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Let $M$ be a Stein manifold with smooth, strictly pseudoconvex boundary, and $x$ a point on its boundary. Is there a holomorphic function $f$ on $M$, smooth on the boundary, with strict maximum of $|f|$ in $x$?

This question was treated for domains in ${\mathbb C}^n$, but I could not find any reference for bigger generality.

Update. The set of maxima is called "Shilov boundary". For domains in ${\mathbb C}^n$, Shilov boundary is the set of strictly pseudoconvex points on the boundary, as follows from the papers

Richard F. Basener, Peak Points, Barriers and Pseudoconvex Boundary Points, Proceedings of the American Mathematical Society Vol. 65, No. 1 (Jul., 1977), pp. 89-92. https://www.jstor.org/stable/2041997

M. Hakim and N. Sibony, Frontiere de Silov et spectre de A(D) pour des domaines faiblement pseudoconvexes, C. R. Acad. Sci. Paris 281, Serie A (1975), 959-962.

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  • $\begingroup$ I think the term for the set of points where such peaks occur is Shilov boundary. $\endgroup$
    – Ben McKay
    Commented Oct 8, 2021 at 10:45
  • $\begingroup$ Many thanks! That's right, now I also remember that term. More stuff to google, then $\endgroup$
    – Misha Verbitsky
    Commented Oct 8, 2021 at 13:04
  • $\begingroup$ ...and for a domain in C^n, Shilov boundary is the set of strictly pseudoconvex points, jstor.org/stable/2041997 Peak Points, Barriers and Pseudoconvex Boundary Points Richard F. Basener Proceedings of the American Mathematical Society Vol. 65, No. 1 (Jul., 1977), pp. 89-92 $\endgroup$
    – Misha Verbitsky
    Commented Oct 8, 2021 at 16:57

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