  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.

Differential Geometry