New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

TL;DR

This paper presents two novel proofs demonstrating that shrinking breathers in Ricci flow on closed manifolds are necessarily gradient Ricci solitons, utilizing properties of singularity models and ancient solutions.

Contribution

It introduces two new proofs of Perelman's theorem, expanding the understanding of shrinking breathers in Ricci flow through alternative approaches.

Findings

Shrinking breathers are gradient Ricci solitons.

Singularity models of type I are shrinking gradient Ricci solitons.

Rescaled limits of non-collapsed type I ancient solutions are shrinking gradient Ricci solitons.

Abstract

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being shrinking gradient Ricci solitons.

Equations

$$G_{j+1}(t)=\alpha \cdot {\phi }^{\ast }{G}_j({\alpha }^{-1}(t-{\tilde{t}}_{j+1})+{\tilde{t}}_j),t\in [{\tilde{t}}_{j+1},{\tilde{t}}_{j+2}],$$

$$G_{j-1}(t)={\alpha }^{-1}\cdot ({\phi }^{-1}{)}^{\ast }{G}_j(\alpha (t-{\tilde{t}}_{j-1})+{\tilde{t}}_j),t\in [{\tilde{t}}_{j-1},{\tilde{t}}_j],$$

$$G_j(t)={\alpha }^j\cdot ({\phi }^j{)}^{\ast }g\left({\alpha }^{-j}\left(t-{\tilde{t}}_0-\frac{1-{\alpha }^j}{1-\alpha }({\tilde{t}}_1-{\tilde{t}}_0)\right)+{\tilde{t}}_0\right),$$

$$G_{-j}(t)={\alpha }^{-j}\cdot ({\phi }^{-j}{)}^{\ast }g\left({\alpha }^j\left(t-{\tilde{t}}_1+\frac{{\alpha }^{-(j+1)}-1}{{\alpha }^{-1}-1}({\tilde{t}}_1-{\tilde{t}}_0)\right)+{\tilde{t}}_0\right).$$

$$K=\underset{(x,t)\in M\times [{\tilde{t}}_0,{\tilde{t}}_1]}{\sup }|\mathrm{Rm}{|}_g(x,t)\in (0,\infty ).$$

$$\underset{x\in M}{\sup }|\mathrm{Rm}{|}_{G_j}(x,t)\le {\alpha }^{-j}K\le \frac{(1-\alpha {)}^{-1}({\tilde{t}}_1-{\tilde{t}}_0)K}{T-t},$$

$$\underset{t\in [{\tilde{t}}_j,{\tilde{t}}_{j+1}]}{\sup }\underset{x\in M}{\sup }|\mathrm{Rm}{|}_{G_j}(x,t)={\alpha }^{-j}K.$$

$$\underset{x\in M}{\sup }|\mathrm{Rm}{|}_G(x,t)\le \frac{(1-\alpha {)}^{-1}({\tilde{t}}_1-{\tilde{t}}_0)K}{T-t}$$

$$\underset{x\in M}{\sup }|\mathrm{Rm}{|}_{G_{-j}}(x,t)\le {\alpha }^{j}K,$$

$$\frac{1}{|t|}\ge \frac{1}{|{\tilde{t}}_{-j}|}=\frac{1}{|{\tilde{t}}_1-\frac{{\alpha }^{-(j+1)}-1}{{\alpha }^{-1}-1}({\tilde{t}}_1-{\tilde{t}}_0)|}\ge \frac{{\alpha }^{-1}-1}{2({\tilde{t}}_1-{\tilde{t}}_0)}\cdot {\alpha }^{j+1}.$$

$$\underset{x\in M}{\sup }|\mathrm{Rm}{|}_G(x,t)\le \frac{2({\tilde{t}}_1-{\tilde{t}}_0)K}{(1-\alpha )|t|}$$

