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# Communications in Analysis and Geometry

## Volume 24 (2016)

### Number 3

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

Pages: 605 – 643

DOI: https://dx.doi.org/10.4310/CAG.2016.v24.n3.a7

#### Authors

#### 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 $u_{xt} = F(u, u_x)$. 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 $u$.

#### Keywords

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

#### 2010 Mathematics Subject Classification

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

Published 22 June 2016