Comptes Rendus
no. 15-16
Differential Geometry
Regularity of the Kähler–Ricci flow
[Régularité du flot de Kähler–Ricci]

Présenté par : Jean-Pierre Demailly

Gang Tian1, 2 ; Zhenlei Zhang3
1 BICMR, Peking University, Yiheyuan Road 5, Beijing 100871, China
2 Department of Mathematics, Princeton University, NJ 02139, USA
3 Capital Normal University, Xisanhuan North Road 105, Beijing 100048, China
Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 635-638.

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial C0 estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.

Dans cette courte note, nous annonçons un théorème de régularité pour le flot de Kähler–Ricci sur une variété compacte de Fano (cʼest-à-dire une variété kählérienne à première classe de Chern positive) et son application à lʼétude du comportement limite du flot de Kähler–Ricci sur les variétés de Fano de dimension 3. Par ailleurs, nous présentons une estimation C0 partielle du flot de Kähler–Ricci sous lʼhypothèse de régularité, qui étend des travaux antérieurs concernant les métriques de Kähler–Einstein et les solitons de Kähler–Ricci régressifs. La preuve détaillée paraîtra ailleurs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.07.005

Gang Tian 1, 2 ; Zhenlei Zhang 3

1 BICMR, Peking University, Yiheyuan Road 5, Beijing 100871, China
2 Department of Mathematics, Princeton University, NJ 02139, USA
3 Capital Normal University, Xisanhuan North Road 105, Beijing 100048, China 
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Gang Tian; Zhenlei Zhang. Regularity of the Kähler–Ricci flow. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 635-638. doi : 10.1016/j.crma.2013.07.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.07.005/

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  • Gang Tian; Zhenlei Zhang Regularity of Kähler–Ricci flows on Fano manifolds, Acta Mathematica, Volume 216 (2016) no. 1, p. 127 | DOI:10.1007/s11511-016-0137-1
  • Rafe Mazzeo; Yanir Rubinstein; Natasa Sesum Ricci flow on surfaces with conic singularities, Analysis PDE, Volume 8 (2015) no. 4, p. 839 | DOI:10.2140/apde.2015.8.839

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