Asymptotics of Willmore minimizers with prescribed small isoperimetric ratio

TL;DR

This paper analyzes the asymptotic behavior of Willmore minimizers with small isoperimetric ratios, revealing a catenoidal neck connecting two spheres in the limit, with implications for geometric analysis and cell membrane models.

Contribution

It provides a detailed blowup analysis of Willmore minimizers as the isoperimetric ratio approaches zero, showing the formation of a catenoidal neck connecting two spheres.

Findings

Convergence to a double sphere as isoperimetric ratio tends to zero

Formation of a catenoidal neck in the limit

Full blowup analysis of the geometric limit

Abstract

We consider surfaces in ${\mathbb R}^3$ of type ${\mathbb S}^2$ which minimize the Willmore functional with prescribed isoperimetric ratio. The existence of smooth minimizers was proved by Schygulla (Archive Rational Mechanics and Analysis, 2012). In the singular limit when the isoperimetric ratio converges to zero, he showed convergence to a double round sphere in the sense of varifolds. Here we give a full blowup analysis of this limit, showing that the two spheres are connected by a catenoidal neck. Besides its geometric interest, the problem was studied as a simplified model in the theory of cell membranes, see e.g. Berndl, Lipowsky, Seifert (Physical Review A, 1991).