IJMMS 29:12 (2002) 749–752 PII. S0161171202007597 http://ijmms.hindawi.com
© Hindawi Publishing Corp.
CONDITION R AND FEFFERMAN’S MAPPING THEOREM
PATRICK DARKO Received 14 May 2001
It is shown that Fefferman’s mapping theorem extends to the case of manifolds, that is a biholomorphic map between two strictly pseudoconvex manifolds extends smoothly to their boundaries.
2000 Mathematics Subject Classification: 32H40, 32T15.
1. Introduction. A central question in complex analysis is “does every proper holo- morphic mappingf:D→Dof bounded domainsD,Dwith smooth boundaries in Cnextend smoothly to the boundary ofD?”
The answer has been known to be “yes” in dimension one for a long time. In higher dimensions, in case D and D are strictly pseudoconvex and f is biholomorphic, Fefferman’s famous mapping theorem [8] answers the question in the affirmative.
Bell and Ligocka [4] simplified the proof of Fefferman’s mapping theorem and ex- tended the theorem to a wide class of pseudoconvex domains.
In [3], Fefferman’s mapping theorem was extended to smoothly bounded pseudo- convex subdomains of Stein manifolds that satisfy conditionR. Thereafter, the ques- tion was asked whether all smoothly bounded pseudoconvex domains satisfy con- dition R. Recently, Barrett [2] and Christ [5] have shown that this question has an answer in the negative. But that was not the end of conditionR, because the case of strictly pseudoconvex manifolds that are not Stein had not been determined. At first it was thought (because of the work of Barrett [1]) that one could not do without the assumption of Steinness.
In this note, we show that a strictly pseudoconvex manifold need not be Stein be- fore it satisfies conditionR; and following the work of Bedford et al. [3], we extend Fefferman’s mapping theorem to all strictly pseudoconvex manifolds.
2. Preliminaries. LetΩbe a relatively compact domain in ann-dimensional com- plex manifoldX. The spaceL2(n,0)(Ω)is defined to be the set of(n,0)formsωsuch that
ω2=
−1n2
Ωω∧ω¯ (2.1)
is finite. The spaceL2(n,0)(Ω)is a Hilbert space with inner product given by
(ω, η)=
−1n2
Ωω∧η.¯ (2.2)
750 PATRICK DARKO
The Bergman-Kobayashi projectionPΩassociated toΩis the orthogonal projection ofL2(n,0)(Ω)ontoH(n,0)(Ω), the closed subspace ofL2(n,0)(Ω)consisting of holomorphic (n,0)forms. IfΩ has a smooth boundary,Ωsatisfies condition R if the Bergman- Kobayashi projection associated toΩmapsC(n,0)∞ (Ω¯)intoC(n,0)∞ (Ω¯).
To make use of the proof in [3], we show that ifΩabove has smooth boundary and it is strictly pseudoconvex, thenΩsatisfies conditionR; and, in addition, ifp0is a point inXnear the boundary∂ΩofΩ, then there arenfunctionsg1, . . . , gnthat are holomorphic in a neighborhood of ¯Ωand that form a coordinate system atp0.
Our main result is the following theorem.
Theorem 2.1. LetX1and X2 be n-dimensional complex manifolds andΩ1X1, Ω2X2strictly pseudoconvex subdomains with smooth boundaries. Letf :Ω1→Ω2
be a biholomorphic mapping betweenΩ1andΩ2. Thenf extends smoothly to aC∞ diffeomorphism ofΩ¯1andΩ¯2.
3. Condition R. To establish conditionR for smoothly bounded strictly pseudo- convex subdomains of complex manifolds, we need a result of Gunning and Rossi [9]
which we met on the way to proving theorems in [6,7]. Their result is the following theorem.
Theorem3.1. LetΩbe a strictly pseudoconvex domain in a complex manifoldY. There are a Stein manifoldXand a proper holomorphic mappingπ:Ω→Xwith the following properties:
(i) π:ᏻXᏻΩ;
(ii) there are finitely many points x1, . . . , xz inX such thatπ−1(xj)is a compact subvariety ofΩof positive dimension, andπ:Ω\
π−1(xj)X\{x1, . . . , xz}. The first statement means that the rings of holomorphic functionsᏻXandᏻΩonXand Ω, respectively, are isomorphic under the map induced byπ. The second means that Ω\∪π−1(xj)andX\{x1, . . . , xz}are biholomorphic.
Now from the proof ofTheorem 3.1as given in [9], it is clear that there is a strictly pseudoconvex neighborhoodΩof ¯Ωsuch thatΩcan replaceΩinTheorem 3.1so that the compact set∪π−1(xj)corresponding toΩis contained inΩ.
