Pseudo-spherical Surfaces of Low Differentiability

TL;DR

This paper explores low-differentiability surfaces with constant negative curvature, introducing a new class called C^{1M} surfaces and extending classical results to this broader context.

Contribution

It introduces the C^{1M} class of K=-1 surfaces, connecting them to Toda's algorithm and extending Hartman-Wintner's results on surface differentiability.

Findings

C^0 potentials correspond to C^{1M} surfaces with K=-1

Extended Hartman-Wintner results to C^{1M} surfaces

Proved a C^{1M} version of Hilbert's Theorem

Abstract

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.