1 Introduction

The Ricci flow on a manifold M can be regarded as an orbit in the space \(\text {Met}(M)\Big /(\text {Diff}\oplus \text {Scal})\), where \(\text {Met}(M)\) stands for the space of all the Riemannian metrics on M and \(\text {Diff}\oplus \text {Scal}\) denotes the group of self-diffeomorphisms of M and scalings (with positive factors) in \(\text {Met}(M)\). The breathers are the periodic orbits in this space.

Definition 1

A metric g(t) evolving by the Ricci flow on a Riemannian manifold M is called a breather, if for some \(t_1<t_2\), there exists an \(\alpha >0\) and a diffeomorphism \(\phi :M\rightarrow M\), such that \(\alpha g(t_1)=\phi ^*g(t_2)\). If \(\alpha =1\), \(\alpha <1\), or \(\alpha >1\), then the breather is called steady, shrinking, or expanding, respectively.

As a special case of the periodic orbits, the Ricci solitons, moving by diffeomorphisms and scalings, are the static orbits in the space \(\text {Met}(M)/(\text {Diff}\oplus \text {Scal})\).

Definition 2

A gradient Ricci soliton is a 3-tuple (M, g, f), where (M, g) is a Riemannian manifold and f is a smooth function on M called the potential function, which satisfies

$$\begin{aligned} Ric+\nabla ^2f=\frac{\lambda }{2}g, \end{aligned}$$

where \(\lambda =0\), \(\lambda =1\), or \(\lambda =-1\), corresponding to the cases of steady, shrinking, or expanding solitons, respectively.

It is well understood that when moving by the 1-parameter family of diffeomorphisms generated by the potential function, along with a scaling factor, the pull-back metric of the soliton satisfies the Ricci flow equation, and this Ricci flow is called the canonical form of the Ricci soliton; one may refer to [3] for more details.

Perelman proved that on a closed manifold, any periodic orbit in \(\text {Met}(M)/\text {Diff}\) must be static.

Theorem 3

(Perelman’s no breather theorem) A steady, shrinking, or expanding breather on a closed manifold is (the canonical form of) a steady, shrinking, or expanding gradient Ricci soliton, respectively. In particular, in the steady or expanding case, the breather is also Einstein. In dimension 3, this was originally proved by Ivey.

We extend the no shrinking breather theorem to the complete noncompact case.

Theorem 4

Every complete noncompact shrinking breather with bounded curvature is (the canonical form of) a shrinking gradient Ricci soliton.

Our main technique is the \({\mathcal {L}}\)-geometry, an important technique for the Ricci flow established by Perelman. In Sect. 2, we give a brief introduction to the \({\mathcal {L}}\)-functional. In Sect. 3, we prove Theorem 4.

2 Perelman’s \({\mathcal {L}}\)-Geometry

The definitions and results in this section can be found in Perelman [8] and Naber [7]. We consider a backward Ricci flow \((M,g(\tau ))\), \(\tau \in [0, T]\), satisfying

$$\begin{aligned} \frac{\partial }{\partial \tau }g(\tau )=2 Ric(g(\tau )). \end{aligned}$$
(1)

Let \(\gamma (\tau ):[0,\tau _0]\rightarrow M\) be a smooth curve. The \({\mathcal {L}}\)-functional of \(\gamma \) is defined by

$$\begin{aligned} {\mathcal {L}}(\gamma ):=\int _0^{\tau _0}\sqrt{\tau }\Big (R(\gamma (\tau ),\tau )+|{\dot{\gamma }}(\tau )|^2_{g(\tau )}\Big )d\tau . \end{aligned}$$
(2)

The reduced distance between two space-time points \((x_0,0)\), \((x_1,\tau _1)\), where \(\tau _1>0\), is defined by

$$\begin{aligned} l_{(x_0,0)}(x_1,\tau _1):=\frac{1}{2\sqrt{\tau _1}}\inf _\gamma {\mathcal {L}}(\gamma ), \end{aligned}$$
(3)

where the \(\inf \) is taken among all the (piecewise) smooth curves \(\gamma : [0,\tau _1]\rightarrow M\) such that \(\gamma (0)=x_0\) and \(\gamma (\tau _1)=x_1\). When regarded as a function of \((x_1,\tau _1)\), \(l_{(x_0,0)}(\cdot ,\cdot )\) is called the reduced distance based at\((x_0,0)\). When the base point is understood, we also write \(l_{(x_0,0)}\) as l. It is well known that the reduced volume based at \((x_0,0)\)

$$\begin{aligned} {\mathcal {V}}_{(x_0,0)}(\tau ):=\int _M(4\pi \tau )^{-\frac{n}{2}}e^{-l_{(x_0,0)}(\cdot ,\tau )}dg(\tau ) \end{aligned}$$
(4)

is monotonically decreasing in \(\tau \). We often write \({\mathcal {V}}_{(x_0,0)}(\tau )\) as \({\mathcal {V}}(\tau )\) for simplicity. We also remark here that the integrand \((4\pi \tau )^{-\frac{n}{2}}e^{-l}\) of the reduced volume is a subsolution to the conjugate heat equation

$$\begin{aligned} \frac{\partial }{\partial \tau }u-\Delta u+Ru=0, \end{aligned}$$

in the barrier sense or in the sense of distributions.

