Limits of conformal immersions under a bound on a fractional normal curvature quantity

Abstract

We consider limits of weakly converging $W^{1,2}$-maps $\Phi_k$ from a ball $B \subset \mathbb{R}^2$ into $\mathbb{R}^3$ which are conformal immersions. Under the assumption that a normal curvature term is small, namely if for the normal map $u$ we have for some $s \in (\frac{1}{2},1)$ $$ \int_{B} \int_{B} \left | \frac{u_k(x) \wedge u_k(y)}{|x-y|^{s}} \right|^{\frac{2}{s}}\, \frac{dx\, dy}{|x-y|^{2}} < \varepsilon $$ then we show that we can either pass to the limit and obtain an almost everywhere immersion $\Phi$ or $\Phi$ collapses and is constant. This is in the spirit of the results by T. Toro, and S. M\"uller and V. Sverak, and F. H\'elein, who obtained similar statements under the stronger assumptions that the second fundamental form is bounded (but also stronger result: a locally bi-Lipschitz parametrization). The fractional normal curvature assumption is vaguely reminiscent of curvature energies such as the scaling-invariant limits of tangent-point energies for surfaces as considered by Strzelecki, von der Mosel et al. and we hope that eventually the analysis in this work can be used to define weak immersions with these kind of energy bounds.