PREFACE open problem list on the geometry and classification of Ricci soli- tons. (2) Introduction to the Kahler-Ricci flow. Long-time existence of the Kahler-Ricci flow on Kahler manifolds with first Chern class hav- ing a sign. Convergence of the Kahler-Ricci flow on Kahler man- ifolds with negative first Chern class. Construction of the Koiso solitons and other U(n)-invariant solitons. Differential Harnack es- timates and their applications under the assumption of nonnegative bisectional curvature. A survey of uniformization-type results for complete noncompact Kahler manifolds with positive curvature. (3) Proof of the global version of Hamilton's Cheeger-Gromov-type compactness theorem for the Ricci flow. We take care to follow Hamilton and prove the compactness theorem for the Ricci flow in the category of pointed solutions with the convergence in C°° on compact sets. Outline of the proof of the local version of the aforementioned result. Application to the existence of singularity models. (4) A unified approach to Perelman's monotonicity formulas for en- ergy and entropy and the expander entropy monotonicity formula. Perelman's A-invariant and application to the second proof of the no nontrivial steady or expanding breathers result. Other entropy results due to Hamilton and Bakry-Emery. (5) Proof of the no local collapsing theorem assuming only an upper bound on the scalar curvature. Relation of no local collapsing and Hamilton's little loop conjecture. Perelman's \i- and [/-invariants and application to the proof of the no shrinking breathers result. Discussion of Topping's diameter control result. Relation between the variation of the modified scalar curvature and the linear trace Harnack quadratic. Second variation of energy and entropy. (6) Theory of the reduced length. Comparison between the reduced length for static metrics and solutions of the Ricci flow. The C- length, L-, L-, and ^-distances and the first and second variation formulas for the £-length. Existence of ^-geodesies and estimates for their speeds. Formulas for the gradient and time-derivative of the //-distance function and its local Lipschitz property. Formu- las for the Laplacian and Hessian of L and differential inequalities for L, L, and £ including a space-time Laplacian comparison the- orem. Upper bound for the spatial minimum of £. Formulas for £ on Einstein and gradient Ricci soliton solutions. £-Jacobi fields, the £-Jacobian, and the ^-exponential map, and their properties. Estimate for the time-derivative of the £-Jacobian. Bounds for £, its space-derivative, and its time-derivative. Properties of Lipschitz functions applied to £ and equivalence of notions of supersolutions in view of differential inequalities for £.
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