  A Wick rotation in the lapse (not in time) is introduced that interpolates
between Riemannian and Lorentzian metrics on real manifolds admitting a
co-dimension one foliation. The definition refers to a fiducial foliation but
covariance under foliation changing diffeomorphisms is ensured. In particular,
the resulting complex metrics are admissible in the sense of Kontsevich-Segal
in all (fiducial and non-fiducial) foliations. This setting is used to
construct a Wick rotated heat semigroup, which remains well-defined into the
near Lorentzian regime. Among the results established for the Wick rotated
version are: (i) Existence as an analytic semigroup uniquely determined by its
sectorial generator. (ii) Construction of an integral kernel that is jointly
smooth in the semigroup time and both spacetime arguments. (iii) Existence of
an asymptotic expansion for the kernel's diagonal in (shifted) powers of the
semigroup time whose coefficients are the Seeley-deWitt coefficients evaluated
on the complex metrics. (iv) Emergence of a Schr\"{o}dinger evolution group in
the strict Lorentzian limit. The toolbox includes local regularity results for
admissible complex metrics.
