Stability Properties of Rotational Catenoids in the Heisenberg Groups

TL;DR

This paper investigates the stability and Morse index of rotational catenoids in the Heisenberg group, revealing how their stability properties depend on a parameter and extending the analysis to higher dimensions.

Contribution

It provides the first detailed analysis of the stability and Morse index of rotational catenoids in the Heisenberg group, including bounds and asymptotic behavior.

Findings

Catenoids have Morse index at least 3.

The index is bounded above by a function of parameter a.

The index tends to infinity as a increases.

Abstract

In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $a$. We also show that the index of $\cC_a$ tends to infinity with $a$. Finally, we study the rotationally symmetric stable domains on the higher dimensional catenoids.