^{1}

A connected real analytic hypersurface

The goal of this paper is to provide the complete details of an alternative direct proof of a recent theorem due to Kossovskiy and Shafikov ([

Let

It is known (

Let

The set of Levi nondegenerate points of

If

Surprisingly, Example 6.2 in [

In [

In the previous theorem,

Let

Classically, writing

Moreover, since

Now, it is known ([

More generally,

One may either show-check that such a definition regives the standard definition of Levi nondegeneracy (

Although we could then spend time to reprove it properly, the following fact—here admitted—is well known.

If a connected real analytic hypersurface

Suppose to begin with for

In [

Introduce the expression

Unfortunately, it is essentially impossible to print in a published article what one obtains after a full expansion of these two derivations; other instances of this phenomenon appear in [

Nevertheless, by thinking a bit, one convinces oneself that, after full expansion and reduction to a common denominator, one obtains a kind of expression that we will denote in summarized form as

Before providing rigorous details to explain the latter assertion, a further comment is in order.

The explicit sphericity formula brings the information that denominator places are occupied by nondegeneracy conditions, so that division is allowed only at points where these conditions are satisfied, but numerators happen to be polynomial, a computational fact which hence enables one to jump across degenerate points through the “bridge-numerator” from one nondegenerate point to another nondegenerate point.

Now, the local version of Theorem

With

Take a (possibly much) smaller bidisc

Now, the Levi nondegeneracy of

Precisely because in [

But looking just at the numerator of the equation which expresses that

Take now any other Levi nondegenerate point:

To finish the proof of Theorem

Assume therefore that

By connectedness of

Since

We briefly summarize the quite similar arguments, relying upon [

The Levi determinant becomes

When one does apply Hachtroudi’s results to CR geometry (instead of Chern-Moser’s results, which is up to now not sufficiently explicit to be applied), the signature of the Levi forms disappears for the following reason.

The infinite-dimensional local Lie (pseudo)group of biholomorphic transformations

But when one passes to the extrinsic complexification, one replaces

It is then clear that all

Consequently, when one applies the main theorem of [

The author declares that they have no conflicts of interest.

The author thanks Valerii Beloshapka, Ilya Kossovskiy, and Rasul Shafikov and is especially grateful to Alexander Sukhov for introducing him to the subject of the interactions between CR geometry and partial differential equations (

^{5}