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

Nabil Kahouadji, Niky Kamran, Keti Tenenblat

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

We consider the class of differential equations that describe pseudo-spherical surfaces of the form ut = F(u, ux,uxx) and uxt = F(u, ux). 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 ℝ3 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.

Original languageEnglish (US)
Pages (from-to)605-643
Number of pages39
JournalCommunications in Analysis and Geometry
Volume24
Issue number3
DOIs
StatePublished - 2016

Keywords

  • Evolution equations
  • Isometric immersions
  • Nonlinear hyperbolic equations
  • Pseudo-spherical surfaces

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Geometry and Topology
  • Statistics, Probability and Uncertainty

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