Asymptotic Analysis of Branched Willmore Surfaces

Abstract

We consider a closed Willmore surface properly immersed in ${\mathbb{R}}^m$ (m>2) with square-integrable second fundamental form, and with one point-singularity of finite arbitrary integer order. Using the "conservative" reformulation of the Willmore equation introduced in a previous paper by the second author, we show that, in an appropriate conformal parametrization, the gradient of the Gauss map of the immersion has bounded mean oscillations if the singularity has order one, and is bounded if the order is at least two. We develop around the singular point local asymptotic expansions for the immersion, its first and second derivatives, and for the mean curvature vector. Finally, we exhibit an explicit condition ensuring the removability of the point-singularity

Cited by in corpus (9)

• Energy Quantization of Willmore surfaces at the boundary of the Moduli Space
• A Resolution of the Poisson Problem for Elastic Plates
• Rigidity and stability of spheres in the Helfrich model
• Minimal Bubbling for Willmore Surfaces
• Higher Regularity of Weak Limits of Willmore Immersions I
• Morse Index Estimates of Min-Max Willmore Surfaces
• Asymptotics of Willmore minimizers with prescribed small isoperimetric ratio
• Classification of branched Willmore spheres
• Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$