$${\cup }_{t\in [\max \{{\bar{t}}_{\ast }-{\alpha }^{j_{\ast }}{r}_{\ast }^2,{\tilde{t}}_0\},{\bar{t}}_{\ast }]}(B_{g(t)}({\phi }^{-j_{\ast }}(p_{\ast }),{\alpha }^{j_{\ast }/2}{r}_{\ast })\times \{t\}),$$

$${\mathrm{Vol}}_{g({\bar{t}}_{\ast })}{B}_{g({\bar{t}}_{\ast })}({\phi }^{-j_{\ast }}(p_{\ast }),{\alpha }^{j_{\ast }/2}{r}_{\ast })\ge \kappa ({\alpha }^{j_{\ast }/2}{r}_{\ast }{)}^n,$$

$${\mathrm{Vol}}_{G(t_{\ast })}{B}_{G(t_{\ast })}(p_{\ast },r_{\ast })\ge \kappa r_{\ast }^n.$$

$${\lambda }_j={\left(T-\left({\tilde{t}}_0+\frac{1-{\alpha }^j}{1-\alpha }({\tilde{t}}_1-{\tilde{t}}_0)\right)\right)}^{-1}={\alpha }^{-j}(1-\alpha )({\tilde{t}}_1-{\tilde{t}}_0{)}^{-1}\to \infty$$

$$g_j(-1)={\alpha }^{-1}(1-\alpha )({\tilde{t}}_1-{\tilde{t}}_0{)}^{-1}({\phi }^{j-1}{)}^{\ast }g({\tilde{t}}_1)$$

$${\lambda }_j|\mathrm{Rm}{|}_{g_j}(x,-1)=|\mathrm{Rm}{|}_G(x,T-{\lambda }_j^{-1})={\alpha }^{-j+1}|\mathrm{Rm}{|}_g({\phi }^{j-1}(x),{\tilde{t}}_1).$$

$$|\mathrm{Rm}{|}_G({\hat{p}}_j,t_j)=\frac{\alpha (1-\alpha {)}^{-1}({\tilde{t}}_1-{\tilde{t}}_0)|\mathrm{Rm}{|}_g(p_{\ast },{\tilde{t}}_1)}{T-t_j}\to \infty \text{ as }j\to \infty .$$

$$|\mathrm{Rm}{|}_G(p_{j_k},t_{j_k})=\frac{\alpha (1-\alpha {)}^{-1}({\tilde{t}}_1-{\tilde{t}}_0)|\mathrm{Rm}{|}_g({\phi }^{j_k-1}(p_{j_k}),{\tilde{t}}_1)}{T-t_{j_k}}\ge \frac{c_2}{T-t_{j_k}},$$

$$t_k\in ({\tilde{t}}_{-j_k},{\tilde{t}}_{-j_k+1}],$$

$$g_k(0)=|t_k{|}^{-1}G(t_k)=|t_k{|}^{-1}{\alpha }^{-j_k}({\phi }^{-j_k}{)}^{\ast }g({\bar{t}}_k)$$

$${\bar{t}}_k={\alpha }^{j_k}\left(t_k-{\tilde{t}}_1+\frac{{\alpha }^{-(j_k+1)}-1}{{\alpha }^{-1}-1}({\tilde{t}}_1-{\tilde{t}}_0)\right)+{\tilde{t}}_0\in ({\tilde{t}}_0,{\tilde{t}}_1].$$

Full text

Keywords. Ricci flow, Shrinking breathers, Type I solutions, gradient Ricci solitons

MSC (2010). 53C44

P.L. is partially supported by Simons Foundation through Collaboration Grant 229727.

Y.Z. is partially supported by CNSF through Grant 11671141.

