chapter 1 Introduction -- chapter 2 Sobolev inequalities in the Euclidean space -- chapter 3 Basics of Riemann geometry -- chapter 4 Sobolev inequalities on manifolds -- chapter 5 Basics of Ricci flow -- chapter 6 Perelman’s entropies and Sobolev inequality -- chapter 7 Ancient κ solutions and singularity analysis -- chapter 8 Sobolev inequality with surgeries -- chapter 9 Applications to the Poincare´ conjecture.