# Minimal Bubbling for Willmore Surfaces

## TL;DR

This paper constructs explicit minimal bubbles on Willmore surfaces, demonstrating non-compactness above a certain energy threshold and establishing compactness results for specific energy ranges.

## Contribution

It provides a concrete example of minimal bubbles on Willmore surfaces and proves new energy bounds ensuring compactness of Willmore immersions.

## Key findings

- Explicit example of a minimal bubble on a Willmore surface
- Non-compactness for energies above 16π
- Compactness for tori with energy below 12π

## 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 $16 \pi$. 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 $12 \pi$.