  In this paper we address several aspects of flat Bogomolnyi-Prasad-Sommerfeld
(BPS) domain walls together with their Lorentz invariant vacua of 4d N=1
supergravity coupled to a chiral multiplet. The scalar field spans a
one-parameter family of 2d K\"ahler manifolds satisfying a K\"ahler-Ricci flow
equation. We find that BPS equations and the scalar potential deform with
respect to the real parameter related to the K\"ahler-Ricci soliton. In
addition, the analysis using gradient and renormalization group flows are
carried out to ensure the existence of Lorentz invariant vacua related to Anti
de Sitter/Conformal Field Theory (AdS/CFT) correspondence.
