Local equivalence problem for Levi flat hypersurfaces

TL;DR

This paper studies the local equivalence of smooth Levi flat hypersurfaces using a new invariant, characterizing trivial cases and constructing infinitely many non-trivial, almost everywhere analytic examples.

Contribution

Introduces a simple invariant for local equivalence of Levi flat hypersurfaces and uses it to classify trivial and non-trivial cases, including infinite families.

Findings

Invariant characterizes trivial Levi flat hypersurfaces.

Constructs infinitely many non-trivial classes.

Identifies examples that are almost everywhere analytic.

Abstract

In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence: S_1 ~ S_2 if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant for this relation, which allows to prove some characterizations of triviality (i.e. equivalence to the hyperplane). Then, we employ the same invariant to construct infinitely many non-trivial classes, including an infinite family of not equivalent hypersurfaces which are almost everywhere analytic.