On the uniqueness of asymptotic limits of the Ricci flow

We consider a normalization of the Ricci flow on a closed Riemannian manifold given by the evolution equation $\partial_{t}g(t)=-2(Ric(g(t))-\frac{1}{2\tau}g(t))$ where $\tau$ is a fixed positive number. Assuming that a solution for this equation exists for all time, and that the full curvature tensor and the diameter of the manifold are both uniformly bounded along the flow, we prove that up to the action of a 1-parameter family of diffeomorphisms, the flow converges to a unique shrinking gradient Ricci soliton using an idea of Sun and Wang to study the stability of the K\"{a}ler Ricci flow near a K\"{a}hler-Einstein metric. This idea relies on the monotonicity of the Ricci flow along Perelman's $\W$-functional and a {\L}ojasiewicz-Simon inequality for the $\mu$-functional. Our result is an extension of the main theorem in Sesum (2006), where convergence to a unique soliton is proved assuming that the solution of the normalized Ricci flow in consideration is sequentially convergent to a soliton which satisfies a certain integrability condition.