  We consider some aspects of the curved BPS domain walls and their
supersymmetric Lorentz invariant vacua of the four dimensional N=1 supergravity
coupled to a chiral multiplet. In particular, the scalar manifold can be viewed
as a two dimensional K\"ahler-Ricci soliton generating a one-parameter family
of K\"ahler manifolds evolved with respect to a real parameter, $\tau$. This
implies that all quantities describing the walls and their vacua indeed evolve
with respect to $\tau$. Then, the analysis on the eigenvalues of the first
order expansion of BPS equations shows that in general the vacua related to the
field theory on a curved background do not always exist. In order to verify
their existence in the ultraviolet or infrared regions one has to perform the
renormalization group analysis. Finally, we discuss in detail a simple model
with a linear superpotential and the K\"ahler-Ricci soliton considered as the
Rosenau solution.
