Riv. Mat. Univ. Parma, Vol. 5, No. 2, 2014

Morris Kalka[1] and Giorgio Patrizio[2]

Splitting parabolic manifolds

Pages: 443-453
Received: 18 July 2014  
Accepted: 5 September 2014
Mathematics Subject Classification (2010): 32L30, 32F07, 32U10.

Keywords: Monge-Ampère foliations, homogeneous complex Monge-Ampère equation, parabolic manifolds.
Authors addresses:
[1] : Tulane University, Mathematics Department, 6823 St. Charles Ave., New Orleans, LA 70118, USA
[2] : Università degli Studi di Firenze, Dip. Matematica e Informatica "U. Dini", Viale Morgani 67/a, I-50134 Firenze, Italy

Abstract: We study the geometric properties of complex manifolds possessing a pair of plurisubharmonic functions satisfying Monge-Ampère type of condition. The results are applied to characterize complex manifolds biholomorphic to \(\mathbb{C}^N\) viewed as a product of lower dimensional complex Euclidean spaces.

Much of this work was done while Kalka was visiting the University in Florence and G. Patrizio Tulane University. The Authors thank the institutions for their support. G. Patrizio acknowledges the support of MIUR PRIN 2010-11 "Varietà reali e complesse: geometria, topologia e analisi armonica" and the collaboration with GNSAGA of INdAM.


[1] E. Bedford and M. Kalka, Foliations and complex Monge-Ampère equations, Comm. Pure Appl. Math. 30 (1977), 543-571. [MR0481107]
[2] D. Burns, Curvature of Monge-Ampère foliations and parabolic manifolds, Ann. of Math. 115 (1982), 349-373. [MR0647810]
[3] M. Kalka and G. Patrizio, Monge-Ampèere foliations for degenerate solutions, Ann. Mat. Pura Appl. 189 (2010), 381-393. [MR2657415]
[4] M. Kalka and G. Patrizio, Locally Monge-Ampère Parabolic Foliations, Adv. Geom., to appear.
[5] G. Patrizio, A characterization of complex manifolds biholomorphic to a circular domain, Math. Z. 189 (1985), 343-363. [MR0783561]
[6] W. Stoll, The characterization of strictly parabolic manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 87-154. [MR0577327]
[7] P.-M. Wong, On umbilical hypersurfaces and uniformization of circular domains, in "Complex Analysis of several Variables", Proc. Sympos. Pure Math., 41, Amer. Math. Soc., Providence, RI, 1984, 225-252. [MR0740886]

Home Riv.Mat.Univ.Parma