Willmore Lagrangian Submanifolds in Complex Euclidean Space

Abstract

Let $\phi :M\to C^n$ be an $n$-dimensional compact Willmore Lagrangian submanifold in the Complex Euclidean Space $C^n$. Denote by $S$ and $H$ the square of the length of the second fundamental form and the mean curvature of $M$, respectively. Let $p$ be the non-negative function on $M$ defined by $p^2=S–n{H}^2$. Let $K$ and $Q$ be the functions which assign to each point of $M$ the infimum of the sectional curvature and Ricci curvature at the point, respectively. In this paper, we prove some integral inequalities of Simons’ type for $n$-dimensional compact Willmore Lagrangian submanifolds $\phi :M\to C^n$ in the Complex Euclidean Space $C^n$ in terms of $p^2$, $K$, $Q$, and $H$, and give some rigidity and characterization theorems.