A deformation theorem for the Kobayashi metric

Kalka, M.

doi:10.1090/S0002-9939-1976-0412481-4

A deformation theorem for the Kobayashi metric
HTML articles powered by AMS MathViewer

by M. Kalka

Proc. Amer. Math. Soc. 59 (1976), 245-251

DOI: https://doi.org/10.1090/S0002-9939-1976-0412481-4

PDF | Request permission

Abstract:

Let ${M_0}$ be a compact hyperbolic complex manifold. It is shown that the infinitesimal Kobayashi metric is upper semicontinuous in a ${C^\infty }$ deformation parameter $t \in U \subseteq {R^k}$. This is accomplished by proving deformation theorems for holomorphic maps.

References

Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
Phillip A. Griffiths, Differential geometry and complex analysis, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 43–64. MR 0399521
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 125–137. MR 0304694
H. L. Royden, The extension of regular holomorphic maps, Proc. Amer. Math. Soc. 43 (1974), 306–310. MR 335851, DOI 10.1090/S0002-9939-1974-0335851-X
Chester Seabury, Some extension theorems for regular maps of Stein manifolds, Bull. Amer. Math. Soc. 80 (1974), 1223–1224. MR 350072, DOI 10.1090/S0002-9904-1974-13688-8

The Kobayashi pseudo-metric on algebraic manifolds of general type and in deformations of complex manifolds