Nonlinear Analysis 95 (2014) 714–720 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na On the generalized Wintgen inequality for Lagrangian submanifolds in complex space forms Ion Mihai ∗ Department of Mathematics, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania article info Article history: Received 8 February 2013 Accepted 9 October 2013 Communicated by Enzo Mitidieri MSC: 53C40 53C25 Keywords: Wintgen inequality DDVV conjecture Complex space form Invariant submanifold Lagrangian submanifold Slant submanifold abstract The normal scalar curvature conjecture, also known as the DDVV conjecture, was formulated by De Smet et al. (1999). It was proven recently by Lu (2011) and by Ge and Tang (2008) independently. We obtain the DDVV inequality, also known as the generalized Wintgen inequality, for Lagrangian submanifolds in complex space forms. Some applications are given. Also we state such an inequality for slant submanifolds in complex space forms. © 2013 Elsevier Ltd. All rights reserved. 1. Preliminaries For surfaces M 2 of the Euclidean space E 3 , the Euler inequality K ≤∥H∥ 2 is fulfilled, where K is the (intrinsic) Gauss curvature of M 2 and ∥H∥ 2 is the (extrinsic) squared mean curvature of M 2 . Furthermore, K =∥H∥ 2 everywhere on M 2 if and only if M 2 is totally umbilical, or still, by a theorem of Meusnier, if and only if M 2 is (a part of) a plane E 2 or, it is (a part of) a round sphere S 2 in E 3 . In 1979, P. Wintgen [1] proved that the Gauss curvature K , the squared mean curvature ∥H∥ 2 and the normal curvature K ⊥ of any surface M 2 in E 4 always satisfy the inequality K ≤∥H∥ 2 −|K ⊥ | and that actually the equality holds if and only if the ellipse of curvature of M 2 in E 4 is a circle. Example 1. The Whitney 2-sphere satisfies the equality case identically. Example 2. The rotation surface of Vrănceanu is defined by the immersion f : R × (0, 2π) → E 4 , f (u,v) = r (u)(cos u cos v, cos u sin v, sin u cos v, sin u sin v), ∗ Tel.: +40 724757096. E-mail addresses: imihai@fmi.unibuc.ro, ion_mihai@yahoo.com. 0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.10.009