Singularity removability at branch points for Willmore surfaces

Yann Bernard and Tristan Rivière

Vol. 265 (2013), No. 2, 257–311

DOI: 10.2140/pjm.2013.265.257

Abstract

We consider a branched Willmore surface immersed in ℝ^m≥3 with square-integrable second fundamental form. We develop around each branch point local asymptotic expansions for the Willmore immersion, its first, and its second derivatives. Our expansions are given in terms of new integer-valued residues which are computed as circulation integrals around the branch point. We deduce explicit “point removability” conditions guaranteeing that the immersion is smooth through the branch point. These conditions are new, even in codimension one.

Mathematical Subject Classification 2010

Primary: 30C70, 35J35, 35J50, 35J48, 35R01

Secondary: 49Q10, 53A30, 32S25, 58E15, 58E30

Milestones

Received: 27 April 2012

Revised: 10 January 2013

Accepted: 6 May 2013

Published: 28 August 2013