Nonrigid spherical real analytic hypersurfaces in C^2

TL;DR

This paper characterizes when a Levi nondegenerate real analytic hypersurface in C^2 is spherical by deriving a sixth-order PDE for its defining function, highlighting the complexity in the nonrigid case and its relation to Cartan's curvature tensors.

Contribution

It provides an explicit sixth-order PDE condition for sphericity of nonrigid hypersurfaces in C^2, extending previous results from the rigid case.

Findings

Derived an explicit sixth-order PDE for sphericity.

Showed increased complexity in the nonrigid case.

Connected the PDE to Cartan's curvature tensors.

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.