Title: 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.