Note on basics of Ricci flow
Covering:
1. Basic definitions and calculations: Ricci flow, variation of tensors, convergence of manifolds and flows, DeTurck trick and short-time existence
2. Various maximum principles: (Strong) MP of parabolic equation and comparison, MP of tensor, MP on noncompact manifold, MP of system(ODE-PDE theorem)
3. Evolution of curvature tensor, high-order estimate of Shi and long-time existence(curvature blow up)
4. Curvature Operator: Hamilton ODE, conditions preserved by Ricci flow, Hamilton-Ivey estimate, manifolds of nonnegative curvature operator, Hamilton 1982 theorem
5. Nonnegative isotropic curvature, construction of cones, proof of differentiable sphere theorem
6. Entropy of Perelman, monotonicity of entropy, no-local-collapsing
7. Solitons and special solutions, geometry at spatial infinity.
One may refer to Chapter 1 to 5 in Hamiltons Ricci Flow by Bennett Chow, Lectures on the Ricci Flow by Topping and Ricci Flow and the Sphere Theorem by Brendle.