Local Palais-Smale Sequences for the Willmore Functional

TL;DR

This paper investigates the behavior of sequences of weak Willmore immersions with square-integrable second fundamental form, proving their limits are smooth and satisfy a conformal Willmore equation, advancing understanding of Willmore surface regularity.

Contribution

It establishes the smoothness and conformal Willmore property of limits of local Palais-Smale sequences, using divergence form reformulation of the Euler-Lagrange equation.

Findings

Limit immersions are smooth.

Limit immersions satisfy the conformal Willmore equation.

Sequences converge to critical points of the Willmore functional.

Abstract

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.