On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci

Abstract

A connected real analytic hypersurface whose Levi form is nondegenerate in at least one point—hence at every point of some Zariski-dense open subset—is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having a different signature . Up to signature, pseudosphericity then jumps across the Levi degenerate locus and in particular across the nonminimal locus, if there exists any.

1. Introduction

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 ([1]) which relies on the explicit zero-curvature equations obtained in [2, 3], following the lines of a clever suggestion of Beloshapka. In fact, the proof we give here freely brings a more general statement.

Let be a connected real analytic hypersurface with . One says that is pseudospherical at one of its points if it is locally near biholomorphic to some Heisenberg -pseudosphere having, in coordinates , the model quadric equation: for some integer with ; when , one simply says that is spherical. One calls the signature.

It is known (see [1] and the references therein) that a connected real analytic hypersurface of which is Levi nondegenerate at every point is -pseudospherical at one point if and only if it is -pseudospherical at every point. More generally, we establish that propagation of pseudosphericality also holds in presence of Levi degenerate points of arbitrary kind.

Theorem 1. Let be a connected real analytic geometrically smooth hypersurface which is Levi nondegenerate in at least one point (hence in some nonempty open subset). Then one has the following:(a)The set of Levi nondegenerate points of is a Zariski open subset of in the sense that there exists a certain proper—that is, having dimension —locally closed global real analytic subset locating exactly the Levi degenerate points of : (b)If is locally biholomorphic, in a neighborhood of one of its points , to some Heisenberg -pseudosphere having equation, for some integer with (so that necessarily is a Levi nondegenerate point of too), then locally at every other Levi nondegenerate point , the hypersurface is also locally biholomorphic to some Heisenberg -pseudosphere with possibly .

Surprisingly, Example  6.2 in [1] shows that may occur, in the case of a nonminimal hypersurface of with , a local example for which the Levi degenerate locus consists precisely of a complex -dimensional hypersurface contained in .

In [1], a similar theorem was proved, assuming that consists of a complex hypersurface (nonminimal case), while consists only of Levi nondegenerate points.

In the previous theorem, can be arbitrary.

2. Proof in

Let be a connected real analytic hypersurface. Pick a point and choose some affine coordinates centered at : satisfying so that the implicit function theorem represents as in terms of some graphing function which is expandable in convergent Taylor series in some (possibly small) open bidisc: with of course .

Classically, writing and using the analytic implicit function theorem, one solves in terms of , , and getting a representation of as recall that implicitly when one does this, one must consider as 4 independent complex variables, which amounts to complexify them, namely, to introduce the complexified variables: in what follows, we will work with -variables, keeping in mind that they can be replaced by since all objects are convergent Taylor series; so here, is a convergent Taylor series of for , , and , after shrinking if necessary.

Moreover, since one has

Now, it is known ([4, 5])—or it could be taken here as a definition—that is Levi nondegenerate at when the local holomorphic map is of rank 2 at ; of course, one would better think in terms of -variables here.

More generally, is Levi nondegenerate at an arbitrary point close to the origin: when the map is of rank 2 at , which precisely means the nonvanishing of the Jacobian determinant: at

One may either show-check that such a definition regives the standard definition of Levi nondegeneracy (cf. [4, 5]) or prove directly that, as it stands, it really is independent of the choice of coordinates ([6]).

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

Proposition 2. If a connected real analytic hypersurface is Levi nondegenerate in at least one point, then the set of Levi degenerate points of is a proper real analytic subset:

Suppose to begin with for that Since the above map is of rank 2 at , one can solve, following [2], the two equations: for the two variables , and then insert the latter in to get a complex second-order ordinary differential equation: One should notice that the possibility of solving is expressed by the nonvanishing of exactly and precisely the same Jacobian determinant as the one expressing Levi nondegeneracy.

In [2], one deduces from an explicitly known condition on for this second-order equation to be pointwise equivalent to the free particle Newtonian equation that a real analytic hypersurface which is Levi nondegenerate at as above is spherical in the sense—recall the definition—that it is locally biholomorphic to if and only if its complex graphing function satisfies an explicit (not completely developed) equation which we now present.

Introduce the expressionnoticing that its denominator does not vanish at the origin since was assumed (temporarily) to be a Levi nondegenerate point. Introduce also the vector field derivation Then the main and unique theorem of [2] states that is spherical at if and only if identically in .

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 [7, 8].

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 and hence instantly, sphericity of is characterized byOne notices that the complex graphing function is differentiated always at least once.

Interpretation. Then the true thing is that, after erasing the Levi determinant lying at denominator place, if this explicit equation vanishes in some very small neighborhood of some pointof the polydisc of convergence of , then by the uniqueness principle for analytic functions, the concerned polynomial numerator immediately vanishes identically all over , so that sphericity at one point should freely propagate to all other Levi nondegenerate points .

