Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

TL;DR

This paper establishes an epsilon-regularity result for the tracefree curvature of Willmore surfaces, providing pointwise control from L^2 bounds, with applications to gap theorems and regularity of minimal spacelike immersions.

Contribution

It introduces a new epsilon-regularity theorem for tracefree curvature of Willmore surfaces and applies it to derive gap results and regularity for minimal spacelike immersions.

Findings

Pointwise control of tracefree second fundamental form from L^2-norm

Gap theorem for certain Willmore surfaces

Regularity results for minimal spacelike immersions into de Sitter space

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 $S^{4,1}$.