Higher Dimensional Curved Domain Walls on K\"ahler Surfaces

In this paper we study some aspects of curved BPS-like domain walls in higher dimensional gravity theory coupled to scalars where the scalars span a complex K\"ahler surface with scalar potential turned on. Assuming that a fake superpotential has a special form which depends on K\"ahler potential and a holomorphic function, we prove that BPS-like equations have a local unique solution. Then, we analyze the vacuum structure of the theory including their stability using dynamical system and their existence in ultraviolet-infrared regions using renormalization group flow.