Can someone please tell me whether there is any self contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self contained I mean it does not assume that the reader is familiar with Analysis of PDE, rather quotes the required results and have comprehensive appendix on PDE. I would appreciate if the book contain some exercises also.
Thanks in advance.
There is a book by Thierry Aubin called "Some Nonlinear Problems in Riemannian Geometry" which, at least in principle, is exactly what you describe. It covers a lot of classic geometric analysis problems, with a couple of introductory chapters on geometry and PDEs. I say in principle because it might be very hard to read for somebody just starting out; it is probably easier to pick one topic in geometric analysis (you seem to be interested in Ricci flow) and try to learn it very well first.
Another, a lot more reader-friendly source for self-contained geometric analysis, is Jerry Kazdan's notes "Applications of Partial Differential Equations to Some Problems in Differential Geometry" which you can find on his personal webpage. This is where I'd start out.
For Ricci flow, B. Chow, P. Lu and L. Ni's "Hamilton's Ricci Flow" is the most self-contained introduction in my opinion. It is rather light on the PDE side of things, but the first chapter has all the important basic calculations and facts you need laid out to the reader, sometimes as exercices. There are also two appendices on various geometric analysis techniques. I recommend reading that together with B. Chow and D. Knopf's "Introduction to Ricci flow" to get into Ricci flow.