Embedding subsets of tori Properly into 2
Annales de l'Institut Fourier, Volume 57 (2007) no. 5, p. 1537-1555
Let 𝕋 be a complex one-dimensional torus. We prove that all subsets of 𝕋 with finitely many boundary components (none of them being points) embed properly into 2 . We also show that the algebras of analytic functions on certain countably connected subsets of closed Riemann surfaces are doubly generated.
Nous avons fait des progrès sur le problème du plongement des surfaces de Riemann ouvertes dans 2 . Il est connu que pour tout entier naturel d2, le nombre N d :=3d 2+1 est le plus petit entier naturel pour lequel il existe un plongement propre de toute variété de Stein de dimension d dans N d . Le problème du plongement propre des variétés de Stein de dimension 1 dans 2 reste ouvert (il existe du plongement propre dans 3 ). Dans ce texte nous prouvons le résultat suivant  : soit 𝕋 un tore complexe de dimension 1  ; alors il existe un plongement propre de toute partie de 𝕋, dont la frontière a un nombre fini de composantes (aucune d’elle n’étant un point), dans 2 . Nous prouvons aussi que les algèbres de fonctions analytiques sur certaines surfaces de Riemann sont doublement générées.
DOI : https://doi.org/10.5802/aif.2305
Classification:  32H35,  30F99
Keywords: Holomorphic embeddings, Riemann surfaces
@article{AIF_2007__57_5_1537_0,
     author = {Wold, Erlend Forn\ae ss},
     title = {Embedding subsets  of tori Properly into $\mathbb{C}^2$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     pages = {1537-1555},
     doi = {10.5802/aif.2305},
     zbl = {1149.32015},
     mrnumber = {2364141},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_5_1537_0}
}
Embedding subsets  of tori Properly into $\mathbb{C}^2$. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1537-1555. doi : 10.5802/aif.2305. https://aif.centre-mersenne.org/item/AIF_2007__57_5_1537_0/

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