Now we consider an ancient solution \((M,g(\tau ))\), where \(\tau \in [0,\infty )\) is the backward time. The Type I condition is the following curvature bound.

Definition 5

An ancient solution \((M,g(\tau ))\), where \(\tau \in [0,\infty )\) is the backward time, is called Type I if there exists \(C<\infty \), such that

$$\begin{aligned} |Rm|(\tau )\le \frac{C}{\tau }, \end{aligned}$$

for every \(\tau \in (0,\infty )\).

To ensure the existence of a smooth limit, the \(\kappa \)-noncollapsing condition is often required.

Definition 6

A backward Ricci flow is called \(\kappa \)-noncollapsed, where \(\kappa >0\), if for any space-time point \((x,\tau )\), any scale \(r>0\), whenever \(|Rm|\le r^{-2}\) on \(B_{g(\tau )}(x,r)\times [\tau ,\tau +r^2]\), it holds that \({\text {Vol}}_{g(\tau )}(B_{g(\tau )}(x,r))\ge \kappa r^n\).

We will use the following theorem of Naber [7].

Theorem 7

(Asymptotic shrinker for Type I ancient solution) Let \((M,g(\tau ))\), where \(\tau \in [0,\infty )\) is the backward time, be a Type I \(\kappa \)-noncollapsed ancient solution to the Ricci flow. Fix \(x_0\in M\). Let l be the reduced distance based at \((x_0,0)\). Let \(\{(x_i,\tau _i)\}_{i=1}^\infty \subset M\times (0,\infty )\) be such that \(\tau _i\nearrow \infty \) and

$$\begin{aligned} \sup _{i=1}^\infty l(x_i,\tau _i)<\infty . \end{aligned}$$
(5)

Then \(\{(M,\tau _i^{-1}g(\tau \tau _i),(x_i,1))_{\tau \in [1,2]}\}_{i=1}^\infty \) converges, after possibly passing to a subsequence, to the canonical form of a shrinking gradient Ricci soliton.

Remark 1

In Naber’s original theorem, he fixes the base points \(x_i\equiv x_0\). However, it is easy to observe from his proof that so long as (5) holds, all the estimates of l also hold in the same way as in his case. Hence one may apply the blow-down shrinker part of Theorem 2.1 in [7] to the sequence of space-time base points \((x_i,\tau _i)\) and the scaling factors \(\tau _i^{-1}\).

Remark 2

The estimates for l and the monotonicity formula for \({\mathcal {V}}\) in [7] do not depend on the noncollapsing condition. According to Hamilton [4], if the noncollapsing assumption is replaced by

$$\begin{aligned} \inf _{i=1}^\infty {\text {inj}}_{\tau _i^{-1}g(\tau _i)}(x_i)>\delta , \end{aligned}$$
(6)

where \({\text {inj}}_g(x)\) stands for the injectivity radius of the metric g at the point x, and \(\delta >0\) is a constant, then the conclusion of Theorem 7 still holds.

3 Proof of the Main Theorem

Following the argument in Lu and Zheng [6], we construct a Type I ancient solution to the Ricci flow starting from a given shrinking breather. After scaling and translating in time, we consider the backward Ricci flow \((M,g_0(\tau ))_{\tau \in [0,1]}\), where \(g_0(\tau )\) satisfies (1), such that there exists \(\alpha \in (0,1)\) and a diffeomorphism \(\phi :M\rightarrow M\), satisfying

$$\begin{aligned} \alpha g_0(1)=\phi ^*g_0(0). \end{aligned}$$
(7)

Furthermore, we let \(C<\infty \) be the curvature bound, that is,

$$\begin{aligned} \sup _{M\times [0,1]}|Rm|(g(\tau ))\le C. \end{aligned}$$
(8)

