## Journal of Differential Geometry

### Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$

#### Abstract

For all open Riemann surface $\mathcal{N}$ and real number $\theta \in (0, \pi/2)$, we construct a conformal minimal immersion $X = (X_1,X_2,X_3) : \mathcal{N} \to \mathbb{R}^3$ such that $X_3+\tan(\theta)\left|X_1\right| : \mathcal{N} \to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with an arbitrarily prescribed flux map.

Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph.

#### Article information

Source
J. Differential Geom., Volume 90, Number 3 (2012), 351-381.

Dates
First available in Project Euclid: 24 April 2012

https://projecteuclid.org/euclid.jdg/1335273387

Digital Object Identifier
doi:10.4310/jdg/1335273387

Mathematical Reviews number (MathSciNet)
MR2916039

Zentralblatt MATH identifier
1252.53005

#### Citation

Alarcón, Antonio; López, Francisco J. Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$. J. Differential Geom. 90 (2012), no. 3, 351--381. doi:10.4310/jdg/1335273387. https://projecteuclid.org/euclid.jdg/1335273387