On energy gap phenomena of the Whitney sphere and related problems

Abstract

In this paper, we study Lagrangian submanifolds satisfying ${\nabla }^{\ast }T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0or1)$, where $T={\nabla }^{\ast }\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. We obtain some Simons' type integral inequalities and rigidity theorems for such Lagrangian submanifolds. Moreover we study Lagrangian submanifolds in ${\mathbb{C}}^n$ satisfying ${\nabla }^{\ast }{\nabla }^{\ast }T=0$ and introduce a flow method related to them.

18 pages, introduction rearranged and errors corrected.In the last version, we uplodaded a wrong file

References in corpus (1)

• Contact stationary Legendrian surfaces in $S^5$