Maximum modulus sets and reflection sets

Nagel, Alexander; Rosay, Jean-Pierre

Annales de l'Institut Fourier, Tome 41 (1991) no. 2, pp. 431-466.

Nous étudions les ensembles dans la frontière d’un domaine de $C^n$, sur lesquels une fonction holomorphe est de module maximum constant. En particulier, nous montrons que dans une frontière réelle analytique strictement pseudoconvexe, les sous-variétés de dimension maximum permise, qui sont ensembles de module maximum, sont réelles analytiques. Les ensembles de réflexion sont les ensembles le long desquels des collections appropriées de fonctions holomorphes et antiholomorphes coïncident, ils interviennent dans l’étude des ensembles de module maximum.

We study sets in the boundary of a domain in $C^n$, on which a holomorphic function has maximum modulus. In particular we show that in a real analytic strictly pseudoconvex boundary, maximum modulus sets of maximum dimension are real analytic. Maximum modulus sets are related to reflection sets, which are sets along which appropriate collections of holomorphic and antiholomorphic functions agree.

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