1. Introduction

In his pioneer paper [Pe02I], as an application of the $W$-entropy, G. Perelman proves that on closed manifolds the shrinking breathers are shrinking GRS (short for gradient Ricci solitons). In this article we give two new proofs of the Perelman’s theorem for shrinking breathers from the perspective of singularity analysis (Theorem 3.1(i) and 3.3(i)). More precisely, for shrinking breathers we will construct ancient solutions of Ricci flow which have type I singularities at finite singular time and $-\infty$, then we use either the fact that the singularity models of type I solutions are shrinking GRS (A. Naber [Na10], J. Enders, R. Muller, and P. Topping [EMT11]), or the fact that non-collapsed type I ancient solutions have rescaled limits being shrinking GRS (X.D. Cao and Q. Zhang [CZ11]), to finish the proof. We can also say a little bit about steady breathers on noncompact manifolds (Proposition 3.4).

This idea of extending breathers to exist on a larger time interval has been used by M. Feldman, T. Ilmanen, and L. Ni (see the Remark in [FIN05, p.53]). They actually use a result of R. Hamilton about asymptotic limit at time $\infty$ to give an alternative proof that there is no non-trivial expanding breather on closed manifolds ([Iv93], [Pe02I]).

When $M$ is noncompact, under some extra assumption our method also proves that shrinking breathers are GRS (Theorem 3.1(ii) and 3.3(ii)). Note that in [Zh14] Q. Zhang proves a result about shrinking breathers on complete noncompact manifolds by considering $W$-entropy, which has different assumptions from ours. Also note that $W$-entropy on noncompact manifolds is used by M. Rimondi and G. Veronelli [RV16] to prove a result about when a complete noncompact Ricci soliton is a GRS.

Acknowledgement. P.L. wants to thank Professor Li, jiayu and School of Mathematical Sciences at the University of Science and Technology of China, where part of this work is carried out, for their warm hospitality during spring, 2017.

2. Construction of ancient solutions

Let $(M^{n},g(t)),\,t\in(\alpha,\beta)$, be a Ricci flow. $g(t)$ is called a breather if for some $\tilde{t}_{0},\tilde{t}_{1}\in(\alpha,\beta)$ with $\tilde{t}_{0}<\tilde{t}_{1}$ there is a constant $\alpha>0$ and a diffeomorphism $\varphi:M\rightarrow M$ such that the metrics $g(\tilde{t}_{1})=\alpha\varphi^{*}g(\tilde{t}_{0})$. The cases $\alpha<1,\alpha=1$, and $\alpha>1$ correspond to shrinking, steady, and expanding breathers, respectively.

Lemma 2.1**.**

Given a breather $(M^{n},g(t))$ we have the following.

$($i$)$* If the breather is shrinking, there is an ancient solution of Ricci flow $(M^{n},G(t))$ such that $G(t)=g(t)$ when $t\in(\alpha,\beta)$.*

$($ii$)$* If the breather is steady, there is an eternal solution of Ricci flow $(M^{n},G(t))$ such that $G(t)=g(t)$ when $t\in(\alpha,\beta)$.*

$($iii$)$* If the breather is expanding, there is an immortal solution of Ricci flow $(M^{n},G(t))$ such that $G(t)=g(t)$ when $t\in(\alpha,\beta)$.*

Proof.

Let $G_{0}(t)=g(t)$ for $t\in[\tilde{t}_{0},\tilde{t}_{1}]$. First we define $G_{1}(t)=\alpha\cdot\varphi^{*}G_{0}(\alpha^{-1}(t-\tilde{t}_{1})+\tilde{t}_{0})$ for $t\in[\tilde{t}_{1},\tilde{t}_{2}]$ where $\tilde{t}_{2}=\tilde{t}_{1}+\alpha(\tilde{t}_{1}-\tilde{t}_{0})$. It is clear that $G_{1}(t)$ is a solution of Ricci flow. Below we will use the convention that $\frac{1-\alpha^{j}}{1-\alpha}=j$ when $\alpha=1$. Inductively for $j\geq 1$ we define

[TABLE]

where $\tilde{t}_{j+2}=\tilde{t}_{j+1}+\alpha(\tilde{t}_{j+1}-\tilde{t}_{j})$, and for $j\leq 0$ we define

[TABLE]

where $\tilde{t}_{j-1}=\tilde{t}_{j}-\alpha^{-1}(\tilde{t}_{j+1}-\tilde{t}_{j})$.

