×

zbMATH — the first resource for mathematics

A note on the Bergman metric of bounded homogeneous domains. (English) Zbl 1127.32008
The authors study the Bergman metric \(ds_D^2\) on a bounded homogeneous domain \(D\) in \(\mathbb C^n\). They prove first, that there exists a real-analytic potential \(\varphi\) for \(ds_D^2\), whose gradient has constant length \(C\), when measured with respect to the Bergman metric, and, further, that this constant \(C\) is independent of the choice of \(\varphi\). As an application, they conclude that the \(L^2\) \(\overline{\partial} \) cohomology of \(D\) (with respect to the Bergman metric) at the bidegree \((p,q)\) vanishes for \(p+q\neq n\) and has infinite dimension for \(p+q=n\) and is Hausdorff.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Z. Błocki and P. Pflug, Hyperconvexity and Bergman completeness, Nagoya Math. J., 151 (1998), 221-225. · Zbl 0916.32016
[2] M. Carlehed, U. Cegrell and F. Wikström, Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function, Ann. Polon. Math., 71 (1999), 87-103. · Zbl 0955.32034
[3] B.-Y. Chen, Bergman completeness of hyperconvex manifolds, Nagoya Math. J., 175 (2004), 165-170. · Zbl 1061.32010
[4] —-, Infinite dimensionality of the middle \(L^{2}\) -cohomology on non-compact Kähler hyperbolic manifolds, Publ. RIMS., 42 (2006), 683-689. · Zbl 1115.32012
[5] H. Donnelly, \(L_{2}\) cohomology of the Bergman metric for weakly pseudoconvex domains , Illinois J. Math., 41 (1997), 151-160. · Zbl 0880.32007
[6] H. Donnelly and C. Fefferman, \(L^{2}\) -cohomology and index theorem for the Bergman metric, Ann. Math., 118 (1983), 593-618. · Zbl 0532.58027
[7] M. Gromov, Kähler hyperbolicity and \(L_{2}\)-Hodge theory, J. Diff. G., 33 (1991), 263-292. · Zbl 0719.53042
[8] G. Herbort, The Bergman metric on hyperconvex domains, Math. Z., 232 (1999), 183-196. · Zbl 0933.32048
[9] H. Ishi, On the Bergman metric of Siegel domains , · Zbl 0933.22012
[10] S. Kobayashi, Hyperbolic Complex Spaces, Grundlehren der Math. Wiss. 318 , Springer-Verlag, Berlin-Heidelberg-New York, 1998.
[11] —-, Hyperbolic Manifolds and Holomorphic Mappings (2nd edition) An Introduction, World Sci., 2005. · Zbl 0207.37902
[12] K. Nakajima, Some studies on Siegel domains, J. Math. Soc. Japan, 27 (1975), 54-75. · Zbl 0293.32030
[13] È. B. Vinberg, S. G. Gindikin and I. I. Pjateckiĭ-Šapiro, Classification and canonical realization of complex homogeneous bounded domains , Trudy Moskov. Mat. Obšč., 12 (1963), 359-388; Trans. Moscow Math. Soc., 12 (1963), 404-437. · Zbl 0137.05603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.