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We prove that a Pfaffian system with coefficients in the critical space$L^2_\mathrm{loc}$ on a simply connected open subset of $\mathbb{R}^2$ has anon-trivial solution in $W^{1,2}_\mathrm{loc}$ if the coefficients areantisymmetric and satisfy a compatibility condition. As an application of thisresult, we show that the fundamental theorem of surface theory holds forprescribed first and second fundamental forms of optimal regularity in theclasses $W^{1,2}_\mathrm{loc}$ and $L^2_\mathrm{loc}$, respectively, thatsatisfy a compatibility condition equivalent to the Gauss-Codazzi-Mainardiequations. Finally, we give a weak compactness theorem for surface immersionsin the class $W^{2,2}_\mathrm{loc}$.

Optimal regularity for two-dimensional Pfaffian systems and the fundamental theorem of surface theory