[Régularité du flot de Kähler–Ricci]
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.
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.
Accepté le :
Publié le :
@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://www.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://www.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 -
Tian, Gang; Zhang, Zhenlei. Regularity of the Kähler–Ricci flow. Comptes Rendus. Mathématique, Tome 351 (2013) no. 15-16, pp. 635-638. doi : 10.1016/j.crma.2013.07.005. http://www.numdam.org/articles/10.1016/j.crma.2013.07.005/
[1] Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds, Invent. Math., Volume 81 (1985), pp. 359-372
[2] Lower bounds on the Ricci curvature and the almost rigidity of warped products, Ann. Math., Volume 144 (1996), pp. 189-237
[3] On the structure of spaces with Ricci curvature bounded below I, J. Differential Geom., Volume 46 (1997), pp. 406-480
[4] On the structure of spaces with Ricci curvature bounded below II, J. Differential Geom., Volume 54 (2000), pp. 13-35
[5] On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal., Volume 12 (2002), pp. 873-914
[6] Kähler–Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities | arXiv
[7] Kähler–Einstein metrics on Fano manifolds, II: limits with cone angle less than 2π | arXiv
[8] Kähler–Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2 and completion of the main proof | arXiv
[9] Ricci flow on Kähler–Einstein manifolds, Duke Math. J., Volume 131 (2006), pp. 17-73
[10] Space of Ricci flows I, Commun. Pure Appl. Math., Volume 65 (2012), pp. 1399-1457
[11] Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry | arXiv
[12] The entropy formula for the Ricci flow and its geometric applications | arXiv
[13] Relative volume comparison with integral curvature bounds, Geom. Funct. Anal., Volume 7 (1997), pp. 1031-1045
[14] Analysis and geometry on manifolds with integral Ricci curvature bounds. II, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 457-478
[15] Degeneration of Kähler–Ricci solitons on Fano manifolds | arXiv
[16] Bounding scalar curvature and diameter along the Kähler–Ricci flow (after Perelman), J. Inst. Math. Jussieu, Volume 7 (2008), pp. 575-587
[17] On Calabiʼs conjecture for complex surfaces with positive first Chern class, Invent. Math., Volume 101 (1990), pp. 101-172
[18] Kähler–Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997), pp. 1-39
[19] Existence of Einstein metrics on Fano manifolds, Prog. Math., Volume 297 (2012), pp. 119-159
[20] K-stability and Kähler–Einstein metrics | arXiv
[21] Degeneration of Kähler–Ricci solitons, Int. Math. Res. Not. IMRN (2012), pp. 957-985
[22] G. Tian, Z.L. Zhang, Long-time limits of the Kähler–Ricci flow, preprint.
[23] Convergence of Kähler–Ricci flow, J. Amer. Math. Soc., Volume 20 (2007), pp. 675-699
[24] A uniform Sobolev inequality under Ricci flow, Int. Math. Res. Not. IMRN (2007), pp. 1-12
[25] Kähler–Ricci flow on Fano manifolds with vanished Futaki invariants, Math. Res. Lett., Volume 18 (2011), pp. 969-982
Cité par Sources :
