Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$

TL;DR

This paper introduces a method using complex parabolic rotations of holomorphic null curves to generate and analyze minimal surfaces in four-dimensional space, revealing new surfaces foliated by conic sections with various eccentricities.

Contribution

The authors develop a novel deformation technique transforming holomorphic null curves into minimal surfaces in ${\mathbb{R}}^{4}$, discovering new foliations by conic sections and extending classical minimal surface examples.

Findings

Discovered new minimal surfaces foliated by hyperbolas and straight lines in ${\mathbb{R}}^{4}$.

Reconstructed Hoffman-Osserman catenoids in ${\mathbb{R}}^{4}$ foliated by ellipses and circles.

Proved existence of minimal surfaces foliated by ellipses converging to circles at infinity.

Abstract

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^{3} \subset {\mathbb{C}}^{4}$ induced by helicoids in ${\mathbb{R}}^{3}$, we discover new minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections with eccentricity grater than $1$: hyperbolas or straight lines. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^{3}$ induced by catenoids in ${\mathbb{R}}^{3}$, we can rediscover the Hoffman-Osserman catenoids in ${\mathbb{R}}^{4}$ foliated by conic sections with eccentricity smaller than $1$: ellipses or circles. We prove the existence of minimal surfaces in ${\mathbb{R}}^{4}$ foliated by ellipses, which converge to circles at infinity. We construct minimal surfaces in ${\mathbb{R}}^{4}$ foliated by parabolas: conic sections which have eccentricity $1$.