# Perelman-type no breather theorem for noncompact Ricci flows

Liang Cheng, Yongjia Zhang  School of Mathematics and Statistics $\&$ Hubei
Key Laboratory of Mathematical Sciences, Central China Normal University,
Wuhan, 430079, P.R.China<EMAIL_ADDRESS>School of Mathematics,
University of Minnesota, Twin Cities, Minneapolis, MN, 55455, USA
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first show that a complete shrinking breather with Ricci
curvature bounded from below must be a shrinking gradient Ricci soliton. This
result has several applications. First, we can classify all complete
$3$-dimensional shrinking breathers. Second, we can show that every complete
shrinking Ricci soliton with Ricci curvature bounded from below must be
gradient—a generalization of Naber’s result in [11]. Furthermore, we develop a
general condition for the existence of the asymptotic shrinking gradient Ricci
soliton, which hopefully will contribute to the study of ancient solutions.

###### Key words and phrases:

shrinking breathers, gradient Ricci solitons, Perelman’s no breather theorem,
noncompact Ricci flows

###### 2000 Mathematics Subject Classification:

Primary 53C44; Secondary 53C42, 57M50.

Liang Cheng’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

##  1\. Introduction

The first application of Perelman’s powerful entropy formulas, immediately
after they are developed, was the proofs of no breather theorems [15].
Perelman proved that every compact shrinking, steady, or expanding breather
must be a shrinking, steady, or expanding gradient Ricci soliton,
respectively. Let us first of all recall the definitions of breathers and
(gradient) Ricci solitons.

######  Definition 1.1.

A complete solution to the Ricci flow $(M,g(t))_{t\in[0,T]}$ is called a
breather, if there exist $0\leq t_{1}<t_{2}\leq T$, $\alpha>0$, and a self-
diffeomorphism $\phi:M\rightarrow M$, such that

$\displaystyle\alpha g(t_{1})=\phi^{*}g(t_{2}).$

If $\alpha<0$, $\alpha=1$, or $\alpha>1$, then the breather is called
shrinking, steady, or expanding, respectively.

######  Definition 1.2.

A Ricci soliton is a tuple $(M,g,X)$, where $(M,g)$ is a smooth Riemannian
manifold and $X$ is a smooth vector field on $M$ , satisfying

$Ric+\frac{1}{2}\mathcal{L}_{X}g=\frac{\lambda}{2}g,$

where $\lambda$ is a real number. If $\lambda>0$, $\lambda=0$, or $\lambda<0$,
then the Ricci soliton is called shrinking, steady, or expanding,
respectively. Furthermore, after scaling, the constant $\lambda$ can always be
normalized to $1$, $0$, and $-1$ in the shrinking, steady, and expanding
cases, respectively (which we shall always do throughout this paper). Finally,
if the vector field $X$ is integrable, that is, if there exists a smooth
function $f$ such that $X=\nabla f$, then the Ricci soliton is called
gradient, and the function $f$ is called the potential function.

Ricci solitons generate self-similar solutions to the Ricci flow, called the
canonical forms. Let the positive function $\tau(t)$, the one-parameter family
of self-diffeomorphisms $\phi_{t}:M\rightarrow M$, and the evolving metric
$g(t)$ be defined as

$\displaystyle\tau(t)$ $\displaystyle=$ $\displaystyle 1+\lambda t,$
$\displaystyle\frac{\partial}{\partial t}\phi_{t}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\tau(t)}X(\phi_{t}(x)),$ $\displaystyle g(t)$
$\displaystyle=$ $\displaystyle\tau(t)\phi_{t}^{*}g,$

then $g(t)$ moves by the Ricci flow equation.

_Remark:_ In the construction of the canonical form, the completeness of the
vector field $X$ is necessary. By the completeness of a vector field we mean
that every one of its flow curves exists for all time. Zhu-Hong Zhang [22]
proved the completeness of the vector $\nabla f$ for a gradient Ricci soliton
$(M,g,f)$. In the general non-gradient case, the completeness of $X$ is not
yet known to the best of our knowledge, though a Ricci soliton without a
canonical form is inconceivable. This is an assumption we make in our paper.

