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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People's Republic, of China

    Haizhong Li

  2. LAMATH, ISTV 2, Campus du Mont Houy, Université de Valenciennes, 59313, Valenciennes Cedex 9, France

    Luc Vrancken

Authors
  1. Haizhong Li
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  2. Luc Vrancken
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Correspondence to Haizhong Li.

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Partially supported by a research fellowship of the Alexander von Humboldt Stiftung.

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Li, H., Vrancken, L. A basic inequality and new characterization of Whitney spheres in a complex space form. Isr. J. Math. 146, 223–242 (2005). https://doi.org/10.1007/BF02773534

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  • DOI: https://doi.org/10.1007/BF02773534

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