Convergence of the Ricci flow toward a unique soliton

We will consider a {\it $\tau$-flow}, given by the equation $\frac{d}{dt}{g}_{ij}=-2R_{ij}+\frac{1}{\tau }{g}_{ij}$ on a closed manifold $M$, for all times $t\in [0,\mathrm{\infty })$. We will prove that if the curvature operator and the diameter of $(M,g(t))$ are uniformly bounded along the flow and if one of the limit solitons is integrable, then we have a convergence of the flow toward a unique soliton, up to a diffeomorphism.