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.
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$\begingroup$ Just generic advice, you are not going to be able to study geometric analysis without learning some PDE theory as well. It would be like trying to study algebraic geometry without learning some commutative algebra. $\endgroup$– Hollis WilliamsCommented Sep 26, 2020 at 13:05
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2 Answers
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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.
answered Apr 19, 2015 at 9:33
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1$\begingroup$ I looked at those books but it seemed to me that those books use lot of PDE theory. Can they really be read without any prior knowledge of PDE? $\endgroup$– BingoCommented Apr 19, 2015 at 9:42
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3$\begingroup$ well, they are aimed to graduate students, so one should be able (at least in principle) to read them with the help of some standard text on PDE (like Evans') $\endgroup$ Commented Apr 19, 2015 at 9:51
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5$\begingroup$ Since the Ricci flow is a PDE, it's not realistic to learn about it without knowing any PDE theory at all. But you don't need much. My advice is to study the books by Chow et al and consult PDE books only as needed. $\endgroup$ Commented Apr 19, 2015 at 20:19
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These books may also be the sort of thing you are after:
- Peter Topping, Lectures on the Ricci flow
- Ben Andrews and Christopher Hopper, Ricci Flow in Riemannian Geometry A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem
As Dean Yang pointed out in the comments above, being a PDE, the Ricci flow is, not surprisingly, studied by PDE methods. However, you can make a reasonable start far with only knowledge of the maximum principle (it's even described in Topping's book) if your are willing to assume existence/uniqueness. I think each book is fairly self-contained, and while many techniques used are PDE techniques, you can probably read them without knowing they apply to a broader range of PDE.
