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

## TL;DR

This paper establishes the existence of solutions for certain Pfaffian systems with minimal regularity and applies this to prove the fundamental theorem of surface theory under optimal regularity conditions, also providing a compactness result.

## Contribution

It proves existence of solutions for low-regularity Pfaffian systems and extends the fundamental theorem of surface theory to optimal regularity classes.

## Key findings

- Existence of solutions in $W^{1,2}_ ext{loc}$ for Pfaffian systems with $L^2_ ext{loc}$ coefficients.
- Validation of the fundamental theorem of surface theory under minimal regularity assumptions.
- A weak compactness theorem for surface immersions in $W^{2,2}_ ext{loc}$.

## Abstract

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 a non-trivial solution in $W^{1,2}_\mathrm{loc}$ if the coefficients are antisymmetric and satisfy a compatibility condition. As an application of this result, we show that the fundamental theorem of surface theory holds for prescribed first and second fundamental forms of optimal regularity in the classes $W^{1,2}_\mathrm{loc}$ and $L^2_\mathrm{loc}$, respectively, that satisfy a compatibility condition equivalent to the Gauss-Codazzi-Mainardi equations. Finally, we give a weak compactness theorem for surface immersions in the class $W^{2,2}_\mathrm{loc}$.