Sphericity and Analyticity of a strictly pdeusoconvex hypersurface in low regularity I

TL;DR

This paper extends analytic regularizability theory for strictly pseudoconvex hypersurfaces to low regularity cases, introduces new tools for CR geometry, and solves the sphericity problem in low regularity in complex dimension two.

Contribution

It generalizes previous results to hypersurfaces with finite regularity, introduces regularizing (0,1) sections, and solves the sphericity problem in low regularity in complex dimension two.

Findings

Extended regularizability results to low regularity hypersurfaces.

Introduced regularizing (0,1) sections as a new analytical tool.

Proved sphericity of certain low regularity hypersurfaces in C^2.

Abstract

In our earlier work \cite{KZ}, we introduced an analytic regularizability theory for smooth strictly pseudoconvex hypersurfaces in complex space. That is, we found a necessary and sufficient condition for a hypersurface to be CR-equivalent to an analytic target. The condition amount to the holomorphic extension property for a smooth function on a totally real submanifold, both the function and the submanifold being uniquely associated with the given hypersurface. In the present paper, we develop our method further. First, we extend the result in \cite{KZ} to hypersurfaces of finite (possibly low) smoothness. Second, we introduce a new tool for studying CR hypersurfaces in low regularity called {\em regularizing $(0,1)$ sections}. Using the latter key tool, we solve the open problem of checking the {\em sphericity} of a strictly pseudoconvex hypersurface in $\mathbb C^{2}$ in {low regularity}, precisely in the regularity $C^k,\,2\leq k< 7$ which is {\em not} covered by the classical Cartan-Tanaka-Chern-Moser theory. As an application of our theory, we deduce the sphericity of a strictly pseudoconvex hypersurface in $\mathbb C^{2}$ of regularity $C^6$ with vanishing Cartan-Chern CR-curvature.