On a system of fifth-order partial differential equations describing surfaces containing two families of circular arcs

Abstract

Let z = f(x, y) be a germ of a C ⁵-surface at the origin in ℝ³ containing several continuous families of circular arcs. For examples, we have a usual torus with four such families and R. Blum's cyclide with six such families. We introduce the system of fifth-order nonlinear partial differential equations for f which describes such a surface germ completely. As applications, we obtain the analyticity of f, and the finite dimensionality of the solution space of such a system of differential equations. We give a brief survey of [Kataoka K, Takeuchi N. A system of fifth-order partial differential equations describing a surface which contains many circles, UTMS 2012-10 (Preprint series of Graduate School of Mathematical Sciences, the University of Tokyo)] concerning surfaces containing two families of circular arcs.

Keywords:

• circle
• cyclide
• surface
• nonlinear PDE
• fifth-order
• analyticity

AMS Subject Classifications::

• 51F99
• 35G50
• 35J62

Acknowledgements

This work was supported by Global COE Program ‘The Research and training centre for new development in mathematics’, MEXT, Japan, and Grants-in-Aid for Scientific Research, JSPS (No.23654047). This article is devoted to Professor Akira Kaneko, when he retires from Ochanomizu University. The first author learned many things from him concerning the theory of hyperfunctions.