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 .

Minimal Bubbling for Willmore Surfaces

In this paper, we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above $16 \pi $. Additionally, we prove an inequality on the 2nd residue for limits of sequences of Willmore immersions with simple minimal bubbles. Doing so, we exclude some gluing configurations and prove compactness for immersed Willmore tori of energy below $12 \pi $.