  This paper provides a study of some aspects of flat and curved BPS domain
walls together with their Lorentz invariant vacua of four dimensional chiral
N=1 supergravity. The scalar manifold can be viewed as a one-parameter family
of K\"ahler manifolds generated by a K\"ahler-Ricci flow equation.
Consequently, a vacuum manifold characterized by $(m,\lambda)$ where $m$ and
$\lambda$ are the dimension and the index of the manifold, respectively, does
deform with respect to the flow parameter related to the geometric soliton.
Moreover, one has to carry out the renormalization group analysis to verify the
existence of such a vacuum manifold in the ultraviolet or infrared regions. At
the end, we discuss a simple model with linear superpotential on U(n) symmetric
K\"ahler-Ricci orbifolds.
