Vol. 265, No. 2, 2013

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Singularity removability at branch points for Willmore surfaces

Yann Bernard and Tristan Rivière

Vol. 265 (2013), No. 2, 257–311
Abstract

We consider a branched Willmore surface immersed in m3 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
Authors
Yann Bernard
Mathematisches Institut
Albert-Ludwigs-Universität
79004 Freiburg
Germany
Tristan Rivière
Department of Mathematics
ETH Zentrum
8093 Zurich
Switzerland