Rigidity of closed CSL submanifolds in the unit sphere

TL;DR

This paper investigates the geometric properties and rigidity conditions of closed contact stationary Legendrian (CSL) submanifolds in spheres, extending known results from surfaces to higher dimensions and identifying conditions for total geodesicity or flatness.

Contribution

It generalizes Luo's result on CSL surfaces in $ ext{S}^5$ to higher dimensions, providing new rigidity theorems for closed CSL submanifolds.

Findings

Closed CSL submanifolds with bounded second fundamental form are either totally geodesic or flat minimal Legendrian tori.

Extension of gap theorems from surfaces to higher-dimensional CSL submanifolds.

Abstract

A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a stationary point of the volume functional of Legendrian submanifolds in a Sasakian manifold. Much effort has been paid in the last two decades to construct examples of such manifolds, mainly by geometers using various geometric methods. But we have rare knowledge about their geometric properties till now. Recently, Y. Luo (\cite{ Luo2, Luo1}) proved that a closed CSL surface in $\mathbb{S}^5$ with the square length of its second fundamental form belonging to $[0,2]$ must be totally geodesic or be a flat minimal Legendrian torus, which generalizes a related gap theorem of minimal Legendrian surface due to Yamaguchi et al. (\cite{YKM}). In this paper, we will study the general dimensional case of this result.