Canonical metrics on some domains of ℂ n

Fabio Zuddas[1]

  • [1] Università di Parma Dipartimento di Matematica Viale G. P. Usberti 53/A 43124 Parma (Italie)

Séminaire de théorie spectrale et géométrie (2008-2009)

  • Volume: 27, page 143-156
  • ISSN: 1624-5458

Abstract

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The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold M is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains D in ℂ n , the so-called Hartogs domains, which can be equipped with a natural Kaehler metric g . We show that if g is a Kähler-Einstein, constant scalar curvature, extremal or a soliton metric then ( D , g ) is holomorphically isometric to an open subset of the n -dimensional complex hyperbolic space. If D is bounded, we also show the same assertion under the assumption that g is a scalar multiple of the Bergman metric.The results we present are proved in papers joint with A. Loi and A. J. Di Scala ([11], [20]).

How to cite

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Zuddas, Fabio. "Canonical metrics on some domains of $\mathbb{C}^n$." Séminaire de théorie spectrale et géométrie 27 (2008-2009): 143-156. <http://eudml.org/doc/116455>.

@article{Zuddas2008-2009,
abstract = {The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold $M$ is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains $D$ in $\{\mathbb\{C\}\}^n$, the so-called Hartogs domains, which can be equipped with a natural Kaehler metric $g$. We show that if $g$ is a Kähler-Einstein, constant scalar curvature, extremal or a soliton metric then $(D, g)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. If $D$ is bounded, we also show the same assertion under the assumption that $g$ is a scalar multiple of the Bergman metric.The results we present are proved in papers joint with A. Loi and A. J. Di Scala ([11], [20]).},
affiliation = {Università di Parma Dipartimento di Matematica Viale G. P. Usberti 53/A 43124 Parma (Italie)},
author = {Zuddas, Fabio},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {preferred Kähler metric; Kähler-Einstein metric; Kähler metric with constant scalar curvature; Kähler-Ricci solitons; Bergman metric; Hartogs domains},
language = {eng},
pages = {143-156},
publisher = {Institut Fourier},
title = {Canonical metrics on some domains of $\mathbb\{C\}^n$},
url = {http://eudml.org/doc/116455},
volume = {27},
year = {2008-2009},
}

TY - JOUR
AU - Zuddas, Fabio
TI - Canonical metrics on some domains of $\mathbb{C}^n$
JO - Séminaire de théorie spectrale et géométrie
PY - 2008-2009
PB - Institut Fourier
VL - 27
SP - 143
EP - 156
AB - The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold $M$ is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains $D$ in ${\mathbb{C}}^n$, the so-called Hartogs domains, which can be equipped with a natural Kaehler metric $g$. We show that if $g$ is a Kähler-Einstein, constant scalar curvature, extremal or a soliton metric then $(D, g)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. If $D$ is bounded, we also show the same assertion under the assumption that $g$ is a scalar multiple of the Bergman metric.The results we present are proved in papers joint with A. Loi and A. J. Di Scala ([11], [20]).
LA - eng
KW - preferred Kähler metric; Kähler-Einstein metric; Kähler metric with constant scalar curvature; Kähler-Ricci solitons; Bergman metric; Hartogs domains
UR - http://eudml.org/doc/116455
ER -

