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

For all open Riemann surface M and real number $\theta \in (0,\pi \mathrm{/}4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M\to {\mathbb{R}}^3$ such that $X_3+\tan (\theta )\mathrm{\mid }{X}_1\mathrm{\mid }:M\to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with 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.