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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 thenormal 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 showthat 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 theresults by T. Toro, and S. M\"uller and V. Sverak, and F. H\'elein, whoobtained similar statements under the stronger assumptions that the secondfundamental form is bounded (but also stronger result: a locally bi-Lipschitzparametrization).  The fractional normal curvature assumption is vaguely reminiscent ofcurvature energies such as the scaling-invariant limits of tangent-pointenergies for surfaces as considered by Strzelecki, von der Mosel et al. and wehope that eventually the analysis in this work can be used to define weakimmersions with these kind of energy bounds.

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