A system of fifth-order partial differential equations describing a surface which contains many circles

Abstract

Let z=f(x,y) be a germ of a C5-surface at the origin in R3 containing several continuous families of circular arcs. For examples, a usual torus with 4 such families and Blum cyclides with 6 such families, which are special cases of Darboux cyclides. We introduce a system of fifth-order nonlinear partial differential equations for f, and prove that this system describes such a surface germ completely. As applications, we obtain the analyticity of f, the finite dimensionality of the solution space of such a system of differential equations with an upper estimate 21 for the dimension. Further we obtained some local characterization of Darboux cyclides by using this system of equations in our forthcoming paper: K. Kataoka, N. Takeuchi, The non-integrability of some system of fifth-order partial differential equations describing surfaces containing 6 families of circles, RIMS Kokyuroku Bessatsu Kyoto University, in press [1].