  Using the reformulation in divergence form of the Euler-Lagrange equation for
the Willmore functional as it was developed in "Analysis of the Willmore
Functional" by T. Riviere (Invent. Math. 174), 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.
