Logarithmically spiraling helicoids

TL;DR

This paper constructs embedded minimal disks resembling helicoids with axes along logarithmic spiral curves, exhibiting self-similarity and embeddedness within a logarithmic cone, advancing understanding of minimal surface geometries.

Contribution

It introduces a novel class of embedded minimal disks with axes on self-similar logarithmic spiral curves, demonstrating self-similarity and embeddedness properties.

Findings

Surfaces are embedded within a logarithmic cone.

Constructs a new family of minimal disks with spiral axes.

Surfaces exhibit self-similarity inherited from the curves.

Abstract

We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.