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.

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.