Complete Willmore Legendrian surfaces in ${\mathbb{S}}^5$ are minimal Legendrian surfaces

In this paper we continue to consider Willmore Legendrian surfaces and csL Willmroe surfaces in ${\mathbb{S}}^5$, notions introduced by Luo in \cite{Luo}. We will prove that every complete Willmore Legendrian surface in ${\mathbb{S}}^5$ is minimal and construct nontrivial examples of csL Willmore surfaces in ${\mathbb{S}}^5$.