Nonrigid spherical real analytic hypersurfaces in C^2

Abstract

A Levi nondegenerate real analytic hypersurface M of C^2 represented in local coordinates (z, w) in C^2 by a complex defining equation of the form w = Theta (z, \bar z, \bar w) which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Theta satisfies an explicitly written sixth-order polynomial complex partial differential equation. In the rigid case (known before), this system simplifies considerably, but in the general nonrigid case, its combinatorial complexity shows well why the two fundamental curvature tensors constructed by Elie Cartan in 1932 in his classification of hypersurfaces have, since then, never been reached in parametric representation.