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This question was originally posted on MSE. But I would like to post it here to see whether anyone could recommend some reference for me.

I am currently reading the paper "Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds" by Gang Liu and would like to see if anyone could recommend some books on the analytical aspect of Kähler geometry. More specifically, are there any books in analysis on Kähler geometry/ Kähler-Ricci flow that is written in a way like Peter Li's "Geometric Analysis" or Chow-Lu-Ni's "Hamilton's Ricci Flow"?

Your help is very much appreciated.

In the original post, I found Wells' and Ballmann's books to be useful, but not exactly what I am looking for.

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    $\begingroup$ For Kähler-Ricci flow I started (when I was master student )from the lecture note of Jian Song, and Ben Weinkove arxiv.org/abs/1212.3653 . $\endgroup$
    – user160903
    Commented Sep 1, 2020 at 16:29
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    $\begingroup$ The Book entitled : Degenerate Complex Monge-Ampere Equations also useful amazon.com/… $\endgroup$
    – user160903
    Commented Sep 1, 2020 at 16:35
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    $\begingroup$ An Introduction to the Kahler-Ricci Flow Sebastien Boucksom ⋅ Philippe Eyssidieux ⋅ Vincent Guedj bokus.com/bok/9783319008189/… $\endgroup$
    – user160903
    Commented Sep 1, 2020 at 16:37
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    $\begingroup$ Follow the joint papers of Tian with Song about Kähler -Ricci flow $\endgroup$
    – user160903
    Commented Sep 1, 2020 at 16:38
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    $\begingroup$ About the paper of Gang Liu, If you read a Book about "Entire and holomorphic functions "then it can help you. When I was master student I passed a course about Space holomorphic functions II. $\endgroup$
    – user160903
    Commented Sep 1, 2020 at 16:46

2 Answers 2

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I'll give an answer that is specifically tailored towards the Kahler-Ricci flow. Hopefully other answers can give some good materials for geometric analysis on Kahler manifolds more generally. For KR flow, I found the following manuscripts to be very informative. All of them can be found for free online as well, which is a plus.

  1. Boucksom, S. et al (ed.) (2013). An introduction to the Kähler-Ricci flow (Vol. 2086). Cham: Springer.

This is really a collection of chapters, but it has a good overview of various different aspects of KR flow analysis.

  1. Song, J., & Weinkove, B. (2012). Lecture notes on the K" ahler-Ricci flow. arXiv preprint arXiv:1212.3653.

This is a good overview with a focus towards the analytic minimal model program.

  1. Tosatti, V. (2018). KAWA lecture notes on the Kähler–Ricci flow. In Annales de la Faculté des sciences de Toulouse: Mathématiques (Vol. 27, No. 2, pp. 285-376).

This is another good introduction to the flow, with a lot of relevant background.

  1. Lecture notes by Otis Chodosh and Christos Mantoulidis based off a class by Richard Bamler.

These aren't specifically about Kahler-Ricci flow, but I like them because they give details about Uhlenbeck's trick and the intuition for Hamilton-Li-Yau inequalities (Hamilton's original paper on the subject also has good insights, but I suppose that's not really an introduction per se).

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  • $\begingroup$ I am so grateful that you provided so many resources for me! It is difficult to find a book in Kähler geometry that is not algebraic, at least on my own. I am glad I asked on overflow! $\endgroup$
    – ldgo
    Commented Sep 2, 2020 at 17:06
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Ben Weinkove 5 lectures The Kähler–Ricci flow on compact Kähler manifolds which has been collected in

Bray, Hubert L. (ed.); Galloway, Greg (ed.); Mazzeo, Rafe (ed.); Sesum, Natasa (ed.), Geometric analysis. Lecture notes from the graduate minicourse of the 2013 IAS/Park City Mathematics Institute session on geometric analysis, Park City, UT, USA, 2013, IAS/Park City Mathematics Series 22. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study (IAS) (ISBN 978-1-4704-2313-1/hbk; 978-1-4704-2881-5/ebook). xvi, 438 p. (2016). ZBL1343.53002.

and the second chapter of famous book of Bennett Chow et all:

Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei, The Ricci flow: techniques and applications. Part I: Geometric aspects, Mathematical Surveys and Monographs 135. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3946-1/hbk). xxiii, 536 p. (2007). ZBL1157.53034.

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  • $\begingroup$ Thank you too! Weinkove's notes seem great! I have the Chow's Ricci Flow "series", but it only has one chapter about the KRF, I am not sure whether it is enough or not. Anyhow, I still use it as a reference for Ricci flow in general. $\endgroup$
    – ldgo
    Commented Sep 2, 2020 at 17:08

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