J. Aust. Math. Soc. 83 (2007), no. 1, pp. 11–15.

The Monge–Ampère equation and warped products of higher rank

Stefan Bechtluft-Sachs Evangelia Samiou
Department of Mathematics
American University of Beirut
P.O. Box 11-0236
Riad El Solh
Beirut 1107 2020
University of Cyprus
Department of Mathematics and Statistics
P.O. Box 20537
1678 Nicosia
Received 8 April 2005; revised 6 May 2006
Communicated by K. Wysocki


We show that a warped product M_f=\mathbb {R}^n\times _f\mathbb {R} has higher rank and nonpositive curvature if and only if f is a convex solution of the Monge–Ampère equation. In this case we show that M contains a Euclidean factor.

Download the article in PDF format (size 72 Kb)

2000 Mathematics Subject Classification: primary 53C21, 53C24; secondary 35J60
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: pre05231329
indicates author for correspondence


  1. W. Ballmann, ‘Nonpositively curved manifolds of higher rank’, Ann. of Math. 122 (1985), 597–609. MR819559
  2. W. Ballmann, M. Brin and P. Eberlein, ‘Structure of manifolds of nonpositive curvature I’, Ann. of Math. 122 (1985), 171–203. MR799256
  3. W. Ballmann, M. Brin and R. Spatzier, ‘Structure of manifolds of nonpositive curvature II’, Ann. of Math. 122 (1985), 205–235. MR808219
  4. J. Berndt and E. Samiou, ‘Rank rigidity, cones and curvature-homogeneous Hadamard manifolds’, Osaka J. Math. 39 (2002), 383–394. MR1914319
  5. E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian Manifolds of conullity two (World Scientific, Singapore, 1996). MR1462887
  6. K. Burns and R. Spatzier, ‘Manifolds of nonpositive curvature and their buildings’, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 35–59. MR908215
  7. C. E. Gutiérrez, The Monge-Ampère equation, Progr. in Nonlinear Differential Equations Appl., 44 (Birkhaüser Boston, Inc., Boston, MA, 2001). MR1829162
  8. O. Kowalski, F. Tricerri and L. Vanhecke, ‘Curvature-homogeneous riemannian manifolds’, J. Math. Pures Appl. 71 (1992), 471–501. MR1193605
Australian Mathematical Publishing Association Inc.

Valid XHTML 1.0 Transitional