Some remarks on the intrinsic measures of Eisenman

Graham, Ian; Wu, H.

doi:10.1090/S0002-9947-1985-0776396-4

Some remarks on the intrinsic measures of Eisenman
HTML articles powered by AMS MathViewer

by Ian Graham and H. Wu

Trans. Amer. Math. Soc. 288 (1985), 625-660

DOI: https://doi.org/10.1090/S0002-9947-1985-0776396-4

PDF | Request permission

Abstract:

This paper studies the intrinsic measures on complex manifolds first introduced by Eisenman in analogy with the intrinsic distances of Kobayashi. Some standard conjectures, together with several new ones, are considered and partial or complete answers are provided. Most of the counterexamples come from a closer examination of unbounded domains in complex euclidean space. In particular, a large class of unbounded hyperbolic domains are exhibited. Those unbounded domains of finite euclidean volume are also singled out for discussion.

References

Errett Bishop, Conditions for the analyticity of certain sets, Michigan Math. J. 11 (1964), 289–304. MR 168801
S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), 179–195. MR 19983, DOI 10.1090/S0002-9904-1947-08778-4
Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219. MR 470252, DOI 10.1090/S0002-9947-1978-0470252-3
Robert Brody and Mark Green, A family of smooth hyperbolic hypersurfaces in $P_{3}$, Duke Math. J. 44 (1977), no. 4, 873–874. MR 454080
L. A. Campbell, A. Howard, and T. Ochiai, Moving holomorphic disks off analytic subsets, Proc. Amer. Math. Soc. 60 (1976), 106–108 (1977). MR 425186, DOI 10.1090/S0002-9939-1976-0425186-0
James A. Carlson, Some degeneracy theorems for entire functions with values in an algebraic variety, Trans. Amer. Math. Soc. 168 (1972), 273–301. MR 296356, DOI 10.1090/S0002-9947-1972-0296356-0
Klas Diederich and Nessim Sibony, Strange complex structures on Euclidean space, J. Reine Angew. Math. 311(312) (1979), 397–407. MR 549981
Donald A. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the American Mathematical Society, No. 96, American Mathematical Society, Providence, R.I., 1970. MR 0259165
Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219–240. MR 372252, DOI 10.1090/S0002-9947-1975-0372252-8
Hans Grauert and Helmut Reckziegel, Hermitesche Metriken und normale Familien holomorpher Abbildungen, Math. Z. 89 (1965), 108–125 (German). MR 194617, DOI 10.1007/BF01111588
Mark Green and Phillip Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 41–74. MR 609557
R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983
Phillip A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 185–251. MR 0258070
Phillip A. Griffiths, Holomorphic mapping into canonical algebraic varieties, Ann. of Math. (2) 93 (1971), 439–458. MR 281954, DOI 10.2307/1970883
Phillip A. Griffiths, Two theorems on extensions of holomorphic mappings, Invent. Math. 14 (1971), 27–62. MR 293123, DOI 10.1007/BF01418742
Phillip A. Griffiths, Entire holomorphic mappings in one and several complex variables, Annals of Mathematics Studies, No. 85, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. The fifth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N. J., October and November 1974. MR 0447638
Peter Kiernan, Extensions of holomorphic maps, Trans. Amer. Math. Soc. 172 (1972), 347–355. MR 318519, DOI 10.1090/S0002-9947-1972-0318519-8
Peter Kiernan, Holomorphic extension theorems, Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972-1973) Dekker, New York, 1974, pp. 97–107. MR 0352547
Shoshichi Kobayashi and Peter Kiernan, Holomorphic mappings into projective space with lacunary hyperplanes, Nagoya Math. J. 50 (1973), 199–216. MR 326007
Shoshichi Kobayashi, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460–480. MR 232411, DOI 10.2969/jmsj/01940460
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Shoshichi Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), no. 3, 357–416. MR 414940, DOI 10.1090/S0002-9904-1976-14018-9
Shoshichi Kobayashi and Takushiro Ochiai, Mappings into compact manifolds with negative first Chern class, J. Math. Soc. Japan 23 (1971), 137–148. MR 288316, DOI 10.2969/jmsj/02310137
K. Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds, J. Differential Geometry 6 (1971/72), 33–46. MR 301228
Myung H. Kwack, Generalization of the big Picard theorem, Ann. of Math. (2) 90 (1969), 9–22. MR 243121, DOI 10.2307/1970678
Junjiro Noguchi, Meromorphic mappings into a compact complex space, Hiroshima Math. J. 7 (1977), no. 2, 411–425. MR 457791
Donald Alfred Pelles, Holomorphic maps which preserve intrinsic measure, Amer. J. Math. 97 (1975), 1–15. MR 367300, DOI 10.2307/2373658
Reinhold Remmert and Karl Stein, Über dei wesentlichen Singularitäten analytischer Mengen, Math. Ann. 126 (1953), 263–306 (German). MR 60033, DOI 10.1007/BF01343164
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
Bernard Shiffman, Holomorphic and meromorphic mappings and curvature, Math. Ann. 222 (1976), no. 2, 171–194. MR 430323, DOI 10.1007/BF01418326
Nessim Sibony, A class of hyperbolic manifolds, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 357–372. MR 627768
Yum Tong Siu, Extension of meromorphic maps into Kähler manifolds, Ann. of Math. (2) 102 (1975), no. 3, 421–462. MR 463498, DOI 10.2307/1971038
Yum Tong Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), no. 1, 89–100. MR 435447, DOI 10.1007/BF01390170
Wilhelm Stoll, Über die Fortsetzbarkeit analytischer Mengen endlichen Oberflächeninhaltes, Arch. Math. 9 (1958), 167–175 (German). MR 101923, DOI 10.1007/BF01900495
H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. MR 224869, DOI 10.1007/BF02392083
H. Wu, Remarks on the first main theorem in equidistribution theory. I, J. Differential Geometry 2 (1968), 197–202. MR 276500
Hung-hsi Wu, The equidistribution theory of holomorphic curves, Annals of Mathematics Studies, No. 64, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0273070
H. Wu, An elementary method in the study of nonnegative curvature, Acta Math. 142 (1979), no. 1-2, 57–78. MR 512212, DOI 10.1007/BF02395057

Function theory on noncompact Kähler manifolds

AMS Website Down

Transactions of the American Mathematical Society

Some remarks on the intrinsic measures of Eisenman
HTML articles powered by AMS MathViewer

Abstract: