Complete proper minimal surfaces in convex bodies of $R^3$

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Consider a convex domain B of space. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D' are convex domains with D bounded and the closure of D contained in D' then any minimal disk whose boundary lies in the boundary of D, can be approximated in any compact subdomain of D by a complete minimal disk which is proper in D'. We apply these results to study the so called type problem for a minimal surface: we demonstrate that the interior of any convex region is not a universal region for minimal surfaces, in the sense explained by Meeks and Perez.
26 pages, 7 figures

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General Mathematics, Differential Geometry, Primary 53A10; Secondary 49Q05, 49Q10, 53C42

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http://salesiana.dossiersoluciones.com/handle/123456789/71337

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0405507
 http://arxiv.org/abs/math/0405507

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arXiv