## Duke Mathematical Journal

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

#### Abstract

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

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

Dates
First available in Project Euclid: 9 June 2005

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1118341233

Digital Object Identifier
doi:10.1215/S0012-7094-04-12835-0

Mathematical Reviews number (MathSciNet)
MR2145744

Zentralblatt MATH identifier
1082.53009

#### Citation

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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