The following statements follow from some easy calculations. For $j\geq 2$ we have $\tilde{t}_{j}=\tilde{t}_{0}+\frac{1-\alpha^{j}}{1-\alpha}(\tilde{t}_{1}-\tilde{t}_{0})$ and for $j\geq 1$ we have $\tilde{t}_{-j}=\tilde{t}_{1}-\frac{\alpha^{-j-1}-1}{\alpha^{-1}-1}(\tilde{t}_{1}-\tilde{t}_{0})$. For $j\geq 1$

[TABLE]

We define a family of metrics $G(t)$ by piecing $G_{j}(t),j\in\mathbb{Z}$, all together. By the uniqueness of solutions of Ricci flow or more directly by the fact that the left and right time-derivatives of $G(t)$ are matching at each $\tilde{t}_{j}$, we conclude that $G(t)$ is a smooth solution and that $G(t)=g(t)$ for $t\in(\alpha,\beta)$.

$($i$)$ For $\alpha<1$, $G(t)$ is defined on $(-\infty,\tilde{t}_{0}+\frac{1}{1-\alpha}(\tilde{t}_{1}-\tilde{t}_{0}))$, which is an ancient solution.

$($ii$)$ For $\alpha=1$, $G(t)$ is defined on $(-\infty,\infty)$, which is an eternal solution.

$($iii$)$ For $\alpha>1$, $G(t)$ is defined on $(\tilde{t}_{1}-\frac{1}{1-\alpha^{-1}}(\tilde{t}_{1}-\tilde{t}_{0}),\infty)$, which is an immortal solution. ∎

Let $\kappa>0$ be a constant. We call that Ricci flow $(N^{n},h(t)),t\in[\alpha,\beta]$, is $\kappa$-non-collapsed on all scales if for any point $(p_{*},{t}_{*})\in N\times[\alpha,\beta]$ and $r_{*}>0$ which satisfy $|\operatorname{Rm}|_{h}\leq r_{*}^{-2}$ on parabolic ball $\cup_{t\in[\max\{t_{*}-r_{*}^{2},\alpha\},t_{*}]}\left(B_{h(t)}(p_{*},r_{*})\times\{t\}\right)$, the volume $\operatorname{Vol}_{h(t_{*})}B_{h(t_{*})}(p_{*},r_{*})\geq\kappa r_{*}^{n}$. In this definition we allow $\alpha=-\infty$. When $\alpha=-\infty$ the definition agrees with [CZ11, Def. 1.1].

Lemma 2.2**.**

Let $(M^{n},g(t))$ be a nonflat shrinking breather with $g(\tilde{t}_{1})=\alpha\varphi^{*}g(\tilde{t}_{0})$ for some $\alpha\in(0,1)$ and $\tilde{t}_{0}<\tilde{t}_{1}$, and let $G(t)$ be the solution constructed in Lemma 2.1$($i$)$. Assume curvature bound $\sup_{(x,t)\in M\times[\tilde{t}_{0},\tilde{t}_{1}]}|\operatorname{Rm}|_{g}(x,t)<\infty$. Then

$($i$)$* $G(t)$ develops a Type I singularity at time $T=\tilde{t}_{0}+\frac{1}{1-\alpha}(\tilde{t}_{1}-\tilde{t}_{0})$.*

$($ii$)$* $G(t)$ develops a Type I singularity at time $-\infty$, i.e., $\sup_{(x,t)\in M\times(-\infty,\tilde{t}_{0}]}|t||\operatorname{Rm}|_{G}$ $(x,t)<\infty$.*

$($iii$)$* If $g(t),\,t\in[\tilde{t}_{0},\tilde{t}_{1}]$, is $\kappa$-non-collapsed on all scales for some $\kappa>0$, then $G(t),\,t\in(-\infty,\tilde{t}_{0}]$, is $\kappa$-non-collapsed on all scales.*

Proof.

