In this paper we address several aspects of flat Bogomolnyi–Prasad–Sommerfeld (BPS) domain walls together with their Lorentz invariant vacua of four-dimensional supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of two-dimensional Kähler manifolds satisfying a Kähler–Ricci flow equation. We find that BPS equations and the scalar potential deform with respect to the real parameter related to the Kähler–Ricci soliton. In addition, the analysis using gradient and renormalization group flows is carried out to ensure the existence of Lorentz invariant vacua related to anti-de Sitter/conformal field theory correspondence.
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