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

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- by C. K. Cheung and K. T. Kim
- Proc. Amer. Math. Soc.
**125**(1997), 1131-1142 - DOI: https://doi.org/10.1090/S0002-9939-97-03695-2
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## 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

## Bibliographic Information

**C. K. Cheung**- Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02167
- Email: Cheung/MT@hermes.bc.edu
**K. T. Kim**- Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784 Republic of Korea
- Email: kimkt@posmath.postech.ac.kr
- Received by editor(s): October 11, 1995
- Additional Notes: Research of the second named author is supported in part by Grants from Pohang University of Science and Technology (POSTECH), GARC, and BSRI
- Communicated by: Peter Li
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 1131-1142 - MSC (1991): Primary 32H15
- DOI: https://doi.org/10.1090/S0002-9939-97-03695-2
- MathSciNet review: 1363414