Willmore Lagrangian Submanifolds in Complex Euclidean Space

Abstract

Let \(\varphi: M \to {C}^n\) 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 \(p^2 = S – nH^2\). 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 \(\varphi: M \to {C}^n\) in the Complex Euclidean Space \({C}^n\) in terms of \(p^2\), \(K\), \(Q\), and \(H\), and give some rigidity and characterization theorems.