Lindel\"of's theorem for catenoids revisited

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.