We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain of dimension . If is a smooth manifold of dimension and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in with the same smooth -dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in vanishes or if is polynomially convex. Finally, the maximum modulus set of an arbitrary has dimension, in the topological sense, not exceeding .
Nous étudions les sous-ensembles du bord d’un domaine strictement pseudoconvexe de dimension , où la valeur absolue d’une fonction de ou de prend son maximum. Ces ensembles sont les maximum modulus sets du titre. Si est une variété différentiable de dimension réelle , et si est l’ensemble des points où la valeur absolue d’une fonction atteint son maximum, alors est totalement réelle et elle admet une structure feuilletée avec comme feuilles des variétés compactes qui sont des ensembles pics d’interpolation. Il y a une converse partielle dans le cas analytique réel. Deux fonctions de qui ont la même variété différentiable de dimension comme “maximum modulus set”, satisfont une relation analytique, et cette relation est polynomiale si une classe particulière de s’annule ou si est polynomialement convexe. Finalement, pour toute fonction , la dimension topologique de l’ensemble des points où prend son maximum est au plus .
@article{AIF_1981__31_3_37_0, author = {Duchamp, Thomas and Stout, Edgar Lee}, title = {Maximum modulus sets}, journal = {Annales de l'Institut Fourier}, pages = {37--69}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {31}, number = {3}, year = {1981}, doi = {10.5802/aif.837}, mrnumber = {83d:32019}, zbl = {0439.32007}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.837/} }
TY - JOUR AU - Duchamp, Thomas AU - Stout, Edgar Lee TI - Maximum modulus sets JO - Annales de l'Institut Fourier PY - 1981 SP - 37 EP - 69 VL - 31 IS - 3 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.837/ DO - 10.5802/aif.837 LA - en ID - AIF_1981__31_3_37_0 ER -
Duchamp, Thomas; Stout, Edgar Lee. Maximum modulus sets. Annales de l'Institut Fourier, Volume 31 (1981) no. 3, pp. 37-69. doi : 10.5802/aif.837. http://archive.numdam.org/articles/10.5802/aif.837/
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