Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Collapsing of products along the Kähler-Ricci flow
HTML articles powered by AMS MathViewer

by Matthew Gill PDF
Trans. Amer. Math. Soc. 366 (2014), 3907-3924 Request permission

Abstract:

Let $X = M \times E$, where $M$ is an $m$-dimensional Kähler manifold with negative first Chern class and $E$ is an $n$-dimensional complex torus. We obtain $C^\infty$ convergence of the normalized Kähler-Ricci flow on $X$ to a Kähler-Einstein metric on $M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture.
References
  • Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95 (French, with English summary). MR 494932
  • Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359–372. MR 799272, DOI 10.1007/BF01389058
  • Albert Chau, Convergence of the Kähler-Ricci flow on noncompact Kähler manifolds, J. Differential Geom. 66 (2004), no. 2, 211–232. MR 2106124
  • Ph. Delanoë and A. Hirschowitz, About the proofs of Calabi’s conjectures on compact Kähler manifolds, Enseign. Math. (2) 34 (1988), no. 1-2, 107–122. MR 960195
  • Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
  • F. Fong, Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds, to appear in Trans. Amer. Math. Soc., arXiv: 1104.3924.
  • F. Fong, On the collapsing rate of Kähler-Ricci flow with finite-time singularity, preprint, arXiv: 1112.5987.
  • F. Fong, Z. Zhang, The collapsing rate of the Kähler-Ricci flow with regular infinite time singularity, preprint, arXiv: 1202.3199.
  • M. Gross, V. Tosatti, Y. Zhang, Collapsing of abelian fibred Calabi-Yau manifolds, preprint, arXiv: 1108.0967.
  • N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144
  • Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Canonical metrics on the moduli space of Riemann surfaces. II, J. Differential Geom. 69 (2005), no. 1, 163–216. MR 2169586
  • D. H. Phong, Natasa Sesum, and Jacob Sturm, Multiplier ideal sheaves and the Kähler-Ricci flow, Comm. Anal. Geom. 15 (2007), no. 3, 613–632. MR 2379807
  • Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277
  • J. Song, G. Székelyhidi, B. Weinkove, The Kähler-Ricci flow on projective bundles, preprint, arXiv: 1107:2144.
  • Jian Song and Gang Tian, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, 609–653. MR 2357504, DOI 10.1007/s00222-007-0076-8
  • Jian Song and Gang Tian, Canonical measures and Kähler-Ricci flow, J. Amer. Math. Soc. 25 (2012), no. 2, 303–353. MR 2869020, DOI 10.1090/S0894-0347-2011-00717-0
  • J. Song, G. Tian, The Kähler-Ricci flow through singularities, preprint, arXiv: 0803.1613.
  • Jian Song and Ben Weinkove, The Kähler-Ricci flow on Hirzebruch surfaces, J. Reine Angew. Math. 659 (2011), 141–168. MR 2837013, DOI 10.1515/CRELLE.2011.071
  • J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow, preprint, arXiv: 1003.0718.
  • J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow II, preprint, arXiv: 1102.1759.
  • J. Song, B. Weinkove, Lecture notes on the Kähler-Ricci flow.
  • Morgan Sherman and Ben Weinkove, Interior derivative estimates for the Kähler-Ricci flow, Pacific J. Math. 257 (2012), no. 2, 491–501. MR 2972475, DOI 10.2140/pjm.2012.257.491
  • J. Song, Y. Yuan Metric flips with Calabi ansatz, preprint, arXiv: 1011.1608.
  • Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. MR 1064867
  • Gang Tian and Zhou Zhang, On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. MR 2243679, DOI 10.1007/s11401-005-0533-x
  • Valentino Tosatti, Adiabatic limits of Ricci-flat Kähler metrics, J. Differential Geom. 84 (2010), no. 2, 427–453. MR 2652468
  • Hajime Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), no. 1, 123–133. MR 944606, DOI 10.1007/BF01449219
  • Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 53C55
  • Retrieve articles in all journals with MSC (2010): 53C44, 53C55
Additional Information
  • Matthew Gill
  • Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093
  • Address at time of publication: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840
  • MR Author ID: 951451
  • Received by editor(s): June 14, 2012
  • Received by editor(s) in revised form: December 17, 2012
  • Published electronically: November 14, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3907-3924
  • MSC (2010): Primary 53C44; Secondary 53C55
  • DOI: https://doi.org/10.1090/S0002-9947-2013-06073-4
  • MathSciNet review: 3192623