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

Authors:	Email author" target="_blank">Haizhong?Li Email author Luc?Vrancken

Institution:	(1) Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, People's Republic, of China;(2) LAMATH, ISTV 2, Campus du Mont Houy, Université de Valenciennes, 59313 Valenciennes Cedex 9, France

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

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