Canonical metrics on some domains of ℂ n
Fabio Zuddas1
1 Università di Parma Dipartimento di Matematica Viale G. P. Usberti 53/A 43124 Parma (Italie)
Séminaire de théorie spectrale et géométrie, Tome 27 (2008-2009), pp. 143-156.

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]).

DOI : 10.5802/tsg.274
Fabio Zuddas 1

1 Università di Parma Dipartimento di Matematica Viale G. P. Usberti 53/A 43124 Parma (Italie) 
@article{TSG_2008-2009__27__143_0,
     author = {Fabio Zuddas},
     title = {Canonical metrics on some domains of $\mathbb{C}^n$},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {143--156},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {27},
     year = {2008-2009},
     doi = {10.5802/tsg.274},
     mrnumber = {2799150},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.274/}
}
TY  - JOUR
AU  - Fabio Zuddas
TI  - Canonical metrics on some domains of $\mathbb{C}^n$
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2008-2009
SP  - 143
EP  - 156
VL  - 27
PB  - Institut Fourier
PP  - Grenoble
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.274/
DO  - 10.5802/tsg.274
LA  - en
ID  - TSG_2008-2009__27__143_0
ER  - 
%0 Journal Article
%A Fabio Zuddas
%T Canonical metrics on some domains of $\mathbb{C}^n$
%J Séminaire de théorie spectrale et géométrie
%D 2008-2009
%P 143-156
%V 27
%I Institut Fourier
%C Grenoble
%U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.274/
%R 10.5802/tsg.274
%G en
%F TSG_2008-2009__27__143_0
Fabio Zuddas. Canonical metrics on some domains of $\mathbb{C}^n$. Séminaire de théorie spectrale et géométrie, Tome 27 (2008-2009), pp. 143-156. doi : 10.5802/tsg.274. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.274/

[1] Shigetoshi Bando; Toshiki Mabuchi Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 11-40 | MR | Zbl

[2] Eugenio Calabi Extremal Kähler metrics, Seminar on Differential Geometry (Ann. of Math. Stud.), Volume 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290 | MR | Zbl

[3] Huai Dong Cao Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math., Volume 81 (1985) no. 2, pp. 359-372 | MR | Zbl

[4] Huai-Dong Cao; Meng Zhu A note on compact Kähler-Ricci flow with positive bisectional curvature, Math. Res. Lett., Volume 16 (2009) no. 6, pp. 935-939 | MR | Zbl

[5] Shu-Cheng Chang On the existence of nontrivial extremal metrics on complete noncompact surfaces, Math. Ann., Volume 324 (2002) no. 3, pp. 465-490 | MR | Zbl

[6] Shu-Cheng Chang; Chin-Tung Wu On the existence of extremal metrics on complete noncompact 3-manifolds, Indiana Univ. Math. J., Volume 53 (2004) no. 1, pp. 243-268 | MR | Zbl

[7] Albert Chau Convergence of the Kähler-Ricci flow on noncompact Kähler manifolds, J. Differential Geom., Volume 66 (2004) no. 2, pp. 211-232 http://projecteuclid.org/getRecord?id=euclid.jdg/1102538610 | MR | Zbl

[8] Xiuxiong Chen; Gang Tian Uniqueness of extremal Kähler metrics, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 4, pp. 287-290 | MR | Zbl

[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., Volume 33 (1980) no. 4, pp. 507-544 | MR | Zbl

[10] Fabrizio Cuccu; Andrea Loi Global symplectic coordinates on complex domains, J. Geom. Phys., Volume 56 (2006) no. 2, pp. 247-259 | MR | Zbl

[11] Antonio J. Di Scala; Andrea Loi; Fabio Zuddas Riemannian geometry of Hartogs domains, Internat. J. Math., Volume 20 (2009) no. 2, pp. 139-148 | MR | Zbl

[12] Miroslav Engliš Berezin quantization and reproducing kernels on complex domains, Trans. Amer. Math. Soc., Volume 348 (1996) no. 2, pp. 411-479 | MR | Zbl

[13] Charles Fefferman The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., Volume 26 (1974), pp. 1-65 | EuDML | MR | Zbl

[14] Mikhail Feldman; Tom Ilmanen; Dan Knopf Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom., Volume 65 (2003) no. 2, pp. 169-209 http://projecteuclid.org/getRecord?id=euclid.jdg/1090511686 | MR | Zbl

[15] Paul Gauduchon Calabi’s extremal Kähler metrics (in preparation)

[16] A. V. Isaev; S. G. Krantz Domains with non-compact automorphism group: a survey, Adv. Math., Volume 146 (1999) no. 1, pp. 1-38 | MR | Zbl

[17] Shoshichi Kobayashi; Katsumi Nomizu Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1996 (Reprint of the 1969 original, A Wiley-Interscience Publication) | MR

[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) (Proc. Sympos. Pure Math.), Volume 52, Amer. Math. Soc., Providence, RI, 1991, pp. 297-304 | MR | Zbl

[19] Andrea Loi Regular quantizations of Kähler manifolds and constant scalar curvature metrics, J. Geom. Phys., Volume 53 (2005) no. 3, pp. 354-364 | MR | Zbl

[20] Andrea Loi; Fabio Zuddas Canonical metrics on Hartogs domains (to appear in Osaka Journal of Mathematics)

[21] Andrea Loi; Fabio Zuddas Symplectic maps of complex domains into complex space forms, Journal of Geometry and Physics, Volume 58 (2008) no. 7, pp. 888 -899 | MR | Zbl

[22] Toshiki Mabuchi Uniqueness of extremal Kähler metrics for an integral Kähler class, Internat. J. Math., Volume 15 (2004) no. 6, pp. 531-546 | MR | Zbl

[23] Gang Tian Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000 (Notes taken by Meike Akveld) | MR | Zbl

[24] Gang Tian; Xiaohua Zhu A new holomorphic invariant and uniqueness of Kähler-Ricci solitons, Comment. Math. Helv., Volume 77 (2002) no. 2, pp. 297-325 | MR | Zbl

[25] Gang Tian; Xiaohua Zhu Convergence of Kähler-Ricci flow, J. Amer. Math. Soc., Volume 20 (2007) no. 3, p. 675-699 (electronic) | MR | Zbl

[26] An Wang; Weiping Yin; Liyou Zhang; Guy Roos The Kähler-Einstein metric for some Hartogs domains over symmetric domains, Sci. China Ser. A, Volume 49 (2006) no. 9, pp. 1175-1210 | MR | Zbl

[27] Xu-Jia Wang; Xiaohua Zhu Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., Volume 188 (2004) no. 1, pp. 87-103 | MR | Zbl

[28] Shing Tung Yau On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | MR | Zbl

[29] Fangyang Zheng Complex differential geometry, AMS/IP Studies in Advanced Mathematics, 18, American Mathematical Society, Providence, RI, 2000 | MR | Zbl

Cité par Sources :