Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

Abstract

We prove an $\epsilon$-regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $L^2$-norm.Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $S^{4,1}$.

References in corpus (6)

• Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space
• A classification theorem for Helfrich surfaces
• Energy Quantization of Willmore surfaces at the boundary of the Moduli Space
• Minimal Bubbling for Willmore Surfaces
• Conformal Gauss Map Geometry and Application to Willmore Surfaces in Model Spaces
• An $\epsilon$-regularity result with mean curvature control for Willmore immersions and application to minimal bubbling

Cited by in corpus (2)

• Energy quantization for Willmore surfaces with bounded index
• Classification of branched Willmore spheres