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In this paper we continue to consider Willmore Legendrian surfaces and csLWillmroe 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$\mathbb{S}^5$.

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