  We localize the entropy functionals of G. Perelman and generalize his
no-local-collapsing theorem and pseudo-locality theorem. Our generalization is
technically inspired by further development of Li-Yau estimates along the Ricci
flow. It has various applications, including to show the continuous dependence
of the Ricci flow with respect to the initial metric in Gromov-Hausdorff
topology with Ricci curvature bounded below, and to show the compactness of the
moduli of K\"ahler manifolds with bounded scalar curvature and a rough locally
almost Euclidean condition.
