Bubble classification of immersions at the boundary of the moduli space with $8\pi$ Willmore energy

TL;DR

This paper analyzes the bubbling behavior of genus-$p$ immersions with diverging conformal classes and energy approaching $8 heta$, revealing a pattern of spheres and catenoids, and applies this to constrained Willmore minimizers.

Contribution

It provides a classification of bubbling patterns for sequences of immersions near the boundary of the moduli space with specific energy levels.

Findings

Limit configurations include two spheres and multiple catenoids.

The classification applies to constrained Willmore minimizers approaching boundary conditions.

The analysis uses M"obius transformations and strong convergence in $W^{2,2}_{loc}$.

Abstract

We study the asymptotic bubbling behavior of sequences of weak genus-$p$ immersions with diverging conformal classes and limiting Willmore energy of $8\pi$. After applying suitable M\"obius transformations, in a strong $W^{2,2}_{\mathrm{loc}}$-limit, we obtain two round spheres at the largest scale and $p+1$ catenoids at the smallest scales. Moreover, we apply this classification to sequences of isoperimetrically, conformally and normalized-total-mean-curvature constrained Willmore minimizers when the constraints approach the boundary of the domain where minimizers exist, respectively.