Addendum to "Contact stationary Legendrian surfaces in $\mathbb{S}^5$"[Pacific Math. J. 293(2018), no.1, 101-120]

TL;DR

This paper extends previous work on contact stationary Legendrian surfaces in the 5-sphere, proving that totally umbilical such surfaces are actually totally geodesic, refining their geometric classification.

Contribution

It establishes that totally umbilical contact stationary Legendrian surfaces in -sphere are necessarily totally geodesic, adding a key detail to their geometric characterization.

Findings

Totally umbilical contact stationary Legendrian surfaces are totally geodesic.

Refinement of classification of Legendrian surfaces in -sphere.

Extension of previous results on second fundamental form bounds.

Abstract

In \cite{Luo}, the present author proved that if $L$ is a contact stationary Legendrian surface in $\mathbb{S}^5$ with the canonical Sasakian structure and the square length of its second fundamental form belongs to $[0,2]$. Then we have that $L$ is either totally umbilical or is a flat minimal Legendrian torus. In this addendum we further prove that if $L$ is a totally umbilical contact stationary Legendrian surface in $\mathbb{S}^5$ , then $L$ is totally geodesic.