Communications in Analysis and Geometry

Volume 19 (2011)

Number 3

Local Palais–Smale sequences for the Willmore functional

Pages: 563 – 599



Yann Bernard (Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg, Germany)

Tristan Rivière (Department of Mathematics, ETH Zentrum, Zürich, Switzerland)


Using the reformulation in divergence form of the Euler–Lagrange equation for the Willmore functional as it was developed in the second author's paper, we study the limit of a local Palais--Smale sequence of weak Willmore immersions with locally square-integrable second fundamental form. We show that the limit immersion is smooth and that it satisfies the conformal Willmore equation: it is a critical point of the Willmore functional restricted to infinitesimal conformal variations.

Published 3 October 2011