Analysis of the Wu metric II: The case of non-convex Thullen domains

Cheung, C.; Kim, K.

doi:10.1090/S0002-9939-97-03695-2

Analysis of the Wu metric II: The case of non-convex Thullen domains
HTML articles powered by AMS MathViewer

by C. K. Cheung and K. T. Kim PDF

Proc. Amer. Math. Soc. 125 (1997), 1131-1142 Request permission

Abstract:

We present an explicit description of the Wu invariant metric on the non-convex Thullen domains. We show that the Wu invariant Hermitian metric, which in general behaves as nicely as the Kobayashi metric under holomorphic mappings, enjoys better regularity in this case. Furthermore, we show that the holomorphic curvature of the Wu metric is bounded from above everywhere by $-1/2$. This leads the Wu metric to be a natural solution to a conjecture of Kobayashi in the case of non-convex Thullen domains.

References

L. Ahlfors, An extension of Schwarz’s lemma, Trans. Am. Math. Soc. 43 (1938), 359–364.
Kazuo Azukawa, Negativity of the curvature operator of a bounded domain, Tohoku Math. J. (2) 39 (1987), no. 2, 281–285. MR 887943, DOI 10.2748/tmj/1178228330
Kazuo Azukawa and Masaaki Suzuki, The Bergman metric on a Thullen domain, Nagoya Math. J. 89 (1983), 1–11. MR 692340, DOI 10.1017/S0027763000020213
Stefan Bergman, The kernel function and conformal mapping, Second, revised edition, Mathematical Surveys, No. V, American Mathematical Society, Providence, R.I., 1970. MR 0507701
John S. Bland, The Einstein-Kähler metric on $\{|\textbf {z}|^2+|w|^{2p}<1\}$, Michigan Math. J. 33 (1986), no. 2, 209–220. MR 837579, DOI 10.1307/mmj/1029003350
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
Shiu Yuen Cheng and Shing Tung Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1980), no. 4, 507–544. MR 575736, DOI 10.1002/cpa.3160330404
C. K. Cheung and Kang-Tae Kim, Analysis of the Wu metric. I. The case of convex Thullen domains, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1429–1457. MR 1357392, DOI 10.1090/S0002-9947-96-01642-X
Chi-Keung Cheung and H. Wu, Some new domains with complete Kähler metrics of negative curvature, J. Geom. Anal. 2 (1992), no. 1, 37–78. MR 1140897, DOI 10.1007/BF02921334
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
K. T. Hahn and P. Pflug, The Kobayashi and Bergman metrics on generalized Thullen domains, Proc. Amer. Math. Soc. 104 (1988), no. 1, 207–214. MR 958068, DOI 10.1090/S0002-9939-1988-0958068-4
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
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
K. Kim and J. Yu, Boundary behavior of the Bergman curvature in the strictly pseudoconvex polyhedral domains, Pacific J. Math. (to appear).
Paul F. Klembeck, Kähler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ. Math. J. 27 (1978), no. 2, 275–282. MR 463506, DOI 10.1512/iumj.1978.27.27020
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
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
Yung-chen Lu, Holomorphic mappings of complex manifolds, J. Differential Geometry 2 (1968), 299–312. MR 250243
P. Pflug and W. Zwonek, The Kobayashi metric for non-convex complex ellipsoids, Complex Variables, Theory and its appl. 29 (1996), 59–71.
H. L. Royden, The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv. 55 (1980), no. 4, 547–558. MR 604712, DOI 10.1007/BF02566705
B. Wong, On the holomorphic curvature of some intrinsic metrics, Proc. Amer. Math. Soc. 65 (1977), no. 1, 57–61. MR 454081, DOI 10.1090/S0002-9939-1977-0454081-7
H. Wu, A remark on holomorphic sectional curvature, Indiana Univ. Math. J. 22 (1972/73), 1103–1108. MR 315642, DOI 10.1512/iumj.1973.22.22092
H. Wu, Old and new invariant metrics on complex manifolds, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 640–682. MR 1207887
Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 486659, DOI 10.2307/2373880