References

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  1. Shigetoshi Bando, Toshiki Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry, Sendai, 1985 10 (1987), 11-40, North-Holland, Amsterdam Zbl0641.53065MR946233
  2. Eugenio Calabi, Extremal Kähler metrics, Seminar on Differential Geometry 102 (1982), 259-290, Princeton Univ. Press, Princeton, N.J. Zbl0487.53057MR645743
  3. Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), 359-372 Zbl0574.53042MR799272
  4. Huai-Dong Cao, Meng Zhu, A note on compact Kähler-Ricci flow with positive bisectional curvature, Math. Res. Lett. 16 (2009), 935-939 Zbl1193.53141MR2576682
  5. Shu-Cheng Chang, On the existence of nontrivial extremal metrics on complete noncompact surfaces, Math. Ann. 324 (2002), 465-490 Zbl1028.53041MR1938455
  6. Shu-Cheng Chang, Chin-Tung Wu, On the existence of extremal metrics on complete noncompact 3-manifolds, Indiana Univ. Math. J. 53 (2004), 243-268 Zbl1067.53023MR2048993
  7. Albert Chau, Convergence of the Kähler-Ricci flow on noncompact Kähler manifolds, J. Differential Geom. 66 (2004), 211-232 Zbl1082.53070MR2106124
  8. Xiuxiong Chen, Gang Tian, Uniqueness of extremal Kähler metrics, C. R. Math. Acad. Sci. Paris 340 (2005), 287-290 Zbl1069.32017MR2121892
  9. Shiu Yuen Cheng, 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), 507-544 Zbl0506.53031MR575736
  10. Fabrizio Cuccu, Andrea Loi, Global symplectic coordinates on complex domains, J. Geom. Phys. 56 (2006), 247-259 Zbl1085.53070MR2173895
  11. Antonio J. Di Scala, Andrea Loi, Fabio Zuddas, Riemannian geometry of Hartogs domains, Internat. J. Math. 20 (2009), 139-148 Zbl1178.32017MR2493356
  12. Miroslav Engliš, Berezin quantization and reproducing kernels on complex domains, Trans. Amer. Math. Soc. 348 (1996), 411-479 Zbl0842.46053MR1340173
  13. Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65 Zbl0289.32012MR350069
  14. Mikhail Feldman, Tom Ilmanen, Dan Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom. 65 (2003), 169-209 Zbl1069.53036MR2058261
  15. Paul Gauduchon, Calabi’s extremal Kähler metrics 
  16. A. V. Isaev, S. G. Krantz, Domains with non-compact automorphism group: a survey, Adv. Math. 146 (1999), 1-38 Zbl1040.32019MR1706680
  17. Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry. Vol. II, (1996), John Wiley & Sons Inc., New York Zbl0091.34802MR1393941
  18. Claude LeBrun, Complete Ricci-flat Kähler metrics on C n need not be flat, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) 52 (1991), 297-304, Amer. Math. Soc., Providence, RI Zbl0739.53053MR1128554
  19. Andrea Loi, Regular quantizations of Kähler manifolds and constant scalar curvature metrics, J. Geom. Phys. 53 (2005), 354-364 Zbl1080.53067MR2108536
  20. Andrea Loi, Fabio Zuddas, Canonical metrics on Hartogs domains Zbl05770024
  21. Andrea Loi, Fabio Zuddas, Symplectic maps of complex domains into complex space forms, Journal of Geometry and Physics 58 (2008), 888-899 Zbl1154.53050MR2426246
  22. Toshiki Mabuchi, Uniqueness of extremal Kähler metrics for an integral Kähler class, Internat. J. Math. 15 (2004), 531-546 Zbl1058.32017MR2078878
  23. Gang Tian, Canonical metrics in Kähler geometry, (2000), Birkhäuser Verlag, Basel Zbl0978.53002MR1787650
  24. Gang Tian, Xiaohua Zhu, A new holomorphic invariant and uniqueness of Kähler-Ricci solitons, Comment. Math. Helv. 77 (2002), 297-325 Zbl1036.53053MR1915043
  25. Gang Tian, Xiaohua Zhu, Convergence of Kähler-Ricci flow, J. Amer. Math. Soc. 20 (2007), 675-699 (electronic) Zbl1185.53078MR2291916
  26. An Wang, Weiping Yin, Liyou Zhang, Guy Roos, The Kähler-Einstein metric for some Hartogs domains over symmetric domains, Sci. China Ser. A 49 (2006), 1175-1210 Zbl1114.32011MR2284205
  27. Xu-Jia Wang, Xiaohua Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004), 87-103 Zbl1086.53067MR2084775
  28. Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), 339-411 Zbl0369.53059MR480350
  29. Fangyang Zheng, Complex differential geometry, 18 (2000), American Mathematical Society, Providence, RI Zbl0984.53002MR1777835

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