Scholar
9. Garabedian, P. R. and Spencer, D. C., A complex tensor calculus for Kàhler manifolds, Acta Math. 89 (1953), 279–331.Google Scholar
10. Grauert, H., Charakterisierung der Holomorphiegebiete durch die vollstàndige Kàhlersche metrik, Math. Ann. 131 (1956), 38–75.Google Scholar
11. Harvey, F. R. and Blaine Lawson, H., Jr., Boundaries of complex analytic varieties, Bull. Amer. Math. Soc. 80 (1974), 180–183.Google Scholar
12. Kelley, J. L. and Isaac Namioka et al., Linear topological spaces (D. Van Nostrand, Princeton, N. J., 1963).Google Scholar
13. Kohn, J. J. and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math. 81 (1965), 451–472.Google Scholar
14. Malgrange, B., Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier Grenoble 6 (1955-56), 271–355.Google Scholar
15. Martinelli, E., Sopra una dimonstrazione di R. Fueter per un teorema di Hartogs, Comm. Mat. Helv. 15 (1942), 340–349.Google Scholar
16. Morrey, C. B., Jr., Multiple integrals in the calculus of variations (Springer-Verlag, New York, 1966).Google Scholar
17. Morrow, J. and Kodaira, K., Complex manifolds (Holt, Rinehart and Winston, New York, 1971).Google Scholar
18. de Rham, G., Variétés differentiables (Hermann, Paris, 1960).Google Scholar
19. Royden, H., The boundary values of analytic and harmonic functions, Math. Z. 78 (1962), 1–24.Google Scholar
20. Serre, J.-P., Une théorème de dualité, Comm. Mat. Helv. 29 (1955), 9–26.Google Scholar
21. Spencer, D. C., Cauchy's formula on Kàhler manifolds, Proc. Nat. Acad. Sci. 38 (1952), 76–80.Google Scholar
22. Weinstock, B., Continuous boundary values of analytic functions of several complex variables, Proc. Amer. Math. Soc. 21 (1969), 463–466.Google Scholar
23. Weinstock, B., An approximation theorem for d-closed forms of type (n, n — 1), Proc. Amer. Math. Soc. 26 (1970), 625–628.Google Scholar
24. Wells, R. O., Jr., Differential analysis on complex manifolds. (Prentice-Hall, Englew∞d Cliffs, N. J., 1973).Google Scholar
25. Yano, K. and Salomon Bochner, Curvature and Betti numbers, Annals of Mathematics Studies 32 (Princeton University Press, Princeton, N. J., 1953).Google Scholar