  The elliptic Einstein-DeTurck equation may be used to numerically find
Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics
are considered by analytically continuing to Euclidean time. Ricci-DeTurck flow
is a constructive algorithm to solve this equation, and is simple to implement
when the solution is a stable fixed point, the only complication being that
Ricci solitons may exist which are not Einstein. Here we extend previous work
to consider the Einstein-DeTurck equation for Riemannian manifolds with
boundaries, and those that continue to static Lorentzian spacetimes which are
asymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using
a maximum principle we prove that Ricci solitons do not exist in these cases
and so any solution is Einstein. We also argue that Ricci-DeTurck flow
preserves these classes of manifolds. As an example we simulate Ricci-DeTurck
flow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum
principle dictates there are no soliton solutions, and we give strong numerical
evidence that there exists a stable fixed point of the flow which continues to
a smooth static Lorentzian Einstein metric. Our asymptotics are such that this
describes the classical gravity dual relevant for the CFT on a Schwarzschild
background in either the Unruh or Boulware vacua. It determines the leading
O(N^2) part of the CFT stress tensor, which interestingly is regular on both
the future and past Schwarzschild horizons.
