## Carleman approximation of maps into Oka manifolds

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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.## References

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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
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4847-4861 - MSC (2010): Primary 32E30, 32V40, 32E10
- DOI: https://doi.org/10.1090/proc/14595
- MathSciNet review: 4011518