Stability of Helicoids in Hyperbolic Three-Dimensional Space

TL;DR

This paper investigates the stability properties of minimal helicoids in hyperbolic 3-space, establishing a critical parameter value that determines their stability or instability.

Contribution

It identifies a specific critical constant for the stability transition of minimal helicoids in hyperbolic space, providing a precise stability criterion.

Findings

H_a is stable for 0<=a<=2.17966

H_a is unstable with index one for a>2.17966

Stability depends on the parameter a in hyperbolic space

Abstract

For a family of minimal helicoids H_a in the hyperbolic 3-space, there exists a constant a_c=2.17966 such that the following statements are true: (1) H_a is a globally stable minimal surface if 0<=a<=a_c, and (2) H_a is an unstable minimal surface with index one if a>a_c.