Lindel\"of's theorem for catenoids revisited

TL;DR

This paper investigates the maximal stable domains on minimal catenoids across Euclidean, hyperbolic, and product spaces, focusing on Lindel"of's property and stable domains on related surfaces.

Contribution

It revisits Lindel"of's theorem for catenoids, analyzing stability properties and maximal domains in various geometric contexts, including hyperbolic space and catenoid-cousins.

Findings

Half-vertical catenoids are not maximal stable domains in the studied spaces.

Stable domains on catenoid-cousins in hyperbolic space are characterized.

The work extends Lindel"of's classical results to new geometric settings.

Abstract

In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in $H^2 \times R$. We in particular investigate whether half-vertical catenoids are maximal stable domains (\emph{Lindel\"of's property}). We also consider stable domains on catenoid-cousins in hyperbolic space. Our motivations come from Lindel\"of's 1870 paper on catenoids in Euclidean space.