IfXcorresponds toΩinTheorem 3.1andX=π (Ω), then clearly ifΩhas a smooth boundary thenXis a Stein strictly pseudoconvex manifold with a smooth boundary, and therefore, as is well-known,Xsatisfies conditionR.
We can regardΩas imbedded inX. Then it is clear thatL2(n,0)(Ω)=L2(n,0)(X)and H(n,0)(Ω)=H(n,0)(X). Therefore the Bergman-Kobayashi projectionsPX and PΩ are equal, and it is not difficult to see (using Sobolev spaces) thatΩsatisfies conditionR.
4. Local coordinates near the boundary. Again fromTheorem 3.1we get the last theorem that we need in the proof ofTheorem 2.1.
Theorem4.1. LetΩbe a strictly pseudoconvex subdomain of a complex manifoldY. Then near the boundary∂ΩofΩ, local coordinates are given by holomorphic functions in a neighborhood ofΩ¯.
CONDITIONRAND FEFFERMAN’S MAPPING THEOREM 751 Proof. As indicated inSection 3, from the proof ofTheorem 3.1as given in [9] it is clear that there is a strictly pseudoconvex neighborhoodΩof ¯Ωsuch thatΩcan replaceΩinTheorem 3.1so that the compact set∪π−1(xj)corresponding toΩis contained inΩ. Now ifp0is a point inΩnear the boundary∂ΩofΩ, letπ (p0)have holomorphic functionsg1, . . . , gnon the Stein manifoldXthat form local coordinates atπ (p0). Theng1◦π , . . . , gn◦πform local coordinates atp0, which are holomorphic in a neighborhood of ¯Ω.
5. Proof of Theorem2.1. The proof of Theorem 2.1relies on the following two lemmas whose proofs are in [3].
Lemma5.1. Ifωis a holomorphic(n,0)form inC(n,0)∞ (Ω¯2), thenf∗ωis inC(n,0)∞ (Ω¯1).
Lemma5.2. Ifωis a holomorphic(n,0)form inC(n,0)∞ (Ω¯2)that vanishes to at most finite order at any boundary point ofΩ2, thenf∗ωvanishes to at most finite order at any boundary point ofΩ1.
Now to proveTheorem 2.1, we initiate the proof ofTheorem 2.1in [3]:
Letp0be a boundary point ofΩ1and letz1, . . . , znbe holomorphic coordinates near p0. We show thatfextends smoothly to∂Ω1nearp0. Let{pi}be a sequence of points inΩ1that converges top0. Then{f (pi)}converges to a pointq0in∂Ω2. Letg1, . . . , gn
benfunctions onΩ2that extend to be holomorphic in a neighborhood of ¯Ω2inX2
and that form a coordinate chart atq0. Define a holomorphic functionunearp0via udz1∧dz2∧···∧dzn=f∗
dg1∧dg2∧···∧dgn
. (5.1)
By Lemmas5.1and5.2uextends smoothly to∂Ω1nearp0anduvanishes to a finite order nearp0.
Ifα=(α1, . . . , αn)is a multi-index, then we definegα=n
i=1gαii.Lemma 5.1implies that the formf∗(gαdg1∧ ··· ∧dgn)extends smoothly to ∂Ω1nearp0 for eachα.
Hence,uandu(gα◦f )extend smoothly to∂Ω1nearp0for eachα, anduvanishes to at most finite order atp0. By the division theorem cited in [3],gi◦fextends smoothly to∂Ω1nearp0for eachi. Hencef extends smoothly to∂Ω1nearp0. Sincep0was arbitrarily chosen, we conclude thatf extends smoothly to all of∂Ω1. Now we can replacef byf−1and then the theorem follows.
References
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[3] E. Bedford, S. Bell, and D. Catlin,Boundary behavior of proper holomorphic mappings, Michigan Math. J.30(1983), 107–111.
[4] S. Bell and E. Ligocka,A simplification and extension of Fefferman’s theorem on biholomor- phic mappings, Invent. Math.57(1980), 283–289.
[5] M. Christ,GlobalC∞irregularity of the∂-Neumann problem for worm domains, J. Amer.
Math. Soc.9(1996), 1171–1185.
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752 PATRICK DARKO
[7] ,Corrigendum: “TheL2∂-problem on manifolds with piecewise strictly pseudoconvex boundaries”, Math. Proc. Cambridge Philos. Soc.123(1998), 191–192.
[8] C. Fefferman,The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math.26(1974), 1–65.
[9] R. C. Gunning and H. Rossi,Analytic Functions of Several Complex Variables, Prentice-Hall, New Jersey, 1965.
Patrick Darko: Department of mathematics, Lincoln University, Lincoln, PA19352, USA
