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

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.

arXiv commentTitle changed. arXiv admin note: text overlap with: arXiv:1211.4227v6, to appear in Pacific J. Math