Stability Properties of Rotational Catenoids in the Heisenberg Groups

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.

Final version to appear in Matemática Contemporânea (Sociedade Brasileira de Matemática)

References in corpus (2)

• Lindelöf's theorem for catenoids revisited
• Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index

Cited by in corpus (3)

• Noether invariants for constant mean curvature surfaces in 3-dimensional homogeneous spaces
• Minimal Graphs in Nil_3 : existence and non-existence results
• Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group