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

In this paper, we continue to consider Willmore Legendrian surfaces and csL Willmore surfaces in $${\mathbb {S}}^5$$ S 5 , notions introduced by Luo (Calc Var Partial Differ Equ 56, Art. 86, 19, 2017. 10.1007/s00526-017-1183-z). We will prove that every complete Willmore Legendrian surface in $${\mathbb {S}}^5$$ S 5 is minimal and find nontrivial examples of csL Willmore surfaces in $${\mathbb {S}}^5$$ S 5 .