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 a comprehensive appendix on PDE. I would appreciate if the book contained some exercises also.
[Math] Self-contained book on Ricci Flow/Geometric Analysis
geometric-analysisreference-requestricci-flowriemannian-geometry
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I'm not even a number theorist, but when I was in grad school I took a one-semester course from Hartshorne, using his book. I felt that it carried a powerful message, more abstract and more general than what I needed for any purpose, but still really interesting. We only made it to the beginning of the third chapter, but that was enough. In fact, without entirely realizing it, I was taking a lot of hard commutative algebra results on faith in the homework, but somehow it was okay.
So I would ask whether Hartshorne is exactly what you want; it exactly explains what you say is missing from your experience. You don't have to and shouldn't read it page by page, confirming every assertion. You can learn from this book using the spiral method. I do not know "Geometry of Schemes", although a book with that title and those authors sounds good too. What I can say is that Hartshorne is especially enthusiastic about Grothendieck's perspective on algebraic geometry. Again, even though I had no career reason to care, I ended up wanting to prove things Grothendieck style: using schemes, with the same argument in characteristic 0 and positive characteristic, with proper schemes as a replacement for projective varieties, etc.
1) There is a great book From Holomorphic Functions to Complex Manifolds by Fritzsche-Grauert.
It is very geometric and gives you the fundamentals on complex manifolds, including specialized topics, from Stein manifolds to compact Kähler manifolds, which in a sense are the two extremities in the spectrum of holomorphic geometry.
An important bonus is that Grauert was arguably the deepest complex analyst in the twentieth century.
It is very geometric but it will also warm your analyst's heart with sections on plurisubharmonic functions, Sobolev spaces and, Neumann operators,...
2) Another, even more analysis rich introduction, to complex geometry is Krantz's Function Theory of Several Complex Variables
There you will find harmonic analysis, regularity of $\bar \partial $ operator, $H^p$ functions and all sorts of integral representations.
3) Both books contain introductions to the indispensable tool of sheaves and their cohomology, which actually had their first applications in complex analysis, and were foreshadowed in Oka's groundbreaking solution to the Levi problem.
There are other good books, by Fuks, Griffiths, Hörmander (mentioned in abx's comment), Huybrechts, Ohsawa, Range, Wells,... but for me the most comprehensive introductions are 1) and 2).
4) Welcome to that enchanting land of complex geometry (full disclosure: that was where I started doing research!) and good luck!
Best Answer
A quick search on Amazon provides at least three titles that are introductory texts to the topic for graduate students.
(1) B. Chow, P. Lu, L. Ni: Hamilton's Ricci Flow, Graduate Studies in Mathematics 77, AMS 2006;
(2) B. Chow, D. Knopf: The Ricci Flow: An Introduction, Mathematical Surveys and Monographs 110, AMS 2004;
(3) B. Chow and others: The Ricci Flow: Techniques and Applications: Geometric Aspects, Mathematical Surveys and Monographs 135, AMS 2007.