  In the context of the holographic correspondence, we introduce a purely
extrinsic renormalization prescription, exemplified with the case of a
minimally-coupled scalar field in AdS space. The counterterms depend only on
the field and its radial derivatives. This would seem to conflict with the
Dirichlet variational principle, but we show that consistency follows from the
fact that the asymptotic structure of asymptotically locally AdS spacetimes
requires not only the leading, but also all of the subleading non-normalizable
modes to be fixed as a boundary condition. Crucially, as seen from a path
integral definition of the bulk partition function involved in the standard
GKPW formula, this condition is valid away from the saddle. We find that the
extrinsic renormalization prescription is maximally efficient when the scalar
field is massless, which is suggestive of a connection with the Kounterterm
method for renormalization of pure gravity.