If we regard the Ricci flows as orbits in the space

$\displaystyle\text{Met}(M)\Big{/}\text{Diff},$

where $\text{Met}(M)$ stands for the space of all complete smooth Riemannian
metrics on $M$, and Diff stands for the group of all self-diffeomorphisms and
scalings, then the breathers are periodic orbits and the Ricci solitons are
static orbits. Therefore, the no breather theorem is tantamount to saying that
the periodic orbits must also be static.

Perelman’s proofs of no breather theorems, being applications of the
$\mathcal{F}$ functional and the $\mathcal{W}$ functional, elegant as they
are, rely on the existence of minimizers of these functionals, which is not
always true in the general noncompact case, as indicated in [20] by a
counterexample. Qi.S Zhang [19] also showed that the minimizer of
$\mathcal{W}$ exists in the noncompact case assuming certain conditions at
infinity, and thereby proved a Perelman-type no shrinking breather theorem for
asymptotically flat manifolds with positive scalar curvature.

Lu and Zheng [13] came up with the idea of constructing an ancient solution
with a given shrinking breather, and presented a new proof of Perelman’s no
shrinking breather theorem in the compact case. Their method was also capable
of dealing with noncompact case under certain additional assumptions. In [21],
the second author applied the method of Lu and Zheng in combination with a
reduced distance estimate, and extended Perelman’s no shrinking breather
theorem to the noncompact case assuming only bounded sectional curvature.

In this present work, we further extend this Perelman-type no shrinking
breather theorem to almost the fullest generality, that is, we assume only a
lower bound for the Ricci curvature.

######  Theorem 1.3.

A complete shrinking breather with Ricci curvature bounded from below must be
a shrinking gradient Ricci soliton.

Let us then discuss some applications of Theorem 1.3. First of all, we recall
the well-known result that 3-dimensional shrinking gradient Ricci solitons are
classified as: $\mathbb{R}^{3}$, a quotient of $\mathbb{S}^{3}$, and a
quotient of $\mathbb{S}^{2}\times\mathbb{R}$; see [15], [12] and [5]. On the
other hand, since one can always construct an ancient solution with a
shrinking breather, it follows from Chen [6] that a $3$-dimensional shrinking
breather always has nonnegative sectional curvature; see Lemma 3.2 and
Corollary 3.3. Taking all these facts into account, we obtain the following
classification of 3-dimensional complete shrinking breathers.

######  Corollary 1.4.

Any $3$-dimensional complete shrinking breather must be a gradient shrinking
Ricci soliton, and hence is isometric to $\mathbb{R}^{3}$, a quotient of
$\mathbb{S}^{3}$, or a quotient of $\mathbb{S}^{2}\times\mathbb{R}$.

Furthermore, by using a result of Munteanu and Wang, we can also derive the
following corollary on shrinking breathers with nonnegative sectional
curvature; see Corollary 3 in [10].

######  Corollary 1.5.

A complete shrinking breather with nonnegative sectional curvature can be
written as a product $\mathbb{R}^{n-k}\times N^{k}$, where $N$ is a compact
shrinking gradient Ricci soliton with nonnegative sectional curvature and
positive Ricci curvature.

Recall that the canonical form of a (not necessarily gradient) Ricci soliton
is self-similar, and hence a breather as well. Combining this observation with
Theorem 1.3, we obtain the following corollary.

######  Corollary 1.6.

A complete shrinking Ricci soliton with Ricci curvature bounded from below
must be gradient; here we also assume that the vector field $X$ in Definition
1.2 is complete.

This is a generalization of Naber [11], where he proved that a shrinking Ricci
soliton with bounded sectional curvature must also be gradient.

Besides the applications presented above, we hereby remark that the method we
developed might be useful in the study of general ancient solutions. Perelman
[15] first discovered the asymptotic shrinking gradient Ricci soliton for
$\kappa$-noncollapsed ancient solutions with bounded and nonnegative curvature
operator. Naber [11] proved the same result with curvature operator
nonnegativity replaced by the Type I curvature bound. Recall that an ancient
solution to the Ricci flow $(M,g(\tau))_{\tau\in[0,\infty)}$, where $\tau$ is
the backward time, is called Type I, if its curvature satisfies the following
bound

