  While there are various results on the long-time behavior of the Willmore
flow, the Helfrich flow with non-zero spontaneous curvature as its natural
generalization is not yet well-understood. Past results for the gradient flow
of a locally area- and volume-constrained Willmore flow indicate the existence
of finite-time singularities which corresponds to the scaling-behavior of the
underlying energy. However, for a non-vanishing spontaneous curvature, the
scaling behavior is not quite as conclusive.
  Indeed, in this article, we find that a negative spontaneous curvature
corresponds to finite-time singularities of the locally constrained Helfrich
flow if the initial surface is energetically close to a round sphere.
Conversely however, in the case of a positive spontaneous curvature, we find a
positive result in terms of the convergence behavior: The locally
area-constrained Helfrich flow starting in a spherical immersion with suitably
small Helfrich energy exists globally and converges to a Helfrich immersion
after reparametrization. Moreover, this energetic smallness assumption is given
by an explicit energy threshold depending on the spontaneous curvature and the
local area constraint of the energy.
