Paper · Oct 29, 2022

Ricci Flow method in the existence problem of the K"ahler-Einstein metrics

Liu Chao

math.AParXiv:2210.16620v2
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As on arXiv

Abstract

This note illustrates the Ricci flow method based on the Cao.H.D's paper[1] and Yau.S.T's paper[4], and tries to explain the method in detail, especially in some calculations. Jian Song and Weinkove's note[9] used some other estimates to obtain the result, this paper will explain some of their estimates as well. This note was a seminar lecture note in 2022 summer when the author was giving lectures on the geometry analysis seminar reasearching the Ricci flow method. The part of the imporatnt zero order estimate is going to be added in a few days. The level of the author is limited, if there are any errors, please do not hesitate to advise. Any comments will be grateful.

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Submitted
Oct 29, 2022
Last revised
Nov 20, 2022 (v2)
Subjects
math.AP, math.DG
Ricci flowEinstein metrics

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)7Nakagawa 2022 Vaisman manifolds and transversally Kähler-Einstein metrics (Vladimir Slesar and Gabriel-Eduard Vîlcu, 2022)8Slesar 2022 Linear stability of compact shrinking Ricci solitons (Huai-Dong Cao and Meng Zhu, 2023)3Cao 2023 Dynamical Stability and Instability of Poincaré–Einstein Manifolds (Klaus Kroencke and Louis Yudowitz, 2023)5Kroencke 2023 Novel Boundary Conditions for the Ricci Flow (Rasmus Jouttijärvi, 2024)1Jouttijärvi 2024 Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds (Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso, 2025)6Conti 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)2Bhattacharya 2025 Ricci solitons as critical points of quadratic curvature functionals (Atreyee Bhattacharya and Sayoojya Prakash, 2026)4Bhattacharya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Novel Boundary Conditions for the Ricci Flow

    Rasmus Jouttijärvi

    Topics Einstein metrics

  2. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics

  3. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu

    Topics Einstein metrics · Shared terms “Cao”

  4. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics

  5. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz

    Topics Einstein metrics

  6. Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso

    Topics Einstein metrics

  7. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura

    Topics Einstein metrics

  8. Vaisman manifolds and transversally Kähler-Einstein metrics

    Vladimir Slesar and Gabriel-Eduard Vîlcu

    Topics Einstein metrics · Shared terms “illustrate”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.