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Über dieses Buch

This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.

The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman’s celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation).
As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman’s ideas: the Kähler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman’s surgeries.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This book is the first comprehensive reference on the Kähler–Ricci flow. It provides an introduction to fully non-linear parabolic equations, to the Kähler–Ricci flow in general and to Perelman’s estimates in the Fano case, and also presents the connections with the Minimal Model program.
Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj

Chapter 2. An Introduction to Fully Nonlinear Parabolic Equations

Abstract
These notes contain a short exposition of selected results about parabolic equations: Schauder estimates for linear parabolic equations with Hölder coefficients, some existence, uniqueness and regularity results for viscosity solutions of fully nonlinear parabolic equations (including degenerate ones), the Harnack inequality for fully nonlinear uniformly parabolic equations.
Cyril Imbert, Luis Silvestre

Chapter 3. An Introduction to the Kähler–Ricci Flow

Abstract
These notes give an introduction to the Kähler–Ricci flow. We give an exposition of a number of well-known results including: maximal existence time for the flow, convergence on manifolds with negative and zero first Chern class, and behavior of the flow in the case when the canonical bundle is big and nef. We also discuss the collapsing of the Kähler–Ricci flow on the product of a torus and a Riemann surface of genus greater than one. Finally, we discuss the connection between the flow and the minimal model program with scaling, the behavior of the flow on general Kähler surfaces and some other recent results and conjectures.
Jian Song, Ben Weinkove

Chapter 4. Regularizing Properties of the Kähler–Ricci Flow

Abstract
These notes present a general existence result for degenerate parabolic complex Monge–Ampère equations with continuous initial data, slightly generalizing the work of Song and Tian on this topic. This result is applied to construct a Kähler–Ricci flow on varieties with log terminal singularities, in connection with the Minimal Model Program. The same circle of ideas is also used to prove a regularity result for elliptic complex Monge–Ampère equations, following Székelyhidi–Tosatti.
Sébastien Boucksom, Vincent Guedj

Chapter 5. The Kähler–Ricci Flow on Fano Manifolds

Abstract
In these lecture notes, we aim at giving an introduction to the Kähler–Ricci flow (KRF) on Fano manifolds. It covers mostly the developments of the KRF in its first 20 years (1984–2003), especially an essentially self-contained exposition of Perelman’s uniform estimates on the scalar curvature, the diameter, and the Ricci potential function for the normalized Kähler–Ricci flow (NKRF), including the monotonicity of Perelman’s μ-entropy and κ-noncollapsing theorems for the Ricci flow on compact manifolds. The lecture notes is based on a mini-course on KRF delivered at University of Toulouse III in February 2010, a talk on Perelman’s uniform estimates for NKRF at Columbia University’s Geometry and Analysis Seminar in Fall 2005, and several conference talks, including “Einstein Manifolds and Beyond” at CIRM (Marseille—Luminy, fall 2007), “Program on Extremal Kähler Metrics and Kähler–Ricci Flow” at the De Giorgi Center (Pisa, spring 2008), and “Analytic Aspects of Algebraic and Complex Geometry” at CIRM (Marseille— Luminy, spring 2011).
Huai-Dong Cao

Chapter 6. Convergence of the Kähler–Ricci Flow on a Kähler–Einstein Fano Manifold

Abstract
The goal of these notes is to sketch the proof of the following result, due to Perelman and Tian–Zhu: on a Kähler–Einstein Fano manifold with discrete automorphism group, the normalized Kähler–Ricci flow converges smoothly to the unique Kähler–Einstein metric. We also explain an alternative approach due to Berman–Boucksom–Eyssidieux–Guedj–Zeriahi, which only yields weak convergence but also applies to Fano varieties with log terminal singularities.
Vincent Guedj

Backmatter

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