  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 a Perelman's modified backward Ricci type
flow with some special restriction conditions. The restriction conditions are
motivated by convexity results for Perelman's $\mathcal{W}$-functional over
convex subsets inside adequate subspaces of Riemannian metrics. We show indeed
that the Soliton-Ricci flow is generated by the gradient flow of the
restriction of Perelman's $\mathcal{W}$-functional over such subspaces.
Assuming long time existence of the Soliton-Ricci flow we show exponentially
fast convergence to a shrinking Ricci soliton provided that the
Bakry-Emery-Ricci tensor is uniformly strictly positive with respect to the
evolving metric.

Differential Geometry;Analysis of PDEs;Complex Variables