Duke Mathematical Journal

Complete proper minimal surfaces in convex bodies of $\mathbbR^3$

Francisco Martín and Santiago Morales

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Consider a convex domain $B$ of $\mathbbR^3$. We prove that there exist complete minimal surfaces that 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 that 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 of $\mathbbR^3$ is not a universal region for minimal surfaces, in the sense explained by Meeks and Pérez in [9].

Article information

Duke Math. J. Volume 128, Number 3 (2005), 559-593.

First available in Project Euclid: 9 June 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]


Martín, Francisco; Morales, Santiago. Complete proper minimal surfaces in convex bodies of ℝ 3 . Duke Math. J. 128 (2005), no. 3, 559--593. doi:10.1215/S0012-7094-04-12835-0. http://projecteuclid.org/euclid.dmj/1118341233.

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