Global homotopy formulas on q concave CR manifolds for small degrees
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Global homotopy formulas on q-concave CR manifolds for small degrees Christine Laurent-Thiebaut and Jurgen Leiterer Prepublication de l'Institut Fourier no 706 (2007) www-fourier.ujf-grenoble.fr/prepublications.html Abstract Using functional analysis, we derive from local homotopy formulas for the tangential Cauchy-Riemann operator a global homotopy formula for compact CR manifolds without loss of regularity. Keywords: Homotopy formula, Tangential Cauchy-Riemann equation, compact CR mam- ifolds Resume Par des methodes d'analyse fonctionnelle, nous construisons, a partir de formules d'ho- motopies locales pour l'operateur de Cauchy-Riemann tangentiel, une formule d'homotopie globale pour les varietes CR compactes sans perte de regularite. Mots-cles : formule d'homotopie, equation de Cauchy-Riemann tangentielle, varietes CR compactes 2000 Mathematics Subject Classification : 32V20
Voir
- formule d'homotopie globale pour les varietes cr compactes sans perte de regularite
- cauchy-riemann tangentielle
- compact cr
- motopies locales pour l'operateur de cauchy-riemann tangentiel
- global homotopy
- without loss
- concave cr
- acting between
Global
homotopy formulas onqvec-noac small degrees
CR
manifolds
ChristineLaurent-Thie´bautandJu¨rgenLeiterer
Pr´epublicationdel’InstitutFourierno706 (2007) www-fourier.ujf-grenoble.fr/prepublications.html
Abstract
Using functional analysis, we derive from local homotopy formulas for the tangential Cauchy-Riemann operator a global homotopy formula for compact CR manifolds without loss of regularity.
Keywords formula, Tangential Cauchy-Riemann equation, compact CR mam-: Homotopy ifolds
R´esume´
Pardesme´thodesd’analysefonctionnelle,nousconstruisons,a`partirdeformulesd’ho-motopieslocalespourl’ope´rateurdeCauchy-Riemanntangentiel,uneformuled’homotopie globalepourlesvari´ete´sCRcompactessansperteder´egularit´e.
Mots-cle´s’delumrof:atqu´ee,pitomohoRCseirav´te´hc-yiRmeoidnCeuantielle,anntange compactes
2000 Mathematics Subject Classification: 32V20
for
2
ChristineLaurent-Thi´ebautandJ¨urgenLeiterer
P. Polyakov [P3, P4] proved global homotopy formulas for∂bonCRmanifolds and used them to study the embedding problem forCR together local formulas, whichmanifolds. Gluing were first introduced by Henkin [H] and Airapetian/Henkin [AH] and then further developed by Polyakov [P1, P2, P3, P4], Polyakov first constructs a global formula which is not yet a homotopy formula, but ”almost”, up to a compact perturbation. Then the main work is to eliminate this compact perturbation. This is successfully done by Polyakov, but with some loss of smoothness. In the present paper we use and develop a general functional analytic construction from [L] to eliminate such compact perturbations without any loss of smoothness. May be, this can be used to improve the results about the embedding problem forCRmanifolds obtained by Polyakov [P4]. This paper is written in the language of functional analysis. No integral formulas are used, but we use the results obtained elsewhere by integral formulas.
1 Notations used throughout the paper f In this papefr,Mis a complexfmanifold of complex dimensionnandEis a holomorphicfvector bundle of classCMs onM. Further,M⊆Mfis a generic, compactCRsubmanifoldf∞of ,ki the real codimension ofMinM, andOis the trivial complex line bundle onM. IfU⊆Mis an open set, then, for 0≤r≤n−k, the following notations are used: -Cn∞,r(U Eehst´rcefoapectheF)isE-valued (n r)-forms onUwhich are of classC∞, endowed with theC∞-topology. -Zn∞,r(U E) is the subspace of all closed forms inCn∞,r(U E), endowed with the same topology. -C,nl+rα(U E),l∈N, 0≤α <1, is the Banach space ofltimes differentiableE-valued (n r)-forms whose derivatives up to orderladmit extensions toUuounswhic¨Hloahernoitedcr with exponentα, endowed with theCl+α-topology. -Zln+,rα(U E) is the subspace of all closed forms inCnl+,rα(U E), endowed with the same topology.
- Ifr≥1, thenB,nl+rα→l(M E) is the space of allf∈ Cr,ln(M E) such thatf=dufor some u∈ Cnl+rα−1(M E). Sometimes we write also , Bn∞,r(M E) :=Bn∞,r→∞(M E) :=dCn∞,r−1(M E). - (Domd)0n,r(M E) is the space of allf∈ Cn0,r(M E) such that alsodfis continuous on M.
2 The main result
If 0< α <1 andqis an integer with 1≤q≤n−k, then we shall say thatconditionH(α q) is satisfiedif, for each point inM, there exist a neighborhoodUand linear operators Tr:Cn0,r(MO)→ Cn0,r−1UO1≤r≤q with the following two properties: (i) For alll∈Nand 1≤r≤q, TrC,rln(MO)⊆ Cnl,+rα−1(U O)
