Local isometric immersions of pseudo-spherical surfaces and k-th order evolution equations

Abstract

We consider the class of evolution equations that describe pseudo-spherical surfaces of the form u\_t = F (u, $\partial$u/$\partial$x, ..., $\partial$^k u/$\partial$x^k), k $\ge$ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature-1. We investigate the following problem: given such a metric, is there a local isometric immersion in R 3 such that the coefficients of the second fundamental form of the surface depend on a jet of finite order of u? By extending our previous result for second order evolution equation to k-th order equations, we prove that there is only one type of equations that admit such an isometric immersion. We prove that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions u are universal, i.e., they are independent of u. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.