We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's W-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow represents the gradient flow of the restriction of Perelman's W-functional over such subspaces.

Over Fano manifolds we introduce a flow of Kähler structures with formal limit at infinite time a Kähler-Ricci soliton. This flow corresponds to Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci flow exists for all times and the Bakry-Emery-Ricci tensor preserves a positive uniform lower bound with respect to the evolving metric. In this case we show that the corresponding Soliton-Kähler-Ricci flow converges exponentially fast to a Kähler-Ricci soliton.