Department of Engineering, Physics and Mathematics, Mid Sweden University, S-851 70 Sundsvall, Sweden
Received date: January 10, 2007; Revised date: March 27, 2008
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We study the problem whether CR functions on a sufficiently pseudoconcave CR manifold M extend locally across a hypersurface of M. The sharpness of the main result will be discussed by way of a counter-example.
It is a very classical fact that for a strictly (pseudo)convex real hypersurface H in holomorphic functions extend from the concave side across H. In the present note we will study the corresponding question for CR functions on embedded CR manifolds from a strictly local point of view.
All manifolds will be assumed to be smooth. Recall that a submanifold M of is called CR manifold if the dimension of the complex tangent space does not depend on denoting multiplication by the complex unit of In this case, the complex tangent spaces form a bundle whose complex rank m is called CR dimension of M, shortly m = CRdimM. A CR manifold is called generic if its CR dimension is as small as real/complex linear algebra allows, i.e. if m = n − codimM. A C1-function f on M is called CR function if is J-linear. Locally one may express this by a system of m independent linear first-order differential equations, allowing to interpret the CR property in distributional sense. The space of continuous CR distributions on M will be denoted by CR(M).
The nonintegrability of TcM is measured by the nonvanishing of the Levi form. For define the vector-valued Levi form by mod where is an arbitrary smooth section of extending X. It is easily verified that the expression is tensorial and yields a well defined mapping be the fiber of the characteristic bundle For nonzero we define the directional Levi form by Now a generic CR manifold M is called strictly/weakly q-concave if for every nonzero the hermitian form has at least q negative/nonpositive eigenvalues. Finally M is called strictly pseudoconvex if is strictly definite for some nonzero (see  for more on CR geometry).
Throughout we will work in the following setting: M will denote a smooth generic CR manifold passing through the origin and H a smooth real hypersurface of intersecting M transversally in the origin. Hence is a smooth hypersurface of M near 0. For simplicity we will assume that the intersection is even J-generic, meaning that and are transverse, or equivalently, that is itself a generic CR manifold near 0. For a distinguished local side , we will consider subdomains whose boundary contains a neighborhood of 0 in . We ask whether CR functions on U+ extend to a uniform Mneighborhood of 0.
Such extension cannot hold for strictly pseudoconvexM. In this case M can be, after a convenient holomorphic coordinate change, locally imbedded into some strictly convex hypersurface. This gives us plentiful functions with isolated peak points, destroying any hope for extension (independently of the shape of H). If we assume H to be strictly pseudoconvex, a similar reason excludes extension from domains U+ lying on the convex side. Our aim is to strive for weak assumptions on M guaranteing extension under the hypothesis H is strictly pseudoconvex and U+ lies on the concave side.
It can be seen that M is weakly 1-concave precisely if it is nowhere strictly pseudoconvex . It is known that in this case one has extension phenomena for certain Dirichlet-type problems [3,4,10]. Interestingly, weak 1-concavity of M is not enough for our Cauchy-type problem (see Section 3). Our main result is the following.
Theorem 1.1. Let be a smooth generic weakly 2-concave CR manifold of CR dimension m intersecting a smooth strictly pseudoconvex hypersurface J-generically in the origin. Let be a relative domain, lying on the pseudoconcave side of H and containing in its closure a neighborhood of the origin in .
Then there is an open neighborhood V of the origin in M such that every continuous CR functions on U+ uniquely extends to a continuous CR function on
In the strictly 2-concave case, this was proved in  by means of adapted integral formulas. Our approach will be very different, focusing on the geometry of related envelopes of holomorphy. In the weakly 2-concave case, Theorem 1.1 is even new for hypersurfaces. Here the reader may consult  for refinements for J-degenerate intersections.
In Section 3 we will see that Theorem 1.1 fails if M is only weakly 1-concave. Note that the CR orbits of M near 0 may be very complicated (see ). It is worth observing that in our situation we need no assumption on CR orbits. Compare this to global results in [2,10], where the situation is very different. Finally we remark that it should be a subtle task to sharpen the condition on significantly. Our arguments extend to the case where H is weakly pseudoconvex but satisfies a certain finite-type condition at 0 (see Remark 2.1). However, even for extendability of holomorphic functions from a given side of a real hypersurface of , finding a geometric characterization is a long-standing open problem.
After a quadratic holomorphic coordinate change, we may assume that H is strictly convex near the origin. After a unitary rotation, M writes as a smooth graph where The strategy is first to prove an extension result for holomorphic functions, and to conclude then by approximation techniques.
Part 1: Holomorphic extension. First we assume that we are to extend functions holomorphic in a thin ambient domain containing Let (X, π) be the envelope of holomorphy of . Recall that X is an n-dimensional complex manifold, a locally biholomorphic map, and can be viewed as a subdomain of X via a canonical embedding satisfying the lifting property The fact that X is the maximal domain over to which all holomorphic functions on extend simultaneously translates as follows: (i) is a topological isomorphism from (extension) and (ii) X is a Stein manifold (maximality). Since X is Stein there is a strictly plurisubharmonic function such that is relatively compact in X for all (see [6,7,9] for envelopes). Holomorphic extension from to a neighborhood of 0 is the content of the following claim: The mapping α extends as a lifting to an M-neighborhood V of the origin whose size depends on U+, but not on the particular shape of
Let h(z) be a complex linear defining function of such that and Re(h) increases along the direction pointing into the convex side. For ε > 0 small, we consider the family is very small, then convexity of H and J-genericity of is a weakly 1-concave generic CR submanifold of Bc of CR dimension m−1 > 0 (topologically an (dimM −2)-ball), (ii) is either empty, an isolated point or a compact ball. The latter means in particular that the boundaries of stay in We may furthermore assume The idea is now to use a version of the continuity principle for subfamilies of the
To prove the claim it suffices to show that, for the union Mc lifts to X denoting the straight segment in between If this is not the case, then there is a maximal half-open segment where such that lifts to X. Maximality of implies that sup Since the distance of the boundaries of the to is positive, this implies that is nonconstant and has a maximum in the interior whenever c is close to ˜c. This contradicts the subsequent maximum principle, and the claim follows.
