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We address some aspects of four dimensional chiral N=1 supersymmetrictheories on which the scalar manifold is described by K\"ahler geometry and canfurther be viewed as K\"ahler-Ricci soliton generating a one-parameter familyof K\"ahler geometries. All couplings and solutions, namely the BPS domainwalls and their supersymmetric Lorentz invariant vacua turn out to be evolvedwith respect to the flow parameter related to the soliton. Two models arediscussed, namely N=1 theory on K\"ahler-Einstein manifold and U(n) symmetricK\"ahler-Ricci soliton with positive definite metric. In the first case we findthat the evolution of the soliton causes topological change andcorrespondingly, modifies the Morse index of the nondegenerate vacua realizedin the parity transformation of the Hessian matrix of the scalar potentialafter hitting singularity, which is natural in the global theory and fornondegenerate Minkowskian vacua of the local theory. However, such situation isnot trivial in anti de Sitter (AdS) vacua. In an explicit model, we find thatthis geometric (K\"ahler-Ricci) flow can also change the index of the vacuumbefore and after singularity. Finally in the second case, since around theorigin the metric is diffeomorphic to $ {\mathrm{\lC P}}^{n -1}$, we have toconsider it in the asymptotic region. Our analysis shows that no indexmodification of vacua is present in both global and local theories.

Kaehler-Ricci Flow, Morse Theory, and Vacuum Structure Deformation of N=1 Supersymmetry in Four Dimensions