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

## TL;DR

This paper constructs new classes of minimal surfaces in three-dimensional space that properly project into the plane, with controlled geometric properties and prescribed flux, expanding the understanding of minimal surface embeddings.

## Contribution

It introduces a method to construct conformal minimal immersions with specific projection properties and prescribed flux on any open Riemann surface.

## Key findings

- Constructed minimal surfaces with proper projections into b2.
- Produced hyperbolic minimal surfaces with boundary above a negative sublinear graph.
- Achieved control over flux maps in the constructed surfaces.

## Abstract

For all open Riemann surface M and real number $\theta \in (0,\pi/4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$ such that $X_3+\tan(\theta) |X_1|:M \to \mathbb{R}$ 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.