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 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 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.
Accepted:
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@article{CRMATH_2013__351_15-16_635_0, author = {Tian, Gang and Zhang, Zhenlei}, title = {Regularity of the {K\"ahler{\textendash}Ricci} flow}, journal = {Comptes Rendus. Math\'ematique}, pages = {635--638}, publisher = {Elsevier}, volume = {351}, number = {15-16}, year = {2013}, doi = {10.1016/j.crma.2013.07.005}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2013.07.005/} }
TY - JOUR AU - Tian, Gang AU - Zhang, Zhenlei TI - Regularity of the Kähler–Ricci flow JO - Comptes Rendus. Mathématique PY - 2013 SP - 635 EP - 638 VL - 351 IS - 15-16 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2013.07.005/ DO - 10.1016/j.crma.2013.07.005 LA - en ID - CRMATH_2013__351_15-16_635_0 ER -
%0 Journal Article %A Tian, Gang %A Zhang, Zhenlei %T Regularity of the Kähler–Ricci flow %J Comptes Rendus. Mathématique %D 2013 %P 635-638 %V 351 %N 15-16 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2013.07.005/ %R 10.1016/j.crma.2013.07.005 %G en %F CRMATH_2013__351_15-16_635_0
Tian, Gang; Zhang, Zhenlei. 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. http://archive.numdam.org/articles/10.1016/j.crma.2013.07.005/
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