$\displaystyle|Rm_{g(\tau)}|\leq\frac{C}{\tau},$

for all $\tau\in(0,\infty)$, where $C$ is a constant independent of $\tau$. At
this point, we do not know many cases in which the asymptotic shrinking
gradient Ricci soliton exists. In the smooth category, there are two major
cases: the Type I case and the case of “curvature positivity”. The “curvature
positivity” condition can be as weak as PIC-2, in which case the existence of
asymptotic shrinking gradient Ricci soliton is guaranteed by an extension of
Hamilton’s trace Harnack proved by Brendle [4]; this, of course, covers
Perelman’s original bounded and nonnegative curvature operator case. Indeed,
in both of the two cases mentioned above, Hamilton’s trace Harnack is the
crucial element—one may either estimate it using the Type I curvature bound,
or directly use its positivity in the case of “curvature positivity”. As to
the non-smooth category, Bamler’s recent ground-breaking work shows that every
singularity model, being a metric flow, always has an asymptotic metric
soliton (which he calls tangent flow at infinity); see [1]—[3].

In contradistinction from the former conditions on the global geometry, we
consider the condition of the existence of asymptotic shrinking gradient Ricci
soliton from another perspective—what is the requirement on the local geometry
to effectuate the existence as well as the soliton structure of the blow-down
limit? As natural as it appears, we have defined a condition called locally
uniformly Type I; see Definition 4.1. This condition means that the ancient
solution, along a sequence of space-time points with time going to negative
infinity, has locally uniform curvature bound, injectivity radii lower bound,
as well as reduced distance bound, and all these bounds are in a “Type I
fashion”, plus a technical assumption—time-wise Ricci curvature lower bound.
This condition is sufficient to render certain the existence of an asymptotic
shrinking gradient Ricci soliton, and may arguably be a necessary condition
also. Indeed, in all the known smooth cases listed above—Type I and “curvature
positivity”—the locally uniformly Type I condition is satisfied.

Other than the existence of the asymptotic shrinking gradient Ricci soliton,
we also discover that a locally uniformly Type I ancient solution is
$\kappa$-noncollapsed on all scales, where $\kappa$ depends on the Gaussian
density of the asymptotic shrinking gradient Ricci soliton. This observation,
largely owing to the work of Yokota [18], is surprising, since we only assume
the noncollapsedness along a sequence of space-time points on a specific
sequence of scales, yet we conclude that such an ancient solution is
noncollapsed everywhere on all scales. This idea was first used by Perelman
[15] to prove that there exists a universal $\kappa$ for 3-dimensional
$\kappa$-solutions.

The key estimates which we obtain are local uniform estimates for the reduced
distance as well as its gradient. These estimates do not follow from a direct
generalization of the methods in [15] and [11]. In particular, we do not have
the following inequality

$\displaystyle|\nabla l|^{2}+R\leq\frac{Cl}{\tau},$

which is a consequence of analyzing Hamilton’s trace Harnack. The idea of our
estimates, is to use local curvature bound to control the reduced distance and
its gradient. These estimates, though very coarse, yet are sufficient, in
combination with the monotonicity of the reduced volume, to construct the
soliton structure on the limit space of the blow-down sequence. Furthermore,
unlike the aforementioned cases, we are not able to prove that the reduced
volume converges to the Gaussian density of the asymptotic shrinking gradient
Ricci soliton; the latter is less than or equal to the limit of the former by
Fatou’s lemma. This results from the lack of the estimate

$\displaystyle l(x,\tau)\sim\frac{1}{\tau}d_{g(\tau)}^{2}(x,p_{\tau}),$

where $p_{\tau}$ is a moving point such that $l(p_{\tau},\tau)$ is bounded
independent of $\tau$. This estimate is valid in all two cases mentioned
above. This no loss of reduced volume property is not necessary for our
application, yet we doubt whether it actually could happen.

To conclude the introduction section, we remark that in an upcoming work of
the second author with Bamler, Chow, Deng, and Ma [23], it is proved that the
canonical form of a noncollapsed steady gradient Ricci soliton with
nonnegative Ricci curvature always admits an asymptotic shrinking gradient
Ricci soliton.