Define

[TABLE]

$($i$)$ For $j\geq 1$ and $t\in[\tilde{t}_{j},\tilde{t}_{j+1}]$ we have

[TABLE]

Equation (2.2) implies that $T$ is a singular time. It follows from (2.1) that

[TABLE]

for $t\in[\tilde{t}_{1},T)$, and $G(t)$ develops a Type I singularity at time $T$.

$($ii$)$ For $j\geq 1$ and $t\in(\tilde{t}_{-j},\tilde{t}_{-j+1}]$ we have

[TABLE]

and for $j$ large enough we have

[TABLE]

Hence

[TABLE]

for $t$ sufficiently close to $-\infty$, and $G(t)$ develops a Type I singularity at time $-\infty$.

$($iii$)$ For any point $(p_{*},{t}_{*})\in M\times(-\infty,\tilde{t}_{0}]$, there is a $j_{*}\geq 1$ such that ${t}_{*}\in(\tilde{t}_{-j_{*}},\tilde{t}_{-j_{*}+1}]$. If there is a $r_{*}>0$ such that $|\operatorname{Rm}|_{G}(x,t)\leq r_{*}^{-2}$ for $(x,t)\in\cup_{t\in[t_{*}-r_{*}^{2},t_{*}]}$ $\left(B_{G(t)}(p_{*},r_{*})\times\{t\}\right)$, by the definition of $G_{-j_{*}}(t)$ and the curvature scaling we have $|\operatorname{Rm}|_{g}\leq\alpha^{-j_{*}}r_{*}^{-2}$ on

[TABLE]

where $\bar{t}_{*}=\alpha^{j_{*}}(t_{*}-\tilde{t}_{1})+\tilde{t}_{0}+\frac{\alpha^{-(j_{*}+1)}-1}{\alpha^{-1}-1}(\tilde{t}_{1}-\tilde{t}_{0})$. By the assumption of the $\kappa$-non-collapsing of $g(t)$, we have

[TABLE]

hence

[TABLE]

This proves that $G(t)$ is $\kappa$-non-collapsed on all scales. Now $($iii$)$ is proved. ∎

3. Proof of main theorems

In this section we adopt the notations used in Lemma 2.2. Now we give the first new proof of Perelman’s theorem that on closed manifolds shrinking breathers are GRS.

Theorem 3.1**.**

Let $(M^{n},g(t))$ be a nonflat complete shrinking breather. Assume one of the following,

$($i$)$* $M$ is closed; or*

$($ii$)$* $M$ is noncompact with bounded curvature and there are a strictly increasing subsequence $\{j_{k}\}$ of natural numbers and a sequence $\{{p}_{j_{k}}\}$ of points in $M$ such that $($iia$)$ ${p}_{j_{k}}$ converges to some point ${p}_{\infty}$, $($iib$)$ $|\operatorname{Rm}|_{g}(\varphi^{j_{k}-1}({p}_{j_{k}}),\tilde{t}_{1})\geq c_{1}$ for some constant $c_{1}>0$, and $($iic$)$ $\varphi^{j_{k}-1}({p}_{\infty})$ converges to a point ${p}^{\prime}_{\infty}\in M$;*

then $(M,g(t))$ is a shrinking GRS.

Proof.

Define

[TABLE]

and the rescaled Ricci flow $g_{j}(t)=\lambda_{j}G(T+\lambda_{j}^{-1}t)$. We compute

[TABLE]

and for any $x\in M$

[TABLE]

(i) When $M$ is closed, fix $p_{*}\in M$ where $|\operatorname{Rm}|_{g}(p_{*},\tilde{t}_{1})\neq 0$, let $\hat{p}_{j}=\varphi^{-j+1}(p_{*})$ and $t_{j}=T-\lambda_{j}^{-1}$ for $j\geq 2$. It follows from (3.2) that

[TABLE]

