Communications in Analysis and Geometry

Volume 24 (2016)

Number 3

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

Pages: 605 – 643



Nabil Kahouadji (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.; and Department of Mathematics, Northeastern Illinois University, Chicago, Il., U.S.A.)

Niky Kamran (Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Keti Tenenblat (Department of Mathematics, Universidade de Brasília, Brazil)


We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u_t = F(u, u_x, u_{xx})$ and $u_{xt} = F(u, u_x)$. We answer the following question: Given a pseudospherical 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 $u$.


evolution equations, nonlinear hyperbolic equations, pseudo-spherical surfaces, isometric immersions

2010 Mathematics Subject Classification

35L60, 37K25, 47J35, 53B10, 53B25

Published 22 June 2016