Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow

Author: Yanir A. Rubinstein
Journal: Trans. Amer. Math. Soc. 361 (2009), 5839-5850
MSC (2000): Primary 32Q20; Secondary 14J45, 32L10, 32W20, 53C25, 58E11
Published electronically: May 7, 2009
MathSciNet review: 2529916
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Šešum, and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kähler-Einstein metrics.

References [Enhancements On Off] (What's this?)

  • [A1] Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bulletin des Sciences Mathématiques 102 (1978), 63-95. MR 494932 (81d:53047)
  • [A2] -, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d'une inégalité, Journal of Functional Analysis 57 (1984), 143-153. MR 0749521 (85k:58084)
  • [BM] Shigetoshi Bando, Toshiki Mabuchi, Uniqueness of Kähler-Einstein metrics modulo connected group actions, in Algebraic Geometry, Sendai, 1985 (T. Oda, Ed.), Advanced Studies in Pure Mathematics 10, Kinokuniya, 1987, 11-40. MR 946233 (89c:53029)
  • [Ca] Huai-Dong Cao, Deformations of Kähler metrics to Kähler-Einstein metrics on compact manifolds, Inventiones Mathematicae 81 (1985), 359-372. MR 799272 (87d:58051)
  • [CLT] Xiu-Xiong Chen, Peng Lu, Gang Tian, A note on uniformization of Riemann surfaces by Ricci flow, Proceedings of the American Mathematical Society 134 (2006), 3391-3393. MR 2231924 (2007d:53109)
  • [CT] Xiu-Xiong Chen, Gang Tian, Ricci flow on Kähler-Einstein surfaces, Inventiones Mathematicae 147 (2002), 487-544. MR 1893004 (2003c:53095)
  • [C] Xiu-Xiong Chen, On Kähler manifolds with positive orthogonal bisectional curvature, Advances in Mathematics 215 (2007), 427-445. MR 2355611
  • [Ch1] Bennett Chow, The Ricci flow on the 2-sphere, Journal of Differential Geometry 33 (1991), 325-334. MR 1094458 (92d:53036)
  • [Ch2] Bennett Chow et al., The Ricci flow: Techniques and applications. Part I: Geometric aspects, American Mathematical Society, 2007. MR 2302600
  • [D] Jean-Pierre Demailly, On the Ohsawa-Takegoshi-Manivel $ L^2$ extension theorem, in Complex Analysis and Geometry (P. Dolbeault et al., Eds.), Progress in Mathematics 188, Birkhäuser, 2000, 47-82. MR 1782659 (2001m:32041)
  • [DK] Jean-Pierre Demailly, János Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Annales Scientifiques de l'École Normale supérieure 34 (2001), 525-556. MR 1852009 (2002e:32032)
  • [Di] Wei-Yue Ding, Remarks on the existence problem of positive Kähler-Einstein metrics, Mathematische Annalen 282 (1988), 463-471. MR 967024 (90a:58186)
  • [DT] Wei-Yue Ding, Gang Tian, The generalized Moser-Trudinger inequality, in Nonlinear Analysis and Microlocal Analysis: Proceedings of the International Conference at Nankai Institute of Mathematics (K.-C. Chang et al., Eds.), World Scientific, 1992, 57-70. ISBN 9810209134.
  • [F] Akito Futaki, Kähler-Einstein metrics and integral invariants, Lecture Notes in Mathematics 1314, Springer, 1988. MR 947341 (90a:53053)
  • [G1] Daniel Z.-D. Guan, Quasi-Einstein metrics, International Journal of Mathematics 6 (1995), 371-379. MR 1327154 (96e:53060)
  • [G2] -, Extremal-solitons and $ C^\infty$ convergence of the modified Calabi flow on certain $ CP^1$ bundles, Pacific J. Math. 233 (2007), 91-124. MR 2366370
  • [H1] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, Journal of Differential Geometry 17 (1982), 255-306. MR 664497 (84a:53050)
  • [H2] -, The Ricci flow on surfaces, in Mathematics and general relativity (J. I. Isenberg, Ed.), Contemporary Mathematics 71, American Mathematical Society, 1988, 237-262. MR 0954419 (89k:53029)
  • [K] Joseph J. Kohn, Subellipticity of the $ \bar \partial$-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Mathematica 142 (1979), 79-122. MR 512213 (80d:32020)
