  Coleman and De Luccia (CDL) showed that gravitational effects can prevent the
decay by bubble nucleation of a Minkowski or AdS false vacuum. In their
thin-wall approximation this happens whenever the surface tension in the bubble
wall exceeds an upper bound proportional to the difference of the square roots
of the true and false vacuum energy densities. Recently it was shown that there
is another type of thin-wall regime that differs from that of CDL in that the
radius of curvature grows substantially as one moves through the wall. Not only
does the CDL derivation of the bound fail in this case, but also its very
formulation becomes ambiguous because the surface tension is not well-defined.
We propose a definition of the surface tension and show that it obeys a bound
similar in form to that of the CDL case. We then show that both thin-wall
bounds are special cases of a more general bound that is satisfied for all
bounce solutions with Minkowski or AdS false vacua. We discuss the limit where
the parameters of the theory attain critical values and the bound is saturated.
The bounce solution then disappears and a static planar domain wall solution
appears in its stead. The scalar field potential then is of the form expected
in supergravity, but this is only guaranteed along the trajectory in field
space traced out by the bounce.
