Abstract
We determine conditions under which the quotient of a Kähler-Hadamard manifold by a torsion-free, discrete group of isometries admits a big set of nontrivial holomorphic functions. We also generalize a theorem of G. A. Margulis on centerless of cocompact discrete groups of automorphisms of a Carathéodory hyperbolic manifold by a different approach based on the Carathéodory pseudodistance.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abe Y., Kopfermann K.: Toroidal groups. Lecture Notes in Mathemtaics, vol. 1759. Springer, Berlin (2001)
Andreotti A., Vesentini E.: Carleman estimates for the Laplace-Beltrami equation in complex manifolds. Publ. Math. IHES 25, 81–130 (1965)
Baily W., Borel A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. 84, 442–528 (1966)
Ballman W., Gromov M., Schroeder V.: Manifolds of nonpositive curvature: progress in mathematics, vol. 61. Birkhäser, Boston (1985)
Beardon A., Maskit B.: Limit sets of Kleinian groups and finite sided fundamental polyhedra. Acta. Math. 132, 1–12 (1974)
Bishop R., O’Neill B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)
Bowditch B.H.: Geometrical finiteness with variable negative curvature. Duke Math. J. 77, 227–274 (1995)
Burns D., Shnider S.: Spherical hypersurfaces in complex manifolds. Invent. Math. 33, 223–246 (1976)
Demailly J.-P.: Estimations L 2 pour l’opérateur \({\bar{\partial}}\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Sup. 15, 457–511 (1982)
Eberlein P., O’Neill B.: Visibility manifolds. Pacific J. Math. 46, 45–109 (1973)
de Fabritiis C., Iannuzzi A.: Quotients of the unit ball of \({{\mathbb{C}}^n}\) for a free action of \({{\mathbb{Z}}}\). J. D’Analyse Math. 85, 213–224 (2001)
Fefferman C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)
Grauert H.: On Levi’s problem and the imbedding of real analytic manifolds. Ann. Math. 68, 460–472 (1958)
Grauert H., Remmert R.: Plurisubharmonische Funktionen in komplexen Räumen. Math. Z. 65, 175–194 (1956)
Greene R.E., Wu H.: Function theory on manifolds possess a pole. Lecture Notes in Mathematics, vol. 699. Springer, Berlin (1979)
Hörmander L.: An introduction to complex analysis in several complex variables, 3rd edn. Elsevier, New York (1990)
Jost, J.: Riemannian geometry and geometric alaysis. Universitext, Springer, Berlin (2002)
Kapovich M.: Hyperbolic groups and discrete groups, Progress in Mathematics, vol. 183. Birkhäuser, Boston (2000)
Klembeck P.: Kähler metrics of negative curvature, the Bergman metric near the boundary and the Kobayashi metric on smooth bounded strictly pseudoconvex sets. Indiana Univ. Math. J. 27, 275–282 (1978)
Kohn J.J., Rossi H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. Math. 81, 451–472 (1965)
Li P., Schoen R.: L p and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153, 279–301 (1984)
Ya Lin V.: Liouville coverings of complex spaces, and amenable groups. Math. USSR Sbornik 60, 197–216 (1988)
Marden A.: The geometry of finite generated Kleinian groups. Ann. Math. 99, 383–462 (1974)
Mok N., Siu Y.T., Yau S.T.: The Poincaré-Lelong equation on complete Kähler manifolds. Composito Math. 44, 183–218 (1981)
Mok N.: Projective-algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume. Perspectives in analysis, geometry, and topology. Progr. Math. 296, 331–354 (2012)
Napier T.: Convexity properties of coverings of smooth projective varieties. Math. Ann. 286, 433–479 (1990)
Nikolov N., Pflug P., Zwonek W.: Estimates for invariant metrics on \({\mathbb{C}}\)-convex domains. Trans. Am. Math. Soc. 363, 6245–6256 (2011)
Ohsawa T.: Isomorphism theorems for cohomology groups of weakly 1−complete manifolds. Publ. RIMS Kyoto Univ. 18, 191–232 (1982)
Piatetskii-Shapiro I.I.: Automorphic functions and the geometry of classical domains. Gordon and Breach, New York (1969)
Siu Y.T., Yau S.T.: Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay. Ann. Math. 105, 225–264 (1977)
Siu, Y.T., Yau, S.T.: Compactification of negatively curved complete Kähler manifolds of finite volume. Seminar on Differential Geomentry, Annals of Mathematical Students, Princeton University Press, Princeton (1982)
Sullivan D.: Related aspects of positivity in Riemannian geometry. J. Diff. Geom. 25, 327–351 (1987)
Thurston, W.: Geometry and topology of 3−manifolds. Princeton Lecture Notes (1978–1981)
Vinberg E.B., Gindikin S.G., Piatetskii-Shapiro I.I.: Classification and canonical realization of complex bounded homogeneous domains. Trans. Moscow Math. Soc. 12, 404–437 (1963)
Author information
Authors and Affiliations
Corresponding author
Additional information
B.-Y. Chen was supported by Chinese NSF grant No. 11031008 and Fok Ying Tung Education Foundation grant No. 111004. Partially supported by Chinese NSF grant No. 11171255.