Israel Journal of Mathematics

, Volume 146, Issue 1, pp 223–242 | Cite as

A basic inequality and new characterization of Whitney spheres in a complex space form

  • Haizhong Li
  • Luc Vrancken


LetN n (4c) be ann-dimensional complex space form of constant holomorphic sectional curvature 4c and letx:M n N n (4c) be ann-dimensional Lagrangian submanifold inN n (4c). We prove that the following inequality always hold onM n:\(\left| {\bar \nabla h} \right|^2 \geqslant \frac{{3n^2 }}{{n + 2}}\left| {\nabla ^ \bot \vec H} \right|^2 \) whereh is the second fundamental form andH is the mean curvature of the submanifold. We classify all submanifolds which at every point realize the equality in the above inequality. As a direct consequence of our Theorem, we give, a new characterization of theWhitney spheres in a complex space form.


Fundamental Form Curvature Vector Lagrangian Submanifold Complex Space Form Tangent Vector Field 
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Copyright information

© The Hebrew University Magnes Press 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople's Republic, of China
  2. 2.LAMATH, ISTV 2, Campus du Mont HouyUniversité de ValenciennesValenciennes Cedex 9France

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