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 1 is as follows. Notice that from now on one does not assume anymore that the origin is a Levi nondegenerate point.

Proposition 3. With , , , , and as above, assuming that the real analytic subset of Levi degenerate points is proper, if is spherical at one Levi nondegenerate point then is also spherical at every other Levi nondegenerate point:

Proof. Take a (possibly much) smaller bidisc with to be chosen below, and center new coordinates at that is to say, introduce the new translated coordinates: The graphed complex equation then becomes Of course, the fact that reads and hence, in the new coordinates centered at , the equation of becomes in terms of a new graphing function that visibly satisfies Observation  4. For any integers with one has Proof. Indeed, the constant disappears after just a single differentiation. This completes the proof.
Now, the Levi nondegeneracy of at which reads according to what precedes as reads in the new coordinates as (remember that inequation (17) is independent of coordinates) which means as we know Levi nondegeneracy at .
Precisely because in [2] one needs only this condition to hold in order to associate as was explained above a second-order complex ordinary differential equation for some possibly very small this is where one has to choose with , the possible presence of rather close Levi degenerate points being a constraint in the needed application(s) of the implicit function theorem, one has the impression that one can in principle only determine whether is spherical restrictively in such a very narrow neighborhood of in , when one applies the main result of [2].
But looking just at the numerator of the equation which expresses that is spherical at in the coordinates if one takes into account the above observation, one readily realizes that so that the identical vanishing of the left-hand side for means the identical vanishing of the right-hand side for which lastly yields thanks to the uniqueness principle enjoyed by analytic functions the identical vanishing of the original numerator in the whole initial domain of convergence: Take now any other Levi nondegenerate point: The goal is to prove that is also spherical at . Center similarly new coordinates at : Introduce the new graphed equations: At such a point, since the Levi determinant is nonvanishing, one can for completeness construct the associated second-order complex ordinary differential equations or question directly whether local sphericity holds near by plainly applying the main theorem of [2]; namely, question whether the following equation holds: But then by exactly the same application of the above observation, we know that this last numerator satisfies when and since we already know that the latter right-hand side vanishes, according to the last boxed equation, we conclude that is indeed spherical at .

To finish the proof of Theorem 1 in the case , it remains only to globalize this local propagation of sphericity.

Assume therefore that is a connected real analytic smooth hypersurface, that is , so that the set of Levi nondegenerate points is Zariski open, everywhere dense in . Assume that there exists a spherical point (abbreviation for “an open neighborhood of in is locally biholomorphic to the unit -sphere ”). Take any other point .

By connectedness of , there exists a finite (possibly large) number of points and there exist small radii such that, in the open polydiscs, the local hypersurface is connected and graphed either as or as and such that is open nonempty in .

Since is locally stratified by a finite union of real submanifolds of dimension , this insures that Remember that the main Proposition 3 that precedes has shown that, for every , Then an immediate induction yields propagation from up to thereby shown to be spherical too.

3. Proof in

We briefly summarize the quite similar arguments, relying upon [3], which is unpublished and several times rejected, although useful here.

The Levi determinant becomes It is nonzero at one point if and only if is Levi nondegenerate at and also if and only if one can associate with a completely integrable system of second-order partial differential equations: Hachtroudi ([9]) established that such a system is pointwise equivalent to if and only if

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 acts simultaneously on -variables and on -variables as

But when one passes to the extrinsic complexification, one replaces -variables by new independent variables: considered as the constants of integration for the system of partial differential equations. Hence the local Lie (pseudo)group considered by Hachtroudi becomes enlarged as the group of transformations: in which the transformations on the “constant-of-integration” variables become completely discoupled from the group of transformations on the true variables . By definition (cf. the explanations in [2]), transformations on differential equations, when viewed in the space of solutions, are always of this general form.

It is then clear that all complexified Heisenberg pseudospheres become all pairwise equivalent through such transformations, because one is allowed to replace by without touching ; even the factor can be erased: Therefore, when passing to systems of partial differential equations associated with CR manifolds, Levi form signatures drop.

Consequently, when one applies the main theorem of [3], according to which a Levi nondegenerate having given Levi form signature is pseudospherical if and only if (notation same as in [3]) its Hachtroudi system is equivalent to and moreover if and only if—after translating back to the graphing function the explicit condition of Hachtroudi—the identical vanishing property holds, one can reason exactly as in the preceding section for —noticing that the denominator is the same as , noticing that the numerator is similarly polynomial in the partial derivatives of the graphing function —but when one jumps from a Levi nondegenerate point to another Levi nondegenerate point , from the property of local equivalence at to one can only conclude that the complexification of near is equivalent near to so that the Levi form signature can in principle change—and it really does in Example  6.3 of [1]—through Levi degenerate points.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

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 (cf. [10–12]). Theorem 1 was presented in the Abel Symposium 2013, organized by John-Erik Fornæss, Marius Irgens, and Erlend Fornæss-Wold and announced in [7].