TY - JOUR

T1 - Second-order equations and local isometric immersions of pseudo-spherical surfaces

AU - Kahouadji, Nabil

AU - Kamran, Niky

AU - Tenenblat, Keti

N1 - Funding Information:
Research partially supported by a CRM-ISM post-doctoral Fellowship, by NSERC Grant RGPIN 105490-2011 and by the Ministério de Ciência e Tecnologia, Brazil, CNPq Proc. No. 303774/2009-6.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Evolution equations

KW - Isometric immersions

KW - Nonlinear hyperbolic equations

KW - Pseudo-spherical surfaces

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U2 - 10.4310/CAG.2016.v24.n3.a7

DO - 10.4310/CAG.2016.v24.n3.a7

M3 - Article

AN - SCOPUS:84976328950

SN - 1019-8385

VL - 24

SP - 605

EP - 643

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

IS - 3

ER -