Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

TL;DR

This paper employs the equivariant moving frames method to systematically analyze normal forms and differential invariants of nondegenerate hypersurfaces in C^2, extending classical results and constructing new convergent normal forms at singular umbilic points.

Contribution

It introduces an algorithmic approach using moving frames to recover and extend classical normal form results, identifying a single differential invariant generating the algebra under generic conditions.

Findings

Complete differential invariants for non-umbilic hypersurfaces are identified.

A single order-7 invariant generates the algebra of invariants.

New convergent normal forms are constructed at singular umbilic points.

Abstract

We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M \subset C^2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allow us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p \in M or umbilic in an open neighborhood of it. In the former case, the coefficients of the normal form expansion, when expressed as functions of the jet of the hypersurface at the point, provide a complete system of functionally independent differential invariants that can be used to solve the equivalence problem. We prove that under a suitable genericity condition, the entire algebra of differential invariants for such hypersurfaces can be generated, through the operators of invariant differentiation, by a single real differential invariant of order 7. We then apply moving frames to construct new convergent normal forms for nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic around them.