A note on the Hitchin-Thorpe inequality and Ricci flow on 4-manifolds
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- by Yuguang Zhang and Zhenlei Zhang PDF
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Abstract:
In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.References
- Kazuo Akutagawa, Masashi Ishida, and Claude LeBrun, Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds, Arch. Math. (Basel) 88 (2007), no. 1, 71–76. MR 2289603, DOI 10.1007/s00013-006-2181-0
- M. T. Anderson, Remarks on Perelman’s papers, preprint, available at http://www.math. sunysb.edu/~anderson/perelman.pdf.
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
- 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
- Fuquan Fang and Yuguang Zhang, Perelman’s $\lambda$-functional and Seiberg-Witten equations, Front. Math. China 2 (2007), no. 2, 191–210. MR 2299748, DOI 10.1007/s11464-007-0014-5
- Fuquan Fang, Yuguang Zhang, and Zhenlei Zhang, Non-singular solutions to the normalized Ricci flow equation, Math. Ann. 340 (2008), no. 3, 647–674. MR 2357999, DOI 10.1007/s00208-007-0164-5
- Fuquan Fang, Zhenlei Zhang, and Yuguang Zhang, Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume, J. Geom. Anal. 20 (2010), no. 3, 592–608. MR 2610891, DOI 10.1007/s12220-010-9120-9
- Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
- Richard S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), no. 4, 695–729. MR 1714939, DOI 10.4310/CAG.1999.v7.n4.a2
- Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435–441. MR 350657
- M. Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structure, arXiv:math.0807.2169.
- Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
- D. Kotschick, Monopole classes and Perelman’s invariant of four-manifolds, arXiv:math. DG/0608504.
- Claude LeBrun, Kodaira dimension and the Yamabe problem, Comm. Anal. Geom. 7 (1999), no. 1, 133–156. MR 1674105, DOI 10.4310/CAG.1999.v7.n1.a5
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
- G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109.
- O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Functional Analysis 42 (1981), no. 1, 110–120. MR 620582, DOI 10.1016/0022-1236(81)90050-1
- 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
- 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
- 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
- Yuguang Zhang, Miyaoka-Yau inequality for minimal projective manifolds of general type, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2749–2754. MR 2497488, DOI 10.1090/S0002-9939-09-09838-4
- Zhei-lei Zhang, Compact blow-up limits of finite time singularities of Ricci flow are shrinking Ricci solitons, C. R. Math. Acad. Sci. Paris 345 (2007), no. 9, 503–506 (English, with English and French summaries). MR 2375111, DOI 10.1016/j.crma.2007.09.017
- Zhou Zhang, Scalar curvature bound for Kähler-Ricci flows over minimal manifolds of general type, Int. Math. Res. Not. IMRN 20 (2009), 3901–3912. MR 2544732, DOI 10.1093/imrn/rnp073
Additional Information
- Yuguang Zhang
- Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China
- MR Author ID: 780283
- Email: yuguangzhang76@yahoo.com
- Zhenlei Zhang
- Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China
- MR Author ID: 794099
- Email: zhleigo@yahoo.com.cn
- Received by editor(s): May 26, 2010
- Received by editor(s) in revised form: January 23, 2011
- Published electronically: September 7, 2011
- Additional Notes: The first author was supported by NSFC-10901111 and KM-210100028003
The second author was supported by NSFC-09221010056 - Communicated by: Michael Wolf
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1777-1783
- MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/S0002-9939-2011-11084-0
- MathSciNet review: 2869163