The present paper is organized as follows. In section 2, we review some well
known results about Perelman’s $\mathcal{L}$-geometry. In section 3, we
construct an ancient solution starting with a given shrinking breather, and
subsequently estimate the reduced distance along a sequence of space-time
points. In section 4, we define the notion of locally uniformly Type I ancient
solution. In section 5, we derive the local uniform estimates for the reduced
distance as well as its gradient. In section 6, we prove the existence of
asymptotic shrinking gradient Ricci soliton for locally uniformly Type I
ancient solutions. In section 7, we discuss the possible loss of reduced
volume and the noncollapsedness of locally uniformly Type I ancient solutions.

_Acknowledgement_ : The authors would like to thank Professor Japing Wang for
bringing this problem to us and many useful discussions. The major work of
this paper was accomplished while the first author was visiting University of
Minnesota, Twin Cities.

##  2\. Preliminaries on Perelman’s $\mathcal{L}$-geometry

In this section, we collect some basic properties of Perelman’s reduced
distance and reduced volume. These properties are well-known, and the reader
may easily look them up in the literature. Let
$(M^{n},g(\tau))_{\tau\in[0,T]}$ be a solution to the backward Ricci flow

(2.1) $\displaystyle\frac{\partial g}{\partial\tau}=2Ric_{g(\tau)},$

where $\tau$ stands for the backward time. Throughout the whole paper, unless
otherwise indicated, we always assume that the (backward) Ricci flows in
question admits a lower bound for the Ricci curvature at each time slice.

Perelman’s $\mathcal{L}$-energy for a piecewise $C^{1}$ curve
$\gamma(s):[0,\tau]\to M$, where $\tau\in(0,T]$, is defined as

(2.2)
$\displaystyle\mathcal{L}(\gamma(s))=\int^{\tau}_{0}\sqrt{s}\big{(}R_{g(s)}(\gamma(s))+|\gamma^{\prime}(s)|_{g(s)}^{2}\big{)}ds.$

One may view $\gamma$ as a curve in the Ricci flow space-time connecting
$(\gamma(0),0)$ and $(\gamma(\tau),\tau)\in M\times[0,T]$, satisfying
$(\gamma(s),s)\in M\times\\{s\\}$ for all $s\in[0,\tau]$.

Let $p\in M$ be a fixed base point. For any $(q,\tau)\in M\times(0,T]$, we
define

$L(q,\tau)=\inf\limits_{\gamma}\mathcal{L}(\gamma(s)),$

where the infimum is taken over all piecewise $C^{1}$ curves
$\gamma(s):[0,\tau]\to M$ satisfying $\gamma(0)=p$ and $\gamma(\tau)=q$. The
minimizer is called $\mathcal{L}$-geodesic. By the first variation, the
$\mathcal{L}$-geodesic equation is

(2.3)
$\displaystyle\nabla_{\gamma^{\prime}}\gamma^{\prime}(\tau)-\frac{1}{2}\nabla
R(\gamma(\tau),\tau)+\frac{1}{2\tau}\gamma^{\prime}(\tau)+2Ric(\gamma^{\prime}(\tau))=0.$

Perelman’s $l$-function, also known as the reduced distance function, is
defined as

(2.4) $\displaystyle l(q,\tau)=\frac{L(q,\tau)}{2\sqrt{\tau}},$

and the reduced volume is definded as

(2.5)
$\displaystyle\mathcal{V}(\tau)=\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-l(\cdot,\tau)}dg(\tau).$

We remark here that both (2.4) and (2.5) depend on the choice of $p$, and the
point $(p,0)\in M\times[0,T]$ is called the _base point_ of the functions $l$
and $\mathcal{V}$. When the base point is understood, we often omit it in the
notation. Otherwise the base point is included in the notation as the
subindex:

$\displaystyle l_{(p,0)}\ \ ,\ \mathcal{V}_{(p,0)}.$

For $v\in T_{p}M,$ let $\gamma_{v}$ denote the $\mathcal{L}$-geodesic, that
is, a solution to (2.3), satisfying $\lim\limits_{s\to
0}\sqrt{s}\gamma^{\prime}(s)=v$. If $\gamma_{v}$ exists on $[0,\tau]$, then
the $\mathcal{L}$-exponential map $\mathcal{L}\text{exp}^{\tau}_{p}:T_{p}M\to
M$ at time $\tau$ is defined as

