A parabolic minimal surface in R3 intersects every nonflat properly embedded minimal surface of bounded curvature

TL;DR

This paper proves that a nonconstant conformal harmonic map from the complex plane into three-dimensional space intersects all nonflat, properly embedded minimal surfaces of bounded curvature, extending to more general conformal surfaces.

Contribution

It establishes a universal intersection property for harmonic maps with minimal surfaces, generalizing previous results to broader classes of conformal surfaces.

Findings

Harmonic maps from  intersect all nonflat minimal surfaces of bounded curvature.

The result extends to any open conformal surface without nonconstant bounded subharmonic functions.

The intersection property holds even if the harmonic map has branch points or is not proper.

Abstract

We show that the image of a nonconstant conformal harmonic map $\mathbb C\to \mathbb R^3$, not necessarily proper and possibly with branch points, intersects every properly embedded nonflat minimal surface of bounded curvature in $\mathbb R^3$. The same holds if $\mathbb C$ is replaced by any open conformal surface without nonconstant bounded subharmonic functions.