NDLI: Immersed Solutions of Plateau’s Problem for Piecewise Smooth Boundary Curves with Small Total Curvature

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Immersed Solutions of Plateau’s Problem for Piecewise Smooth Boundary Curves with Small Total Curvature

Content Provider	Springer Nature Link
Author	Desideri, Laura Jakob, Ruben
Copyright Year	2012
Abstract	We provide a new proof of the classical result that any closed rectifiable Jordan curve $${\Gamma \subset \mathbb{R}^3}$$ being piecewise of class C 2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Γ is smaller than 6π. In contrast to the methods due to Osserman (Ann Math 91(2):550–569, 1970), Gulliver (Ann Math 97(2):275–305, 1973) and Alt (Math Z 127:333–362, 1972, Math Ann 201:33–35, 1973), our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau’s problem for polygonal boundary curves, provided by the first author’s accomplishment (The Plateau problem, Fuchsian equations and the Riemann–Hilbert problem. Mémoires de la Soc. Math. Fr. (to appear) arXiv: 1003.0978) of Garnier’s ideas in (Annales scientifiques de l’É.N.S. 45:53–144, 1928).
Ending Page	901
Page Count	11
Starting Page	891
File Format	PDF
ISSN	14226383
e-ISSN	14209012
Journal	Results in Mathematics
Issue Number	3
Volume Number	63
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-03-09
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Minimal surfaces Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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