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.

Automated summary

In this paper, we provide an explicit example of a minimal bubble on a Willmore surface, demonstrating that Willmore immersions of energy exceeding 16π cannot be compact. Additionally, we establish an inequality on the 2nd residue for sequences of Willmore immersions with simple minimal bubbles. As a result, we exclude certain gluing configurations and prove compactness for immersed Willmore tori with energy below 12π.