$\mathcal{L}\text{exp}^{\tau}_{p}(v)=\gamma_{v}(\tau).$

Let $U(\tau)\subset T_{p}M\cong\mathbb{R}^{n}$ denote the maximal domain of
$\mathcal{L}\text{exp}^{\tau}_{p}$. By applying basic ODE theory to the
$\mathcal{L}$-geodesic equation—a linear ODE depending only on the geometry
quantities near the $\mathcal{L}$-geodesic—one may obtain that $U(\tau)$ is an
open set and that $\mathcal{L}\text{exp}^{\tau}_{p}$ is a smooth map from
$U(\tau)$ to $M$. The injectivity domain at time $\tau$ is defined as

$\Omega^{T_{p}M}(\tau)=\Big{\\{}v\in U(\tau)\ \Big{|}\
\gamma_{v}|_{[0,\tau]}:[0,\tau]\to M\text{ is the unique minimal
}\mathcal{L}-\text{geodesic}\text{\ from $p$ to $\gamma_{v}(\tau)$
};\text{$\gamma_{v}(\tau)$ is not conjugate to }p\text{ along\
}\gamma_{v}.\Big{\\}}.$

Correspondingly we also define

$\Omega(\tau)=\Big{\\{}q\in M\ \Big{|}\ \text{There is a unique minimal
}\mathcal{L}-\text{geodesic }\gamma:[0,\tau]\to M\text{\ with\
}\gamma(0)=p,\gamma(\tau)=q;q\text{ is not conjugate to }p\text{ along\
}\gamma.\Big{\\}}.$

It is well known that

$\Omega(\tau)=\mathcal{L}\text{exp}^{\tau}_{p}\big{(}\Omega^{T_{p}M}(\tau)\big{)}.$

The cut-locus is defined as

$C(\tau)=M\backslash\Omega(\tau).$

In Perelman’s study [15] of the reduced geometry, a general assumption is
bounded sectional curvature. However, Ye [17] studied the properties of the
$l$-function and the reduced volume assuming only a lower bound for the Ricci
curvature. We now state these useful results below.

######  Theorem 2.1 (Proposition 2.7, Proposition 2.11, and Lemma 2.14 in
[17]).

Let $(M^{n},g(\tau))_{\tau\in[0,T]}$ be a backward Ricci flow such that the
Ricci curvature of each time-slice is bounded from below. Then the following
hold:

  1. (1)

For any $(q,\tau)\in M\times(0,T]$, there exists a minimal
$\mathcal{L}$-geodesic connecting $(p,0)$ and $(q,\tau)$, that is,
$\mathcal{L}\text{exp}^{\tau}_{p}$ is onto.

  2. (2)

$L$ is locally Lipschitz in space-time.

  3. (3)

For each $\tau\in(0,T)$, $C(\tau)\subset M$ is a closed set of zero Remannian
measure. Consequently $\cup_{0<\tau<T}C(\tau)\times\\{\tau\\}$ is a closed set
of zero measure in $M\times(0,T)$.

Furthermore, we need the following analytic properties of the $l$-function.
They were first discovered by Perelman [15] and vindicated by Ye [17] under
the assumption of Ricci curvature lower bound.

######  Theorem 2.2 ([15], see also Lemma 2.19 and Theorem 2.20 in [17]).

Let $(M^{n},g(\tau))_{\tau\in[0,T]}$ be a backward Ricci flow such that the
Ricci curvature of each time-slice is bounded from below. Let $l$ be the
reduced distance function defined in (2.4). Then on
$\cup_{0<\tau<T}\Omega(\tau)\times\\{\tau\\}$ it holds that:

(2.6) $\displaystyle 2\frac{\partial l}{\partial\tau}+|\nabla
l|^{2}-R+\frac{l}{\tau}=0,$ (2.7)
$\displaystyle\frac{\partial}{\partial\tau}l-\Delta l+|\nabla
l|^{2}-R+\frac{n}{2\tau}\geq 0,$ (2.8) $\displaystyle 2\Delta l-|\nabla
l|^{2}+R+\frac{l-n}{\tau}\leq 0.$

Furthermore, (2.7) and (2.8) both hold on $M\times(0,T)$ in the sense of
distribution. That is to say, for any $0<\tau_{1}<\tau_{2}<T$ and for any
nonnegative Lipshcitz function $\phi$ compactly supported on
$M\times[\tau_{1},\tau_{2}]$, it holds that

