On the regularity of nondegenerate hypo-analytic structures of hypersurface type
Ilya Kossovskiy, Vinícius Novelli
TL;DR
This work extends analytic regularizability results beyond Levi-nondegenerate CR hypersurfaces to general nondegenerate locally integrable structures of hypersurface type by introducing the central CR submanifold \Sigma. It proves that analytic regularizability of the ambient structure $\mathcal{V}$ is equivalent to that of $\Sigma$ when the Levi form is definite, and it shows that regularizability can be inferred from Marson's external CR construction via condition (E). The central-manifold approach effectively reduces the global equivalence problem to CR geometry on $\Sigma$, enabling the use of CR tools such as normal forms and parabolic geometries for classification, and it yields concrete criteria and applications, including results for rigid structures. Overall, the paper provides a unified framework linking regularity, CR geometry, and auxiliary constructions to characterize when nondegenerate locally integrable structures become real-analytic.
Abstract
For a smooth, non-degenerate locally integrable structure of hypersurface type on a manifold $M$, we provide necessary and sufficient conditions for it to be equivalent, near a point, to a real-analytic locally integrable structure (the analytic regularizability), generalizing a recent result of Zaitsev and the first author. First, we discover, in our setting, a (previously unknown) invariant CR submanifold $Σ$ in $M$ of hypersurface type, which we call the central submanifold. We prove that the analytic regularizability of $M$ is equivalent to that of the associated CR manifold $Σ$. Furthermore, as a byproduct of our construction, we show that the central manifold construction reduces the whole (smooth or analytic) equivalence problem for nondegenerate structures with the Levi positivity condition to that of the associated central manifolds, i.e. to CR geometry. Second, we make use of a classical construction due to Marson and show that sufficient for the analytic regularizability of $M$ is the analytic regularizability of the CR manifold $\tilde M$ associated with $M$ in the sense of Marson. We show applications of both regularizability conditions to classes of locally integrable structures.