Article dans une revue
Titre :
The ∂∂ ̄−problem for extensible currents defined in a ring
Auteurs :
M. E. Bodian, I. Hamidine, S. Sambou
Résumé :
In this paper, we solve the \(\partial \bar{\partial }\)-problem for extensible currents defined on \({\mathbb{C}}^{n}{\setminus } \bar{B}\) where B is the unit ball of \({\mathbb{C}}^{n}\). For that, we first solve the equation \(dS=T\) where T is an extendable current defined on \({\mathbb{C}}^{n}{\setminus }\bar{B}\). The resulting solution decomposes without loss of generality into a part \(\bar{\partial }\)-closed and \(\partial\)-closed. The set \({\mathbb{C}}^{n} {\setminus } \bar{B}\) has the geometrical conditions necessary for the resolution of \(\partial\) and \(\bar{\partial }\) for extensible currents (see Sambou in Annales de la Faculté des sciences de Toulouse 6e série, tome 11(1):105–129, 2002). Starting from results known from de Rham cohomology and the convex analogue (see Bodian et al. in C R Math Rep Acad Sci Canada 38:34–37, 2016 and Bodian et al. in Imhotep Math J 3(1):1–4, 2018), then the equation \(dS=T\) admits a solution. The \(\partial \bar{\partial }\) solution then becomes a consequence of the results of the solution of the \(\bar{\partial }\) for the extensible currents obtained in Sambou (Annales de la Faculté des sciences de Toulouse 6e série, tome 11(1):105–129, 2002). In the case of a manifold, we introduce the notion of contractible extension and then we solve the \(\partial \bar{\partial }\) in this context
Journal :
Iran J Sci Technol Trans Sci
Volume :
43
Pages ou Numéro article :
2885–2889
Année :
2019
DOI :
doi.org/10.1007/s40995-019-00759-5
Date de publication :
27 August 2019
Lien de la publication :
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