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Non-singular solutions to the normalized Ricci flow equation

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Abstract

In this paper, we study non-singular solutions to Ricci flow on a closed manifold of dimension at least 4. Amongst other things we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t > 0 with uniformly bounded sectional curvature, then the Euler characteristic \(\chi (M)\ge 0\) . Moreover, the 4-manifold satisfies one of the followings

  1. (i)

    M is a shrinking Ricci soliton;

  2. (ii)

    M admits a positive rank F-structure;

  3. (iii)

    the Hitchin–Thorpe type inequality holds

$$2\chi (M)\ge 3|\tau(M)|$$

where \(\chi (M)\) (resp. \(\tau(M)\)) is the Euler characteristic (resp. signature) of M.

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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China

    Fuquan Fang & Yuguang Zhang

  2. Nankai Institute of Mathematics, Weijin Road 94, Tianjin, 300071, People’s Republic of China

    Zhenlei Zhang

Authors
  1. Fuquan Fang
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  2. Yuguang Zhang
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  3. Zhenlei Zhang
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Correspondence to Fuquan Fang.

Additional information

The first author was supported by a NSF Grant of China and the Capital Normal University.

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Fang, F., Zhang, Y. & Zhang, Z. Non-singular solutions to the normalized Ricci flow equation. Math. Ann. 340, 647–674 (2008). https://doi.org/10.1007/s00208-007-0164-5

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  • DOI: https://doi.org/10.1007/s00208-007-0164-5

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