  The character of quantum corrections to the gravitational action of a
conformally invariant field theory for a self-interacting scalar field on a
manifold with boundary is considered at third loop-order in the perturbative
expansion of the zero-point function. Diagramatic evaluations and higher
loop-order renormalisation can be best accomplished on a Riemannian manifold of
constant curvature accommodating a boundary of constant extrinsic curvature.
The associated spherical formulation for diagramatic evaluations reveals a
non-trivial effect which the topology of the manifold has on the vacuum
processes and which ultimately dissociates the dynamical behaviour of the
quantised field from its behaviour in the absence of a boundary. The first
surface divergence is evaluated and the necessity for simultaneous
renormalisation of volume and surface divergences is shown.
