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  • Null curves and directed immersions of open Riemann surfaces
    Alarcón, Antonio ; Forstnerič, Franc, 1958-
    We study holomorphic immersions of open Riemann surfaces into ▫${\mathbb C}^n$▫ whose derivative lies in a conical algebraic subvariety ▫$A$▫ of ▫${\mathbb C}^n$▫ that is smooth away from the origin. ... Classical examples of such ▫$A$▫-immersions include null curves in ▫${\mathbb C}^3$▫ which are closely related to minimal surfaces in ▫${\mathbb R}^3$▫, and null curves in ▫$SL_2(\mathbb{C})$▫ that are related to Bryant surfaces. We establish a basic structure theorem for the set of all ▫$A$▫-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that ▫$A$▫ is irreducible and is not contained in any hyperplane, we show that every ▫$A$▫-immersion can be approximated by ▫$A$▫-embeddings; this holds in particular for null curves in ▫${\mathbb C}^3$▫. If in addition ▫$A \setminus \{0\}$▫ is an Oka manifold, then ▫$A$▫-immersions are shown to satisfy the Oka principle, including the Runge and the Mergelyan approximation theorems. Another version of the Oka principle holds when ▫$A$▫ admits a smooth Oka hyperplane section. This lets us prove in particular that every open Riemann surface is biholomorphic to a properly embedded null curve in ▫${\mathbb C}^3$▫.
    Vir: Inventiones Mathematicae. - ISSN 0020-9910 (Vol. 196, iss. 3, 2014, str. 733-771)
    Vrsta gradiva - članek, sestavni del
    Leto - 2014
    Jezik - angleški
    COBISS.SI-ID - 16655705

    Povezava(-e):

    http://dx.doi.org/10.1007/s00222-013-0478-8

vir: Inventiones Mathematicae. - ISSN 0020-9910 (Vol. 196, iss. 3, 2014, str. 733-771)
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