  • [Ko] Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Mathematica 180 (1998), 69-117. MR 1618325 (99h:32017)
  • [M] Toshiki Mabuchi, K-energy maps integrating Futaki invariants, Tôhoku Mathematical Journal 38 (1986), 575-593. MR 867064 (88b:53060)
  • [N] Alan M. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Annals of Mathematics 132 (1990), 549-596. MR 1078269 (92d:32038)
  • [P] Nefton Pali, Characterization of Einstein-Fano manifolds via the Kähler-Ricci flow, preprint, arxiv:math.DG/0607581v2.
  • [PSS] Duong H. Phong, Nataša Šešum, Jacob Sturm, Multiplier ideal sheaves and the Kähler-Ricci flow, Comm. Anal. Geom. 15 (2007), 613-632. MR 2379807
  • [R1] Yanir A. Rubinstein, On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood, Journal of Functional Analysis 255, special issue dedicated to Paul Malliavin (2008), 2641-2660. MR 2473271
  • [R2] -, The Ricci iteration and its applications, Comptes Rendus de l'Académie des Sciences Paris 345 (2007), 445-448. MR 2367363
  • [R3] -, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Advances in Mathematics 218 (2008), 1526-1565. MR 2419932
  • [ST] Nataša Šešum, Gang Tian, Bounding scalar curvature and diameter along the Kähler-Ricci flow (after Perelman) and some applications, J. Inst. Math. Jussieu 7 (2008), 575-587. MR 2427424
  • [S] Santiago R. Simanca, Heat flows for extremal Kähler metrics, Annali della Scuola Normale Superiore di Pisa 4 (2005), 187-217. MR 2163555 (2006d:53083)
  • [Si] Yum-Tong Siu, The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group, Annals of Mathematics 127 (1988), 585-627. MR 942521 (89e:58032)
  • [So] Jian Song, The $ \alpha$-invariant on toric Fano manifolds, American Journal of Mathematics 127 (2005), 1247-1259. MR 2183524 (2007a:32027)
  • [T1] Gang Tian, On Kähler-Einstein metrics on certain Kähler manifolds with $ C_1(M)>0$, Inventiones Mathematicae 89 (1987), 225-246. MR 894378 (88e:53069)
  • [T2] -, On stability of the tangent bundles of Fano varieties, International Journal of Mathematics 3 (1992), 401-413. MR 1163733 (93d:32044)
  • [T3] -, Kähler-Einstein metrics with positive scalar curvature, Inventiones Mathematicae 130 (1997), 1-37. MR 1471884 (99e:53065)
  • [TY] Gang Tian, Shing-Tung Yau, Kähler-Einstein metrics on complex surfaces with $ C_1>0$, Communications in Mathematical Physics 112 (1987), 175-203. MR 904143 (88k:32070)
  • [TZ] Gang Tian, Xiao-Hua Zhu, Convergence of Kähler-Ricci flow, Journal of the American Mathematical Society 20 (2007), 675-699. MR 2291916 (2007k:53107)
  • [Y] Shing-Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Communications in Pure and Applied Mathematics 31 (1978), 339-411. MR 480350 (81d:53045)
  • [Ye] Rugang Ye, The logarithmic Sobolev inequality along the Ricci flow, preprint, arxiv:0707.2424v4 [math.DG].
  • [Z] Qi S. Zhang, A uniform Sobolev inequality under Ricci flow, International Mathematics Research Notices (2007), Article ID rnm056. Erratum to: A uniform Sobolev inequality under Ricci flow (2007), Article ID rnm096. MR 2354801; MR 2359549 (2008k:53144)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32Q20, 14J45, 32L10, 32W20, 53C25, 58E11

Retrieve articles in all journals with MSC (2000): 32Q20, 14J45, 32L10, 32W20, 53C25, 58E11

Additional Information

Yanir A. Rubinstein
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Masachusetts 02139-4307
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Received by editor(s): August 30, 2007
Published electronically: May 7, 2009
Dedicated: To Aynat Rubinstein
Article copyright: © Copyright 2009 Yanir A. Rubinstein

American Mathematical Society