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

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 Möbius transformations, in a strong $W_{\mathrm{loc}}^{2,2}$-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.

61 pages, 8 Figures. Comments are welcome!