A Note on Perelman’s No Shrinking Breather Theorem

As an application of his entropy formula, Perelman (The entropy formula for the Ricci flow and its geometric applications, 2002) proved that every compact shrinking breather solution to the Ricci flow is a shrinking gradient Ricci soliton. Zhang (Asian J Math 18(4):727–756, 2014) and Lu and Zheng (J Geom Anal, 1–7, 2017) proved no shrinking breather theorems in the noncompact case under additional conditions. It is a natural question to ask whether one can generalize Perelman’s no shrinking breather theorem to the noncompact case assuming only bounded curvature. This is the result we prove in this paper. Our proof uses Perelman’s L-geometry and an idea of Lu and Zheng (J Geom Anal, 1–7, 2017). The novelty of this paper is that we can remove the technical assumptions in Zhang (Asian J Math 18(4):727–756, 2014) and Lu and Zheng (J Geom Anal, 1–7, 2017).