On energy gap phenomena of the Whitney sphere and related problems

Yong Luo and Jiabin Yin

Vol. 320 (2022), No. 2, 299–318

DOI: 10.2140/pjm.2022.320.299

Abstract

We study Lagrangian submanifolds satisfying ${\nabla <!--FUNCTION\; APPLICATION-->}^{\ast }T=0$ introduced by Zhang (2021) in the complex space forms $N(4c)$ ($c=0$ or $1$), where $T={\nabla <!--FUNCTION\; APPLICATION-->}^{\ast }\tilde{h}$ 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 ${ℂ}^n$ satisfying ${\nabla <!--FUNCTION\; APPLICATION-->}^{\ast }{\nabla <!--FUNCTION\; APPLICATION-->}^{\ast }T=0$ and introduce a flow method related to them.

Keywords

Lagrangian submanifolds, the Whitney sphere, energy gap theorems, conformal Maslov form

Mathematical Subject Classification

Primary: 53C24, 53C42

Milestones

Received: 25 August 2021

Revised: 13 June 2022

Accepted: 16 August 2022

Published: 15 February 2023