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Local envelopes on CR manifolds 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Local envelopes on CR manifolds 1

Egmont PORTEN*

Department of Engineering, Physics and Mathematics, Mid Sweden University, S-851 70 Sundsvall, Sweden

*Corresponding Author:
Egmont PORTEN
Department of Engineering, Physics and
Mathematics, Mid Sweden University
S-851 70 Sundsvall, Sweden
E-mail: [email protected]

Received date: January 10, 2007; Revised date: March 27, 2008

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Abstract

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.

Introduction

It is a very classical fact that for a strictly (pseudo)convex real hypersurface H in equation 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 equation smooth. Recall that a submanifold M ofequation is called CR manifold if the dimension of the complex tangent space equation does not depend on equation denoting multiplication by the complex unit ofequation In this case, the complex tangent spaces form a bundle equation whose complex rank m is called CR dimension of M, shortly m = CRdimM. A CR manifoldequation 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 equation 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 equation define the vector-valued Levi form byequation modequation whereequation is an arbitrary smooth section of equation extending X. It is easily verified that the expression is tensorial and yields a well defined mapping equationequation be the fiber of the characteristic bundleequation For nonzeroequation we define the directional Levi form by equation Now a generic CR manifold M is called strictly/weakly q-concave if for every nonzero equation the hermitian formequation has at least q negative/nonpositive eigenvalues. Finally M is called strictly pseudoconvex if equation is strictly definite for some nonzero equation (see [9] 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 equation intersecting M transversally in the origin. Hence equation is a smooth hypersurface of M near 0. For simplicity we will assume that the intersection is even J-generic, meaning that equation andequation are transverse, or equivalently, that equation is itself a generic CR manifold near 0. For a distinguished local side equation, we will consider subdomains equation whose boundaryequation contains a neighborhood of 0 in equation. 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 [4]. 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 equation be a smooth generic weakly 2-concave CR manifold of CR dimension m intersecting a smooth strictly pseudoconvex hypersurface equation J-generically in the origin. Let equation be a relative domain, lying on the pseudoconcave side of H and containing in its closure a neighborhood of the origin in equation.

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 equation

In the strictly 2-concave case, this was proved in [8] 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 [10] 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 [9]). 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 equation 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 equation, finding a geometric characterization is a long-standing open problem.

Proof of the main result

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 equation whereequation 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 equation containingequation Let (X, π) be the envelope of holomorphy of equation . Recall that X is an n-dimensional complex manifold, equation a locally biholomorphic map, and equation can be viewed as a subdomain of X via a canonical embeddingequation satisfying the lifting propertyequation The fact that X is the maximal domain over equationto which all holomorphic functions on equation extend simultaneously translates as follows: (i) equation is a topological isomorphism fromequation (extension) and (ii) X is a Stein manifold (maximality). Since X is Stein there is a strictly plurisubharmonic function equation such thatequation is relatively compact in X for allequation (see [6,7,9] for envelopes). Holomorphic extension from equation 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 equation

Let h(z) be a complex linear defining function of equation such thatequation and Re(h) increases along the direction pointing into the convex side. For ε > 0 small, we consider the family equation is very small, then convexity of H and J-genericity of equation is a weakly 1-concave generic CR submanifold of Bc of CR dimension m−1 > 0 (topologically an (dimM −2)-ball), (ii) equation is either empty, an isolated point or a compact ball. The latter means in particular that the boundaries of equation stay inequation We may furthermore assumeequation The idea is now to use a version of the continuity principle for subfamilies of the equation

To prove the claim it suffices to show that, for equation the unionequation Mc lifts to X equation denoting the straight segment inequation betweenequation If this is not the case, then there is a maximal half-open segment equation where equation such thatequation equation lifts to X. Maximality of equation implies that sup equation Since the distance of the boundaries of the equation to equation is positive, this implies that equation 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 manifoldequation If equation is a smooth strictly plurisubharmonic function defined near equation then we have equation

This follows from [5]. For the sake of completeness we provide a short argument: If equation the same holds for a generic Morse perturbationequation which we may choose such that equation has no critical points on equation is also Morse. Thenequation has somewhere a quadratic maximum equation touches the strictly pseudoconvex hypersurfaceequation 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 equation such that the intersectionequation with the concave side of M is contained in U+ and thatequation Applying the claim to the points ofequation we see that the restriction of anyequation to equation is bounded. Applying the claim with equation instead of equation , we obtain extension from equation to equation together with an estimateequation The estimate immediately follows from the inclusion equation In fact, if we hadequation thenequation 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 ([1], see also [11]). Since equation is generic near the origin, there is a smooth totally real n-dimensional submanifold equation We may include R into a smooth foliation equation of an M-neighborhood of the origin such that (i)equation are smooth real functions with independent differentials, (ii) the parameter ˆs ranges over some ball Us around the origin in equation and (iii) s1 is a local defining function of equation positive on the (+)-side of equation. Supplementing functions, we obtain real coordinatesequation on an M-neighborhood of 0. By [1], there are arbitrarily small open balls equation and equation all centered in 0, such that continuous CR functions onequation can be approximated by the restrictions of polynomials inequation locally uniformly on compact subsets of equation

For equation given as in Theorem 1.1, we may arrange thatequation Pick furthermore a slightly smaller ball equation 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 equation possesses a holomorphic extensionequation to an ambient neighborhood of equation satisfyingequation

Let now equation be polynomials approximatingequation Thenequation 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 equationequation g and gV glue into the desired extension. Uniqueness follows from general structure theorems [11] 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 equation asequation where f is a continuous CR function onequation 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 equation 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 [8]).

Weakly 1-concave counter-example

Our example will be a modification of the weakly but not strongly 1-concave hypersurface

equation

Note that M0 is foliated by complex lines and the Levi form2 L has one zero eigenvalue at every equation Pick a smooth functionequation which vanishes identically forequation and is strictly convex for t > 0. We claim that the hypersurface

equation

is weakly 1-concave in a neighborhood of the origin. To see this, we observe first that the term equation implies that the Levi form3 of equation is positive in the z2-direction, which is contained in equation Hence we have a positive direction at anyequation close to the origin. Secondly, we note that the slices

equation

are concave graphs over the real equation hyperplane inequation Consequently the Levi form ofequation must have a nontrivial nonpositive eigenvector tangent to M at anyequation This implies the claim.

Next we verify that the hypersurface equation can be embedded into a strictly pseudoconvex hypersurface H transverse to M. Note that the simplest candidate equationequation is only weakly pseudoconvex. Instead we try to construct H as a graphequation satisfying dh(0) = 0. The desired h is hence prescribed alongequation 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 equation do not intersectequation On the other hand, the intersectionequation contains points in an arbitrarily given neighborhood of the origin, if equation is sufficiently close to 0. Hence the functions equation 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. equation as in Theorem 1.1, and consider ambient open neighborhoodsequation We observe that there is always holomorphic extension fromequation through some part of H. To this end, we construct small Bishop discs attached to the generic CR manifold equation (see [9] 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 equation 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 equation 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.

Acknowledgement

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.

1Presented at the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics“, G¨oteborg, Sweden, October 11–13, 2007.

2For hypersurfaces the characteristic bundle is one-dimensional. Hence the total and the directional Levi forms coincide essentially.

3The Levi form of a function is equation The Levi form of a regular level setequation is the restriction of equation

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