Minimal Bubbling for Willmore Surfaces

Abstract

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 16π. Additionnally we prove an inequality on the second residue for limits 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 12π.

Anε-regularity result with mean curvature control for Willmore immersions and application to minimal bubbling.

“…He proved that the configuration displayed in Figure 1.1 was impossible, and that the regularity found increased the compactness ceiling to sequences of energy below or equal 12π. On the other hand, the first part of [19] displays an explicit example of Willmore minimal bubbling (using an Enneper surface to desingularize the branch point of an inverted López surface) which can illustrate this article's studies. -Let Φ k be a sequence of Willmore immersions of a closed surface Σ satisfying the hypotheses of theorem 1.2.…”

“…During the reviewing process, the author applied and furthered these studies (in [19]). He proved that the configuration displayed in Figure 1.1 was impossible, and that the regularity found increased the compactness ceiling to sequences of energy below or equal 12π.…”