Minimal surfaces in R³ properly projecting into R²

For all open Riemann surface N and real number θ ∈ (0, π/2), we construct a conformal minimal immersion X = (X₁,X₂,X₃) : N → R³ such that X₃+tan(θ)|X₁| : N → 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 R³ lying above a negative sublinear graph.