Hence $(\hat{p}_{j},t_{j})$ is an essential blow-up sequence as defined in [EMT11, Def. 1.2]. Let $\hat{p}_{\infty}$ be a limit of some subsequence of $\{\hat{p}_{j}\}$, then $\hat{p}_{\infty}$ a Type I singular point as in [EMT11, Def. 1.2]. We apply [EMT11, Thm 1.1] to $(M,g_{j}(t),\hat{p}_{\infty})$ to conclude that the sequence subconverges in Cheeger-Gromov sense to a normalized nontrivial shrinking GRS in canonical form. It follows from (3.1) that the subsequence of $\{(M,g_{j}(-1),\hat{p}_{\infty})\}$ has the same limit as the corresponding subsequence $(M,\alpha^{-1}(1-\alpha)(\tilde{t}_{1}-\tilde{t}_{0})^{-1}g(\tilde{t}_{1}),\varphi^{j-1}(\hat{p}_{\infty}))$.

Since $M$ is closed, the subsequence of $\{\varphi^{j-1}(\hat{p}_{\infty})\}$ subconverges to some $\tilde{p}_{\infty}\in M$. Hence $(M,\alpha^{-1}(1-\alpha)(\tilde{t}_{1}-\tilde{t}_{0})^{-1}g(\tilde{t}_{1}),\varphi^{j-1}(\hat{p}_{\infty}))$ subconverges to $(M,\alpha^{-1}(1-\alpha)(\tilde{t}_{1}-\tilde{t}_{0})^{-1}g(\tilde{t}_{1}),\tilde{p}_{\infty})$, it follows from the uniqueness of Cheeger-Gromov limit that $g(\tilde{t}_{1})$ is a shrinking GRS.

(ii) When $M$ is noncompact, it follows from (3.2) and the assumption (iia) that

[TABLE]

where $c_{2}=\alpha(1-\alpha)^{-1}(\tilde{t}_{1}-\tilde{t}_{0})c_{1}$. Hence $({p}_{j_{k}},t_{j_{k}})$ is an essential blow-up sequence and $p_{\infty}$ is a Type I singular point by the assumption (iib). Applying [EMT11, Thm 1.1] to $(M,g_{j_{k}}(t),p_{\infty})$ we conclude that the sequence subconverges in Cheeger-Gromov sense to a normalized nontrivial shrinking GRS in canonical form. It follows from (3.1) that the subsequence of $\{(M,g_{j_{k}}(-1),p_{\infty})\}$ has the same limit as the corresponding subsequence of $\{(M,\alpha^{-1}(1-\alpha)(\tilde{t}_{1}-\tilde{t}_{0})^{-1}g(\tilde{t}_{1}),\varphi^{j_{k}-1}(p_{\infty}))\}$.

By the assumption (iic) $(M,\alpha^{-1}(1-\alpha)(\tilde{t}_{1}-\tilde{t}_{0})^{-1}g(\tilde{t}_{1}),\varphi^{j_{k}-1}(p_{\infty}))$ subconverges to $(M,\alpha^{-1}(1-\alpha)(\tilde{t}_{1}-\tilde{t}_{0})^{-1}g(\tilde{t}_{1}),{p}^{\prime}_{\infty})$. By the uniqueness of Cheeger-Gromov limit $g(\tilde{t}_{1})$ is a shrinking GRS. ∎

Remark 3.2**.**

Suppose that there is a $p_{*}\in M$ such that $|\operatorname{Rm}|_{g}(p_{*},\tilde{t}_{1})\geq c_{1}$. We assume that $\varphi^{-j_{k}+1}(p_{*})$ converges to some $p_{\infty}\in M$ for some subsequence $j_{k}$ and that $\varphi^{j_{k}-1}(p_{\infty})$ subconverges to some ${p}^{\prime}_{\infty}\in M$. Then the conditions in Theorem 3.1(ii) are satisfied by taking $p_{j_{k}}=\varphi^{-j_{k}+1}(p_{*})$.

