Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains

Yu, Ji Ye

doi:10.1090/S0002-9947-1995-1276938-5

Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains
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by Ji Ye Yu

Trans. Amer. Math. Soc. 347 (1995), 587-614

DOI: https://doi.org/10.1090/S0002-9947-1995-1276938-5

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Abstract:

The purpose of this paper is to study the existence of weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains in ${\mathbb {C}^n}$ and to explore the connections between the limits and the Levi invariants. The main result extends Graham’s result on strongly pseudoconvex domains to a large class of weakly pseudoconvex domains.

References

Brian E. Blank, Da Shan Fan, David Klein, Steven G. Krantz, Daowei Ma, and Myung-Yull Pang, The Kobayashi metric of a complex ellipsoid in $\textbf {C}^2$, Experiment. Math. 1 (1992), no. 1, 47–55. MR 1181086
Eric Bedford and Sergey Pinchuk, Domains in $\textbf {C}^{n+1}$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), no. 3, 165–191. MR 1120679, DOI 10.1007/BF02921302
Kang-Tae Kim, Domains with noncompact automorphism groups, Recent developments in geometry (Los Angeles, CA, 1987) Contemp. Math., vol. 101, Amer. Math. Soc., Providence, RI, 1989, pp. 249–262. MR 1034985, DOI 10.1090/conm/101/1034985
David W. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), no. 3, 429–466. MR 978601, DOI 10.1007/BF01215657
David Catlin, Boundary invariants of pseudoconvex domains, Ann. of Math. (2) 120 (1984), no. 3, 529–586. MR 769163, DOI 10.2307/1971087

Estimates of the invariant metrics on convex domains

Sanghyun Cho, A lower bound on the Kobayashi metric near a point of finite type in $\textbf {C}^n$, J. Geom. Anal. 2 (1992), no. 4, 317–325. MR 1170478, DOI 10.1007/BF02934584
John P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. MR 657241, DOI 10.2307/2007015

Several complex variables and geometry

Klas Diederich and John E. Fornæss, Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 110 (1979), no. 3, 575–592. MR 554386, DOI 10.2307/1971240
Jean-Pierre Demailly, Un exemple de fibré holomorphe non de Stein à fibre $\textbf {C}^{2}$ ayant pour base le disque ou le plan, Invent. Math. 48 (1978), no. 3, 293–302 (French). MR 508989, DOI 10.1007/BF01390248
Klas Diederich, Gregor Herbort, and Takeo Ohsawa, The Bergman kernel on uniformly extendable pseudoconvex domains, Math. Ann. 273 (1986), no. 3, 471–478. MR 824434, DOI 10.1007/BF01450734
Klas Diederich and Ingo Lieb, Konvexität in der komplexen Analysis, Birkhäuser, Boston, Mass., 1981 (German). Neue Ergebnisse und Methoden. [New results and methods]; DMV Seminar, 2. MR 698605, DOI 10.1007/978-3-0348-5153-4
Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
Franc Forstnerič and Jean-Pierre Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987), no. 2, 239–252. MR 919504, DOI 10.1007/BF01461721
Sidney Frankel, Applications of affine geometry to geometric function theory in several complex variables. I. Convergent rescalings and intrinsic quasi-isometric structure, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 183–208. MR 1128543, DOI 10.1090/pspum/052.2/1128543
John Erik Fornæss and Nessim Sibony, Construction of P.S.H. functions on weakly pseudoconvex domains, Duke Math. J. 58 (1989), no. 3, 633–655. MR 1016439, DOI 10.1215/S0012-7094-89-05830-4
Robert E. Greene and Steven G. Krantz, Stability of the Carathéodory and Kobayashi metrics and applications to biholomorphic mappings, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 77–93. MR 740874, DOI 10.1090/pspum/041/740874
Robert E. Greene and Steven G. Krantz, Deformation of complex structures, estimates for the $\bar \partial$ equation, and stability of the Bergman kernel, Adv. in Math. 43 (1982), no. 1, 1–86. MR 644667, DOI 10.1016/0001-8708(82)90028-7
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
Gregor Herbort, Invariant metrics and peak functions on pseudoconvex domains of homogeneous finite diagonal type, Math. Z. 209 (1992), no. 2, 223–243. MR 1147815, DOI 10.1007/BF02570831
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Monique Hakim and Nessim Sibony, Quelques conditions pour l’existence de fonctions pics dans des domaines pseudoconvexes, Duke Math. J. 44 (1977), no. 2, 399–406 (French). MR 457784
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
M. Kalka, A deformation theorem for the Kobayashi metric, Proc. Amer. Math. Soc. 59 (1976), no. 2, 245–251. MR 412481, DOI 10.1090/S0002-9939-1976-0412481-4
Kang-Tae Kim, Complete localization of domains with noncompact automorphism groups, Trans. Amer. Math. Soc. 319 (1990), no. 1, 139–153. MR 986028, DOI 10.1090/S0002-9947-1990-0986028-X
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, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365

Fonctions plurisubharmonique d’exhaustion bornées et domaines taut

257

Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
Steven G. Krantz, The boundary behavior of the Kobayashi metric, Rocky Mountain J. Math. 22 (1992), no. 1, 227–233. MR 1159955, DOI 10.1216/rmjm/1181072807

Geometric analysis and function spaces

Steven G. Krantz, Invariant metrics and the boundary behavior of holomorphic functions on domains in $\textbf {C}^n$, J. Geom. Anal. 1 (1991), no. 2, 71–97. MR 1113372, DOI 10.1007/BF02938115

Boundary behavior of the Bergman curvature in the strongly pseudoconvex polyhedral domains

László Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427–474 (French, with English summary). MR 660145, DOI 10.24033/bsmf.1948
Jeffery D. McNeal, Local geometry of decoupled pseudoconvex domains, Complex analysis (Wuppertal, 1991) Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991, pp. 223–230. MR 1122183

Peak functions for pseudoconvex domains

Sergey Pinchuk, The scaling method and holomorphic mappings, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 151–161. MR 1128522, DOI 10.1090/pspum/052.1/1128522
I. P. Ramadanov, Some applications of the Bergman kernel to geometrical theory of functions, Complex analysis (Warsaw, 1979) Banach Center Publ., vol. 11, PWN, Warsaw, 1983, pp. 275–286. MR 737768

Carathéodory and Kobayashi metric on convex domains

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
Nessim Sibony, Some aspects of weakly pseudoconvex domains, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 199–231. MR 1128526, DOI 10.1090/pspum/052.1/1128526

A class of hyperbolic manifolds

10

Ji Ye Yu, Multitypes of convex domains, Indiana Univ. Math. J. 41 (1992), no. 3, 837–849. MR 1189914, DOI 10.1512/iumj.1992.41.41044

Invariant metrics and finite type conditions

Stability of the Bergman kernel and metric

Peak functions on weakly pseudoconvex domains

Geometric analysis on weakly pseudoconvex domains