Lemma 2.1. Let D be a relatively compact domain in a smooth generic weakly 1-concave CR manifold If is a smooth strictly plurisubharmonic function defined near then we have
This follows from . For the sake of completeness we provide a short argument: If the same holds for a generic Morse perturbation which we may choose such that has no critical points on is also Morse. Then has somewhere a quadratic maximum touches the strictly pseudoconvex hypersurface in z0 from the convex side and is therefore itself strictly pseudoconvex near z0. The lemma follows.
In the sequel, we will need a simple a-priori estimate: Pick such that the intersection with the concave side of M is contained in U+ and that Applying the claim to the points of we see that the restriction of any to is bounded. Applying the claim with instead of , we obtain extension from to together with an estimate The estimate immediately follows from the inclusion In fact, if we had then would still be holomorphic near B+ without being extendable along V .
Part 2: Approximation. CR extension will now be derived by an application of the Baouendi-Treves approximation theorem (, see also ). Since is generic near the origin, there is a smooth totally real n-dimensional submanifold We may include R into a smooth foliation of an M-neighborhood of the origin such that (i) are smooth real functions with independent differentials, (ii) the parameter ˆs ranges over some ball Us around the origin in and (iii) s1 is a local defining function of positive on the (+)-side of . Supplementing functions, we obtain real coordinates on an M-neighborhood of 0. By , there are arbitrarily small open balls and all centered in 0, such that continuous CR functions on can be approximated by the restrictions of polynomials in locally uniformly on compact subsets of
For given as in Theorem 1.1, we may arrange that Pick furthermore a slightly smaller ball The constructions in Part 1 depend continuously on the data. If λ > 0 is sufficiently small, we can find an M-neighborhood V of 0 such that every function f holomorphic near possesses a holomorphic extension to an ambient neighborhood of satisfying
Let now be polynomials approximating Then converges uniformly, hence also its restriction to V by the a-priori estimate. Thus the limit defines a continuous CR function gV on V . Since Pj approaches g locally uniformly on g and gV glue into the desired extension. Uniqueness follows from general structure theorems  or from a closer inspection of the approximation process. The proof of Theorem 1.1 is complete.
Remark 2.1. a) It is not very essential to work with continuous CR functions. If g is a CR distribution on U+, we may use a method from [1,11], to represent it on as where f is a continuous CR function on is a CR variant of the Laplace operator and k is a sufficiently large integer. Now one first extends f by Theorem 1.1 and obtains the desired extension as We omit the details.
b) The argument still works if H is only weakly pseudoconvex but possesses a supporting holomorphic hyperplane touching it (from the concave side) with finite-order contact at the origin. But, as mentioned in the introduction, it should be hard to obtain a sharp result.
c) One can reduce the number of strictly pseudoconvex directions required for H if one assumes weak q-concavity for M with q > 2 (compare ).
Our example will be a modification of the weakly but not strongly 1-concave hypersurface
Note that M0 is foliated by complex lines and the Levi form2 L has one zero eigenvalue at every Pick a smooth function which vanishes identically for and is strictly convex for t > 0. We claim that the hypersurface
is weakly 1-concave in a neighborhood of the origin. To see this, we observe first that the term implies that the Levi form3 of is positive in the z2-direction, which is contained in Hence we have a positive direction at any close to the origin. Secondly, we note that the slices
are concave graphs over the real hyperplane in Consequently the Levi form of must have a nontrivial nonpositive eigenvector tangent to M at any This implies the claim.
Next we verify that the hypersurface can be embedded into a strictly pseudoconvex hypersurface H transverse to M. Note that the simplest candidate is only weakly pseudoconvex. Instead we try to construct H as a graph satisfying dh(0) = 0. The desired h is hence prescribed along These partial data already imply that H will have positive Levi curvature in the z2-direction. But now it is standard that we can produce a strictly pseudoconvex H by bending H0 near the origin strongly enough along the y3-direction into the pseudoconvex side (without changing HM).
As a matter of fact, the complex hyperplanes do not intersect On the other hand, the intersection contains points in an arbitrarily given neighborhood of the origin, if is sufficiently close to 0. Hence the functions show that there is no local CR extension from M+ to a uniform neighborhood of the origin.
We conclude with a remark on the envelope of V+
Remark 3.1. Fix a domain. as in Theorem 1.1, and consider ambient open neighborhoods We observe that there is always holomorphic extension from through some part of H. To this end, we construct small Bishop discs attached to the generic CR manifold (see  for the disc method). Because of the strict pseudoconvexity of H the interior of the discs will lie in the pseudoconvex side of H. If we deform together with the attached discs into V+, we obtain a one-sheeted part of X (the envelope of holomorphy of V+) which passes through H into the pseudoconvex side and contains the origin in its closure.
Note that the size of this part of X depends sensitively on the thickness of V+. However the disc argument shows that for every V+ the projection of the envelope to contains points on the pseudoconvex side of distance to H bounded from below by some uniform positive constant. Of course the above arguments show that there is no M-neighborhood of the origin lifting simultaneously to all possible X. Intuitively speaking, the trouble is that the X lose the contact to M at the origin.
This research builds on a part of the author’s habilitation thesis, supported by the Humboldt University Berlin. I would like to thank C. Denson Hill, Christine Laurent-Thi´ebaut and J¨urgen Leiterer for inspiring discussions on the topic.