  In [12], the authors studied a particular class of equilibrium solutions of
the Helfrich energy which satisfy a second order condition called the reduced
membrane equation. In this paper we develop and apply a second variation
formula for the Helfrich energy for this class of surfaces.
  The reduced membrane equation also arises as the Euler-Lagrange equation for
the area of surfaces under the action of gravity in the three dimensional
hyperbolic space. We study the second variation of this functional for a
particular example.
