Second-order equations and local isometric immersions of pseudo-spherical surfaces

TL;DR

This paper investigates the dependence of the second fundamental form coefficients on solutions of certain pseudo-spherical surface equations, revealing that except for sine-Gordon, these coefficients are universal functions independent of the solution.

Contribution

It proves that for a class of second-order PDEs describing pseudo-spherical surfaces, the second fundamental form coefficients are generally independent of the solution, except in the sine-Gordon case.

Findings

Coefficients depend on a jet of order zero for sine-Gordon.

For all other equations in the class, coefficients are universal functions of x and t.

Existence of local isometric immersions implies solution independence of coefficients.

Abstract

We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u\_t=F(u,u\_x,u\_{xx})$ and $u\_{xt}=F(u, u\_x)$ given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat \cite{RabeloTenenblat90}. We answer the following question: Given a pseudo-spherical surface determined by a solution $u$ of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in $\mathbb{R}^3$ depend on a jet of finite order of $u$? We show that, except for the sine-Gordon equation, where the coefficients depend on a jet of order zero, for all other differential equations, whenever such an immersion exists, the coefficients are universal functions of $x$ and $t$, independent of $u$.