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Carleman approximation of maps into Oka manifolds


Author: Brett Chenoweth
Journal: Proc. Amer. Math. Soc. 147 (2019), 4847-4861
MSC (2010): Primary 32E30, 32V40, 32E10
DOI: https://doi.org/10.1090/proc/14595
Published electronically: May 29, 2019
MathSciNet review: 4011518
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Abstract: In this paper we obtain a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds. More precisely, we show that under suitable complex analytic conditions on a totally real set $M$ of a Stein manifold $X$, every smooth map $X \rightarrow Y$ to an Oka manifold $Y$ satisfying the Cauchy-Riemann equations along $M$ up to order $k$ can be $\mathscr {C}^k$-Carleman approximated by holomorphic maps $X \rightarrow Y$. Moreover, if $K$ is a compact $\mathscr {O}(X)$-convex set such that $K \cup M$ is $\mathscr {O}(X)$-convex, then we can $\mathscr {C}^k$-Carleman approximate maps which satisfy the Cauchy-Riemann equations up to order $k$ along $M$ and are holomorphic on a neighbourhood of $K$ or merely in the interior of $K$ if the latter set is the closure of a strongly pseudoconvex domain.


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Additional Information

Brett Chenoweth
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia
Email: brett.s.chenoweth@gmail.com

Keywords: Stein manifold, Oka manifold, holomorphic map, Carleman approximation, bounded exhaustion hulls.
Received by editor(s): August 6, 2018
Received by editor(s) in revised form: February 13, 2019
Published electronically: May 29, 2019
Additional Notes: The author was supported by grant MR-39237 from ARRS, Republic of Slovenia, associated to the research program P1-0291 Analysis and Geometry.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2019 American Mathematical Society