In this paper, we show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. Our result improves all existing results of the same type such as Zhang [Asian J. Math. 18 (2014), pp. 727-755], Lu and Zheng [J. Geom. Anal. 28 (2018), pp. 3718-3724], Rimoldi and Veronelli [Calc. Var. Partial Differential Equations 58 (2019), Paper No. 66, 26], and Zhang [J. Geom. Anal. 29 (2019), pp. 2702-2708]. We also discuss some geometric applications of this theorem. For instance, we show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient.
|Original language||English (US)|
|Number of pages||22|
|Journal||Transactions of the American Mathematical Society|
|State||Published - 2021|
Bibliographical noteFunding Information:
Received by the editors December 3, 2020, and, in revised form, February 23, 2021. 2020 Mathematics Subject Classification. Primary 53E20. Key words and phrases. Shrinking breathers, gradient Ricci solitons, Perelman’s no breather theorem, noncompact Ricci flows. The first author’s research partially supported by China Scholarship Council, self-determined research funds of CCNU from the colleges’ basic research and operation of MOE CCNU19QN075 and Natural Science Foundation of Hubei 2019CFB511.
© 2021 American Mathematical Society.
- Gradient Ricci solitons
- Noncompact Ricci flows
- Perelman's no breather theorem
- Shrinking breathers