Here is a list of literature which I compiled when I taught the course
on Ricci flow.

Basic differential geometry:
* Einstein Manifolds (Besse). 
* Riemannian geometry (Gallot S., Hulin D., Lafontaine J.)
* [Sign and geometric meaning of curvature](https://doi.org/10.1007/BF02925201) (Gromov),
also available [here](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/177.pdf)

Textbooks:
* [Lectures on the Ricci Flow](http://www.warwick.ac.uk/~maseq/RFnotes.html) (2006, 133 pp.) Topping P.
* [Hamilton's Ricci Flow](https://bookstore.ams.org/view?ProductCode=GSM/77)  (Chow B., Lu P., Ni L.)<br>

Standard texts:<br>
* Wikipedia: [Poincaré conjecture](http://en.wikipedia.org/wiki/Solution_of_the_Poincar%C3%A9_conjecture) (and links therein)
* Perelman, Grisha (November 11, 2002).  
[The entropy formula for the Ricci flow and its geometric applications](http://arxiv.org/abs/math.DG/0211159).
* Perelman, Grisha (March 10, 2003).
[Ricci flow with surgery on three-manifolds](http://arxiv.org/abs/math.DG/0303109).
* Perelman, Grisha (July 17, 2003).
[Finite extinction time for the solutions to the 
Ricci flow on certain three-manifolds](http://arxiv.org/abs/math.DG/0307245).

* Bruce Kleiner, John Lott. [Notes on Perelman's papers](http://arxiv.org/abs/math/0605667)
* Huai-Dong Cao, Xi-Ping Zhu. [Hamilton-Perelman's Proof of
the Poincaré  Conjecture and the Geometrization Conjecture](http://arxiv.org/abs/math.DG/0612069).

* John W. Morgan, Gang Tian.
[Ricci Flow and the Poincaré Conjecture](http://arxiv.org/abs/math/0607607)

* * * 

It's very obsolete (2007), and does not contain much on short-term
existence of solutions of Ricci flow.