An $\varepsilon$-regularity result with mean curvature control for Willmore immersions and application to minimal bubbling

TL;DR

This paper establishes an $\varepsilon$-regularity result for Willmore immersions with mean curvature control, demonstrating convergence with minimal bubbles and replacing total curvature control with local Willmore energy control.

Contribution

It introduces a new regularity criterion for Willmore immersions based on local Willmore energy, extending previous curvature-based methods.

Findings

Proves convergence of Willmore immersions with minimal bubbles

Replaces total curvature control with local Willmore energy control

Provides a new $\varepsilon$-regularity framework for Willmore surfaces

Abstract

In this paper we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end we replace the total curvature control in T. Rivi\`ere's proof of the $\varepsilon$-regularity for Willmore immersions by a control of the local Willmore energy.