Normal forms of para-CR hypersurfaces

TL;DR

This paper develops normal forms for finite type hypersurfaces in a product space, extending concepts from complex hypersurfaces, and analyzes their automorphisms, with results on convergence and classification.

Contribution

It introduces formal and convergent normal forms for regular and singular para-CR hypersurfaces, linking them to second-order ODEs and studying automorphism groups.

Findings

Established convergence of normal forms for regular hypersurfaces.

Derived normal forms for degenerate (Levi degenerate) cases.

Analyzed automorphism groups of finite type hypersurfaces.

Abstract

We consider hypersurfaces of finite type in a direct product space ${\mathbb R}^2 \times {\mathbb R}^2$, which are analogues to real hypersurfaces of finite type in ${\mathbb C}^2$. We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in ${\mathbb C}^2$. For the regular case, we study formal normal forms and prove convergence by following Chern and Moser. The normal form of such an hypersurface, considered as the solution manifold of a 2nd order ODE, gives rise to a normal form of the corresponding 2nd order ODE. For the degenerate case, we study normal forms for weighted $\ell$-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.