For notational simplicity, we define

$$\begin{aligned} \tau _i=\sum _{j=0}^i\alpha ^{-j}, \end{aligned}$$

where \(i=0,1,2,...\) Evidently, \(\tau _i\nearrow \infty \) since \(\alpha \in (0,1)\), and we can find a \(C_0<\infty \) depending only on \(\alpha \) (for instance, one may let \(C_0=(1-\alpha )^{-1}\)) such that

$$\begin{aligned} \alpha ^{-i}\le \tau _i\le C_0\alpha ^{-i}, \text { for every } i\ge 0. \end{aligned}$$
(9)

For each \(i\ge 1\), we define a Ricci flow

$$\begin{aligned} g_i(\tau ):=\alpha ^{-i}(\phi ^{i})^*g_0\left( \alpha ^i(\tau -\tau _{i-1})\right) ,\ \text {where }\tau \in [\tau _{i-1},\tau _{i}]. \end{aligned}$$
(10)

To see all these Ricci flows are well-concatenated, we apply (7) to observe that

$$\begin{aligned} g_1(\tau _0)= & {} \alpha ^{-1}\phi ^*g_0(0)=g_0(1), \\ g_i(\tau _{i-1})= & {} \alpha ^{-i}(\phi ^i)^*g_0(0)=\alpha ^{-(i-1)}(\phi ^{i-1})^*g_0(1) \\= & {} \alpha ^{-(i-1)}(\phi ^{i-1})^*g_0\left( \alpha ^{i-1}(\tau _{i-1}-\tau _{i-2})\right) =g_{i-1}(\tau _{i-1}). \end{aligned}$$

Therefore, we define an ancient solution

$$\begin{aligned} g(\tau )=\left\{ \begin{array}{ll} g_0(\tau ) &{} \text {for } \tau \in [0,1] \\ g_i(\tau ) &{} \text {for } \tau \in [\tau _{i-1},\tau _{i}]\text { and } i\ge 1.\end{array}\right. \end{aligned}$$
(11)

It then follows from the uniqueness theorem of Chen and Zhu [2] that the ancient solution \(g(\tau )\) is smooth.

Now we proceed to show that \((M,g(\tau ))_{\tau \in [0,\infty )}\), where \(g(\tau )\) is defined in (11), is Type I. We need only to consider the case when \(\tau \ge 1\). Let \(i\ge 1\) be such that \(\tau \in [\tau _{i-1},\tau _i]\). Then

$$\begin{aligned} |Rm(g(\tau ))|=|Rm(g_i(\tau ))|\le \alpha ^{i}\sup _{M\times [0,1]}\left| Rm\Big ((\phi ^i)^*g_0(\tau )\Big )\right| \le C\alpha ^{i}, \end{aligned}$$

where we have used (8), (10), and (11). Then we have

$$\begin{aligned} |Rm(g(\tau ))|\le C\alpha ^{i}\le \frac{C}{\tau }\tau _i\alpha ^{i}\le \frac{B}{\tau }, \end{aligned}$$
(12)

where we have used (9), and \(B=CC_0\) is independent of i.

With all these preparations, we are ready to prove our main theorem.

Proof of Theorem 4

Fix an arbitrary point \(y\in M\) as the base point, and for each \(i\ge 0\) we define

$$\begin{aligned} x_i=\phi ^{-(i+1)}(y). \end{aligned}$$
(13)

In Lu and Zheng [6], they made the assumption that the \(\{x_i\}_{i=1}^\infty \) do not drift away to spatial infinity so that they may apply Theorem 4.1 in [1] to show that \(\{(M,\tau _i^{-1}g(\tau \tau _i),(x_i,1))_{\tau =[1,2]}\}_{i=1}^\infty \) converges, after passing to a subsequence, to the canonical form of a shrinking gradient Ricci soliton. Instead we will show that \(l(x_i,\tau _i)\), where \(i\ge 0\) and l is the reduced distance based at (y, 0), is a bounded sequence. To see this, we let \(\sigma :[0,1]\rightarrow M\) be a smooth curve such that \(\sigma (0)=y\) and \(\sigma (1)= x_0\). Let \(A<\infty \) be such that

$$\begin{aligned} |{\dot{\sigma }}(\tau )|_{g_0(\tau )}\le A, \text { for all } \tau \in [0,1]. \end{aligned}$$
(14)

For each \(i\ge 0\), we define

$$\begin{aligned} \sigma _i(\tau ):=\phi ^{-(i+1)}\circ \sigma (\alpha ^{i+1}(\tau -\tau _{i})),\ \text {where } \tau \in [\tau _i,\tau _{i+1}]. \end{aligned}$$
(15)

