Sphericity and analyticity of a strictly pseudoconvex hypersurface in low regularity

It is well known that the sphericity of strictly pseudoconvex real hypersurface amounts to the vanishing of its Chern-Moser tensor. The latter is computed pointwise in terms of the 6-jet of the hypersurface at a point, and thus requires regularity of the hypersurface of class at least $C^6$. In our joint work with Zaitsev, we apply our recent theorem on the analytic regularizability of a strictly pseudoconvex hypersurface to find a necessary and sufficient condition for the sphericity of a strictly pseudoconvex hypersurfaces of arbitrary regularity exceeding $C^{5/2}$. We as well discuss applications to the analytic regularizability for hypersurfaces of the respective classes. Surprisingly, despite of the seemingly analytic nature of the problem, our technique is geometric and is based on the Reflection Principle in SCV.