Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

Abstract

We consider the class of evolution equations that describe pseudo-spherical surfaces of the form ut = F (u, ∂u/∂x, ..., ∂ k u/∂x k ), k ≥ 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.

Third-order differential equations and local isometric immersions of pseudospherical surfaces

“…Our goal in this section is to analyze the system (12), (13), (14) governing the components a, b, c of the second fundamental form and to obtain necessary conditions for the existence of solution depending on jets of finite order of u. We note that since the coefficients f ij appearing in the classification given in the Section 2, i.e., in Theorems 2.2-2.5, depend only on z 0 , z 1 and z 2 , it follows that the functions ∆ ij defined in (15) depend only on z 0 , z 1 and z 2 .…”

“…Firstly, let use (59) and the Gauss equation in order to obtain b and c in terms of a, f 11 and f 21 . We will then substitute the total derivatives of b and c back into (13) and (14). If (59) holds then substituting c into the Gauss equation leads to…”

“…It is therefore a natural question to ask whether such a remarkable property holds for other equations within the class of differential equations describing pseudospherical surfaces, or whether the sine-Gordon equation in any way special in this regard. In [13] and [14], we investigated this question for k-th order evolution equations u t = F (u, u x , ..., u x k ), (17) and second order hyperbolic equations u xt = F (u, u x ), (18) and proved that there are no other equations than the sine-Gordon equation for which this property holds, except for some special equations for which a, b, c are universal , that is functions of x, t which are independent of u. These results show that the sine-Gordon equation occupies a special position within the class of differential equations of the form (17) and (18) which describe pseudospherical surfaces.…”

“…In Section 2, we recall without proof the classification results of [2] that will be needed to prove Theorem 1.1. The classification splits into branches which are treated on a case-by-case basis in Section 3, starting from the expression of the Codazzi and Gauss equations in terms of the coefficients f ij of the 1-forms ω 1 , ω 2 , ω 3 (see (13), (14)). Finally, we carry out in Section 3.3 the integration of the Codazzi and Gauss equations in the cases in which the components a, b, c of the second fundamental form are universal functions of x and t and obtain explicit expressions for these functions.…”

“…A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In a pair of earlier papers [13], [14] we have shown that this property fails to hold for all k-th order evolution equations ut = F (u, ux, ..., u x k ) and all other second order equations of the form uxt = F (u, ux), except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of x and t which are independent of the choice of solution u. In the present paper, we consider thirdorder equations of the form ut − uxxt = λuuxxx + G(u, ux, uxx), λ ∈ R, which describe pseudospherical surfaces.…”