HIGHER ORDER ASYMPTOTIC BEHAVIOR OF CERTAIN KÄHLER METRICS AND UNIFORMIZATION FOR STRONGLY PSEUDOCONVEX DOMAINS

• Joo, Jae-Cheon (Department of Mathematics and Informatics University of Wuppertal) ;
• Seo, Aeryeong (School of Mathematics Korea Institute for Advanced Study (KIAS))
• Published : 2015.01.01

Abstract

We provide some relations between CR invariants of boundaries of strongly pseudoconvex domains and higher order asymptotic behavior of certain complete K$\ddot{a}$hler metrics of given domains. As a consequence, we prove a rigidity theorem of strongly pseudoconvex domains by asymptotic curvature behavior of metrics.

Acknowledgement

Supported by : National Research Foundation (NRF)

References

1. E. Barletta, On the boundary behavior of the holomorphic sectional curvature of the Bergman metric, Matematiche (Catania) 61 (2006), no. 2, 301-316.
2. S. S. Chern and S. Ji, On the Riemann mapping theorem, Ann. of Math (2) 144 (1996), no. 2, 421-439. https://doi.org/10.2307/2118596
3. K. Hirachi, Scalar pseudo-Hermitian invariants and the Szego kernel on three-dimensional CR manifolds, Complex geometry (Osaka, 1990), 67-76, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993.
4. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. https://doi.org/10.1007/BF02392146
5. C. Fefferman, On the Bergman kernel and biholomorphic mappings of pseudoconvex domains, Bull. Amer. Math. Soc. 80 (1974), no. 4, 667-669. https://doi.org/10.1090/S0002-9904-1974-13539-1
6. C. R. Graham and J. M. Lee, Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Math. J. 57 (1988), no. 3, 697-720. https://doi.org/10.1215/S0012-7094-88-05731-6
7. K. T. Kim and J. Yu, Boundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domains, Pacific J. Math. 176 (1996), no. 1, 141-163. https://doi.org/10.2140/pjm.1996.176.141
8. P. F. Klembeck, Kahler 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. https://doi.org/10.1512/iumj.1978.27.27020
9. Q. K. Lu, On Kaehler manifolds with constant curvature, Acta Math. Sinica 16 (1966), 269-281 (Chinese); English transl. in Chinese Math. Acta 8 (1966), 283-298.
10. N. Mok, The uniformization theorem for compact Kahler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), no. 2, 179-214. https://doi.org/10.4310/jdg/1214441778
11. S. Y. Nemirovskii and R. G. Shafikov, Uniformization of strictly pseudoconvex domains. I, Izv. Mat. 69 (2005), no. 6, 1189-1202. https://doi.org/10.1070/IM2005v069n06ABEH002295
12. S. Y. Nemirovskii and R. G. Shafikov, Uniformization of strictly pseudoconvex domains. II, Izv. Mat. 69 (2005), no. 6, 1203-1210. https://doi.org/10.1070/IM2005v069n06ABEH002296
13. Y.-T. Siu and S.-T. Yau, Complete Kahler manifolds with nonpositive curvature of faster than quadratic decay, Ann. of Math. (2) 105 (1977), no. 2, 225-264. https://doi.org/10.2307/1970998
14. S. I. Pincuk, Proper holomorphic maps of strictly pseudoconvex domains, (Russian) Sibirsk. Mat. Z. 15 (1974), 909-917, 959.
15. N. Seshadri, Volume renormalization for complete Einstein-Kahler metrics, Differential Geom. Appl. 25 (2007), no. 4, 356-379. https://doi.org/10.1016/j.difgeo.2007.02.004
16. Y.-T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. (2) 112 (1980), no. 1, 73-111. https://doi.org/10.2307/1971321
17. Y.-T. Siu and S.-T. Yau, Compact Kahler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189-204. https://doi.org/10.1007/BF01390043
18. S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25-41. https://doi.org/10.4310/jdg/1214434345

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