The local entropy along Ricci flow---Part A: the no-local-collapsing theorems

Abstract: 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 estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the K\"ahler Ricci flow on each minimal projective manifold of general type.