  Kontsevich and Segal (K-S) have proposed a criterion to determine which
complex metrics should be allowed, based on the requirement that quantum field
theories may consistently be defined on these metrics, and Witten has recently
suggested that their proposal should also apply to gravity. We explore this
criterion in the context of gravitational path integrals, in simple
minisuperspace models, specifically considering de Sitter (dS), no-boundary and
Anti-de Sitter (AdS) examples. These simple examples allow us to gain some
understanding of the off-shell structure of gravitational path integrals. In
all cases, we find that the saddle points of the integral lie right at the edge
of the allowable domain of metrics, even when the saddle points are complex or
Euclidean. Moreover the Lefschetz thimbles, in particular the steepest descent
contours for the lapse integral, are cut off as they intrude into the domain of
non-allowable metrics. In the AdS case, the implied restriction on the
integration contour is found to have a simple physical interpretation. In the
dS case, the lapse integral is forced to become asymptotically Euclidean. We
also point out that the K-S criterion provides a reason, in the context of the
no-boundary proposal, for why scalar fields would start their evolution at
local extrema of their potential.
