  We study the effect of non-vanishing surface terms at spatial infinity on the
dynamics of a scalar field in an open FLRW spacetime. Starting from the
path-integral formulation of quantum field theory we argue that classical
physics is described by field configurations which extremize the action
functional in the space of field configurations for which the variation of the
action is well defined. Since these field configurations are not required to
vanish outside a bounded domain, there is generally a non-vanishing
contribution of a surface term to the variation of the action. We then
investigate whether this surface term has an effect on the dynamics of the
action-extremizing field configurations. This question appears to be
surprisingly nontrivial in the case of the open FLRW geometry, since surface
terms tend to grow as fast as volume terms in the infinite volume limit. We
find that surface terms can be important for the dynamics of the field at a
classical and at a quantum level, when there are supercurvature perturbations.
