Local Aronson-Benilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

In this work we derive local gradient and Laplacian estimates of the Aronson-Benilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelmans work we discover some new entropy formulae for these equations.