Below we give the second new proof of the Perelman’s theorem.

Theorem 3.3**.**

Let $(M^{n},g(t))$ be a nonflat complete shrinking breather. Assume one of the following,

$($i$)$* $M$ is closed; or*

$($ii$)$* $M$ is noncompact with bounded curvature and we assume that $(M,g(t))$, $t\in[\tilde{t}_{0},\tilde{t}_{1}]$, is $\kappa$-non-collapsed on all scale and that for the sequence of points $(q_{k},{t}_{k})$ in [CZ11, Thm 4.1], the sequence $\varphi^{-j_{k}}(q_{k})$ subconverges to some point $\hat{q}_{\infty}\in M$ where $j_{k}$ is defined by (3.3) below;*

then $(M,g(t))$ is a shrinking GRS.

Proof.

Note that when $M$ is closed, nonflat solution $(M,g(t))$, $t\in[\tilde{t}_{0},\tilde{t}_{1}]$, is $\kappa$-non-collapsed on all scales for some small $\kappa>0$. Hence it follows from Lemma 2.2 that in either case $($i$)$ or $($ii$)$ solution $(M,G(t))$ is nonflat type I ancient solution and is $\kappa$-non-collapsed on all scales. By [CZ11, Thm 4.1] there is a sequence of points $(q_{k},{t}_{k})$ with ${t}_{k}\rightarrow-\infty$ such that $(M,g_{k}(t),q_{k})$ converges to a nonflat shrinking GRS $(M_{\infty},g_{\infty}(t),q_{\infty})$. Here ${g}_{k}(t)\doteqdot|{t}_{k}|^{-1}G({t}_{k}+s|{t}_{k}|)$. Choose $j_{k}$ so that

[TABLE]

then

[TABLE]

where

[TABLE]

It follows from (3.4) that $\bar{t}_{k}$ subconverges to some $\bar{t}_{\infty}\in[\tilde{t}_{0},\tilde{t}_{1}]$ and that $|{t}_{k}|^{-1}\alpha^{-j_{k}}$ subconverges to a number $c_{1}>0$. Note that to prove the theorem it suffices to prove the subconvergence of $\{(M,g(\bar{t}_{k}),\varphi^{-j_{k}}(q_{k}))\}$.

$($i$)$ When $M$ is closed, then $\varphi^{-j_{k}}(q_{k})$ subconverges to $\bar{q}_{\infty}\in M$. We conclude that $(M,g(\bar{t}_{k}),\varphi^{-j_{k}}(q_{k}))$ subconveregs to $(M,g(\bar{t}_{\infty}),\bar{q}_{\infty})$. By the uniqueness of Cheeger-Gromov limit we have that $(M_{\infty},g_{\infty}(0),q_{\infty})$ is isometric to $(M,c_{1}g(\bar{t}_{\infty}),\bar{q}_{\infty})$. Hence $g(\bar{t}_{\infty})$ is a nonflat shrinking GRS.

$($ii$)$ When $M$ is noncompact, by the assumption that $\varphi^{-j_{k}}(q_{k})$ subconverges to some $\hat{q}_{\infty}$, we can repeat the proof of $($i$)$ to conclude that $g(\bar{t}_{\infty})$ is a shrinking GRS. ∎

G. Perelman proves that steady breathers on closed manifolds are steady GRS (actually Einstein), the following result is of interest when $M$ is noncompact.

Proposition 3.4**.**

Let $(M^{n},g(t)),\,t\in[\tilde{t}_{0},\tilde{t}_{1}]$, be a nonflat complete steady breather with bounded curvature. Assume that the solution $g(t)$ has positive curvature operator and that the scalar curvature $R(\cdot,t)$ attains its supremum over space for each $t$. Then $g(t)$ is a steady GRS.

Proof.

The assumptions enable us to apply [Ha93, Main Thm] to conclude that $(M,G(t))$ in Lemma 2.1(i) is a steady GRS. ∎