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Astérisque - Titles - 322 (2008) 71-92

Titles < 2008 < 322

Géométrie différentielle, physique mathématique, mathématiques et société (II)
Volume en l'honneur de Jean Pierre Bourguignon

Oussama Hijazi (éditeur)
Astérisque 322 (2008), xvi+256 pages
Buy the book
Presentation, Summary

New results and problems on Kähler-Ricci flow
Gang Tian
Astérisque 322 (2008), 71-92

Résumé :
Nouveaux problèmes et résultats sur le flot de Kähler-Ricci
Dans cet article, nous donnons un aperçu rapide d'un programme d'études sur le flot de Kähler-Ricci avec chirurgie et son interaction avec la classification des variétés projectives. Le flot de Kähler-Ricci peut développer des singularités en un temps fini. Il est important de comprendre comment étendre le flot de Kähler-Ricci à travers le temps singulier, c'est-à-dire, comment construire une solution du flot de Kähler-Ricci avec chirurgie. La première tâche de cette article consiste à décrire une procédure de construction de solutions globales pour le flot de Kähler-Ricci avec chirurgie. Cette procédure est plutôt canonique. Nous allons discuter les propriétés de telles solutions avec chirurgie et leurs implications géométriques. Nous allons également discuter leurs limites asymptotiques au temps infini. Les résultats présentés ici proviennent principalement de travaux communs avec Z. Zhang, J. Song et al. Nous allons également présenter certains problèmes ouverts. L'article est plutôt explicatif.

Mots-clefs : Flot de Kähler-Ricci, métriques de Kähler-Einstein

Abstract:
In this paper, I give a brief tour on a program of studying the Kähler-Ricci flow with surgery and its interaction with the classification of projective manifolds. The Kähler-Ricci flow may develops singularity at finite time. It is important to understand how  to extend the Kähler-Ricci flow across the singular time, that is, construct solution of the Kähler-Ricci flow with surgery. The first task of this paper is to describe a procedure of constructing global solutions for the Kähler-Ricci flow with surgery. This procedure is rather canonical. I will discuss properties of such solutions with surgery and their geometric implications. I will also discuss their asymptotic limits at time infinity. The results discussed here were mainly from my joint works  with Z. Zhang, J. Song et al. Some open problems will be also discussed. The paper is mostly expository.

Keywords: Kähler-Ricci flow, Kähler-Einstein metrics

Class. math. : 53C25, 53C44


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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