BPS Domain Walls and Vacuum Structure of N=1 Supergravity Coupled to a Chiral Multiplet

We study BPS domain walls of N=1 supergravity coupled to a chiral multiplet and their Lorentz invariant vacua which can be viewed as critical points of BPS equations and the scalar potential. Supersymmetry further implies that gradient flows of BPS equations controlled by a holomorphic superpotential and the Kaehler geometry are unstable near local maximum of the scalar potential, whereas they are stable around local minimum and saddles of the scalar potential. However, the analysis using RG flows shows that such gradient flows do not always exist particularly in infrared region.