Pseudo-spherical Surfaces of Low Differentiability

We continue our investigations into Toda's algorithm [14,3]; a Weierstrass-type representation of Gauss curvature \(K=-1\) surfaces in \(\mathbb{R}^3\). We show that \(C^0\) input potentials correspond in an appealing way to a special new class of surfaces, with \(K=-1\), which we call \(C^{1M}\). These are surfaces which may not be \(C^2\), but whose mixed second partials are continuous and equal. We also extend several results of Hartman-Wintner [5] concerning special coordinate changes which increase differentiability of immersions of \(K=-1\) surfaces. We prove a \(C^{1M}\) version of Hilbert's Theorem.