We observe that these \(\sigma _i\)’s and \(\sigma \) altogether define a continuous curve in M:

$$\begin{aligned} \sigma _0(\tau _0)= & {} \phi ^{-1}\circ \sigma (0)=\phi ^{-1}(y)=x_0=\sigma (1), \\ \sigma _i(\tau _i)= & {} \phi ^{-(i+1)}\circ \sigma (0)=\phi ^{-i}\circ \sigma (1) \\= & {} \phi ^{-i}\circ \sigma (\alpha ^i(\tau _i-\tau _{i-1}))=\sigma _{i-1}(\tau _i). \end{aligned}$$

We then define \(\gamma _i:[0,\tau _{i+1}]\rightarrow M\), where \(i\ge 0\), as

$$\begin{aligned} \gamma _i(\tau ):=\left\{ \begin{array}{ll} \sigma (\tau ) &{} \text {when } \tau \in [0,1], \\ \sigma _j(\tau ) &{} \text {when } \tau \in [\tau _j,\tau _{j+1}] \text { and } 0\le j\le i. \end{array}\right. \end{aligned}$$

Evidently \(\gamma _i(\tau )\) is piecewise smooth, and \(\gamma _i(0)=y\), \(\gamma _i(\tau _{i+1})=\phi ^{-(i+2)}(y)=x_{i+1}\). We compute for \(i\ge 0\)

$$\begin{aligned} {\mathcal {L}}(\gamma _i)= & {} {\mathcal {L}}(\sigma )+\sum _{j=0}^i\int _{\tau _j}^{\tau _{j+1}}\sqrt{\tau }\Big ( R(\sigma _j(\tau ),\tau )+|\dot{\sigma _j}(\tau )|^2_{g_{j+1}(\tau )}\Big )d\tau \\\le & {} D + \sum _{j=0}^i\int _{\tau _j}^{\tau _{j+1}}\sqrt{\tau }\Big ( \frac{B}{\tau }+A\alpha ^{j+1}\Big )d\tau , \end{aligned}$$

where in the last inequality we have used D, a constant independent of i, to represent \({\mathcal {L}}(\sigma )\), and we have used the Type I condition (12), the definition (15) of \(\sigma _j\), and the assumption (14). Continuing the computation using (9), we have

$$\begin{aligned} {\mathcal {L}}(\gamma _i)\le D+C_1\sum _{j=0}^i \alpha ^{-\frac{j+1}{2}}, \end{aligned}$$

where \(C_1\) is a constant independent of i. It follows from the definition (3) that

$$\begin{aligned} l(x_{i+1},\tau _{i+1})\le & {} \frac{1}{2\sqrt{\tau _{i+1}}}{\mathcal {L}}(\gamma _i) \\\le & {} \frac{1}{2}D\alpha ^{\frac{i+1}{2}}+\frac{1}{2}C_1\sum _{j=0}^i\alpha ^{\frac{j}{2}}\le C_2<\infty , \end{aligned}$$

where \(C_2\) is a constant independent of i and where we have used \(\alpha ^{\frac{1}{2}}\in (0,1)\).

Now we consider the sequence

$$\begin{aligned} \{(M,\tau _{i}^{-1}g(\tau \tau _i),(x_i,1))_{\tau \in [1,\alpha ^{-1}]}\}_{i=1}^\infty . \end{aligned}$$
(16)

We observe that

$$\begin{aligned} \tau _{i}^{-1}g(\tau _i)=\tau _{i}^{-1}\alpha ^{-(i+1)}\Big (\phi ^{i+1}\Big )^*g_0(0), \end{aligned}$$

where \(\tau _{i}^{-1}\alpha ^{-(i+1)}\) is bounded from above and below by constants independent of i, because of (9). Taking into account the definition (13) of \(x_i\), we can use

$$\begin{aligned} {\text {inj}}_{g_0(0)}(y)>0 \end{aligned}$$

to verify the condition (6). It follows from Theorem 7 that (16) converges smoothly to the canonical form of a shrinking gradient Ricci soliton. Furthermore, since \((M,\tau _{i}^{-1}g(\tau _i),x_i)\) and \((M,g_0(0), y)\) differ only by bounded scaling constants and diffeomorphisms that preserve the base points, by the definition of Cheeger–Gromov convergence, such diffeomorphisms do not affect the limit. In other words, there exists a constant \(C_3>0\) such that

$$\begin{aligned} (M,\tau _{i}^{-1}g(\tau _i),x_i)\rightarrow (M,C_3g_0(0),y) \end{aligned}$$

in the pointed smooth Cheeger–Gromov sense. Therefore, \((M,g_0(0),y)\) also has a shrinker structure up to scaling. It then follows from the backward uniqueness theorem of Kotschwar [5] that the shrinking breather \((M,g_0(\tau ))_{\tau \in [0,1]}\) is the canonical form of a shrinking gradient Ricci soliton. \(\square \)