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)



Convergence of general inverse $ \sigma_k$-flow on Kähler manifolds with Calabi ansatz

Authors: Hao Fang and Mijia Lai
Journal: Trans. Amer. Math. Soc. 365 (2013), 6543-6567
MSC (2010): Primary 53C44
Published electronically: August 19, 2013
MathSciNet review: 3105762
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the convergence behavior of the general inverse $ \sigma _k$-flow on Kähler manifolds with initial metrics satisfying the Calabi ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected subvarieties are formed as a result of partial blow up.

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

  • [C] L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4, (1998), 383-402 MR 1658612 (2000b:49004)
  • [Ca] E. Calabi, Extremal Kähler metrics, Seminar on Differetial Geometry, vol. $ \mathbf {102}$ of Ann. Math. Studies, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290 MR 645743 (83i:53088)
  • [Cao] H. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359-372 MR 799272 (87d:58051)
  • [Chen] X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom. $ \mathbf {12}$, no. 4 (2004), 837-852 MR 2104078 (2005h:53116)
  • [Do] S. Donaldson, Moment maps and diffeomorphisms, Asian J. Math. $ \mathbf {3}$, no. 1 (1999), 1-16 MR 1701920 (2001a:53122)
  • [EGZ] P. Eyssidieux, V. Guedj and A. Zeriahi, Singular Käler-Einstein metrics, J. Amer. Math. Soc. $ \mathbf {22}$ (2009), 607-639 MR 2505296 (2010k:32031)
  • [FIK] M. Feldman, T. Ilmanen and D. Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geometry, $ \mathbf {65}$ (2003), 169-209 MR 2058261 (2005e:53102)
  • [FL] H. Fang and M. Lai, On the geometric flows solving Kählerian inverse $ \sigma _k$ equations, arXiv:1203.2571, to appear in Pac. J. Math
  • [FLM] H. Fang, M. Lai and X. Ma, On a class of fully nonlinear flows in Kähler geomety, J. Reine Angew. Math. 653 (2011), 189-220 MR 2794631 (2012g:53134)
  • [H] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. $ \mathbf {17}$ (1982), 255-306 MR 664497 (84a:53050)
  • [K] S. Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69-117 MR 1618325 (99h:32017)
  • [L] C. Li, On rotationally symmetric Kähler-Ricci solitons, arXiv:1004.4049
  • [ST1] J. Song and G. Tian, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. $ \mathbf {170}$ (2007), no. 3, 609-653 MR 2357504 (2008m:32044)
  • [ST2] J. Song and G. Tian, The Kähler-Ricci flow through singularities, arXiv:0909.4898
  • [SW1] J. Song and B. Weinkove, The convergence and singularities of the J-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math. 61 (2008), no. 2, 210-229 MR 2368374 (2009a:32038)
  • [SW2] J. Song and B. Weinkove, The Kähler-Ricci flow on Hirzebruch surfaces, arXiv:0903.1900v2, to appear in J. Reine Ange. Math. MR 2837013 (2012g:53142)
  • [SW3] J. Song and B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow, arXiv:1003.0718v2
  • [SW4] J. Song and B. Weinkove, Lecture notes on Kähler-Ricci flow,$ \sim $ jiansong/publication_files/krflectures112111.pdf
  • [SY] J. Song and Y. Yuan, Metric flips with Calabi Ansatz, Geom. Funct. Anal., DOI: 10.1007/s00039-012-0151-1, 2012 MR 2899688
  • [T] G.Tian, New results and problems on Kähler-Ricci flow, Géométrie différentielle, physique mathématique, mathématiques et société. II. Astérisque $ \mathbf {322}$ (2008), 71-92 MR 2521654 (2011a:53131)
  • [TZ] G. Tian and Z. 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 (2007c:32029)
  • [W] L. Wang, On the regularity theory of fully nonlinear parabolic equations. II. Comm. Pure Appl. Math. 45 (1992), no. $ \mathbf {2}$, 141-178 MR 1139064 (92m:35127)
  • [We1] B. Weinkove, Convergence of the J-flow on Kähler surfaces, Comm. Anal. Geom. 12 (2004), no. $ \mathbf {4}$, 949-965 MR 2104082 (2005g:32027)
  • [We2] B. Weinkove, On the $ J$-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom. 73 (2006), no. $ \mathbf {2}$, 351-358 MR 2226957 (2007a:32026)
  • [Y] S-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge- Ampère equation, I, Comm. Pure Appl. Math. $ \mathbf {31}$ (1978), 339-411 MR 480350 (81d:53045)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44

Retrieve articles in all journals with MSC (2010): 53C44

Additional Information

Hao Fang
Affiliation: Department of Mathematics, University of Iowa, 14 McLean Hall, Iowa City, Iowa 52242

Mijia Lai
Affiliation: Department of Mathematics, University of Rochester, 915 Hylan Building, RC Box 270138, Rochester, New York 14627

Received by editor(s): March 23, 2012
Received by editor(s) in revised form: June 13, 2012
Published electronically: August 19, 2013
Additional Notes: The first author was partially supported by NSF grant DMS1008249.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society