(2.9) $\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\Bigg{(}\nabla
l\cdot\nabla\phi+\Big{(}\frac{\partial}{\partial\tau}l+|\nabla
l|^{2}-R+\frac{n}{2\tau}\Big{)}\phi\Bigg{)}dg(\tau)d\tau\geq 0,$

and, for any $\tau\in(0,T)$ and any nonnegative Lipshcitz function $\phi$
compactly supported on $M$, it holds that

(2.10) $\displaystyle\int_{M}\Bigg{(}-2\nabla l\cdot\nabla\phi+\Big{(}-|\nabla
l|^{2}+R+\frac{l-n}{\tau}\Big{)}\phi\Bigg{)}dg(\tau)\leq 0.$

Formulas (2.9) and (2.10) will be especially important in our future
arguments.

The following is a useful consequence of the first variation formula.

######  Lemma 2.3 (Perelman [15]).

Let $\gamma$ be an $\mathcal{L}$-geodesic starting from $(p,0)$. Then, so long
as $\tau\in(0,T)$ and $\gamma(\tau)\in\Omega(\tau)$, it holds that

(2.11) $\displaystyle\nabla l(\gamma(\tau),\tau)=\gamma^{\prime}(\tau).$

The following monotonicity formula is also a well-known result of Perelman.

######  Theorem 2.4 (Perelman[15], see also Theorem 4.3 and Theorem 4.5 in
[17]).

Let $(M^{n},g(\tau)_{\tau\in[0,T]})$ be a backward Ricci flow such that the
Ricci curvature of each time-slice is bounded from below. Then

  1. (1)

$\mathcal{V}(\tau)\leq 1$ for all $\tau\in(0,T]$,

  2. (2)

$\mathcal{V}(\tau)$ is non-increasing in $\tau$,

where $\mathcal{V}(\tau)$ is the reduced volume defined in (2.5).

##  3\. Construction of the Ancient Solution

In this section, we will use a given shrinking breather to construct an
ancient solution. This method was first applied by Lu and Zheng [13], and
later adopted by the second author [21]. The content of this section is not
essentially different from [13] and [21], and we are including it for the
convenience of the reader. The idea of the construction, so to speak, is to
take infinitely many copies of the given shrinking breather, each scaled by a
proper factor, and then splice them together head-to-tail. The crucial
estimate arising from this construction is this, that if we fix a base point
on the given shrinking breather, then the reduced distance evaluated at the
base point on each copy is uniformly bounded. In other cases, this fact is
also essential to the existence of the asymptotic shrinking gradient Ricci
soliton; see [15] and [11]. This estimate is the first step to the locally
uniform estimates for both the $l$-function and its gradient, which we will
present in section 5.

After rescaling and translating in time, we may assume that the shrinking
breather is a backward Ricci flow $(M^{n},g_{0}(\tau))_{\tau\in[0,1]}$, such
that there exists $\alpha\in(0,1)$ and a diffeomorphism $\phi:M\to M$
satisfying

(3.1) $\alpha g_{0}(1)=\phi^{*}g_{0}(0).$

We first of all observe

######  Lemma 3.1.

$(M,g_{0}(\tau))_{\tau\in[0,1]}$ is smooth on $[0,1]$. In particular,
$g_{0}(\tau)$ is smooth up to the boundary of the interval $[0,1]$.

###### Proof.

According to the localized Shi’s estimates [16], we have that $g_{0}(0)$ must
be a smooth Riemannian metric. Since $(M,g_{0}(0))$ and $(M,g_{0}(1))$ are
isometric, we have that $g_{0}(1)$ is also smooth. Then by the Appendix in
[14], we have that $g_{0}(\tau)$ is smooth up to $\tau=1$. ∎

Let

(3.2) $\tau_{i}=\sum^{i}_{k=0}\alpha^{-k},$

where we conventionally let $\tau_{0}=1$. Then there exists $C_{0}>0$
depending only on $\alpha$ such that

(3.3) $\alpha^{-i}\leq\tau_{i}\leq C_{0}\alpha^{-i},$

for each $i\geq 0$. Next, we define

$g_{i}(\tau)=\alpha^{-i}(\phi^{i})^{*}g_{0}(\alpha^{i}(\tau-\tau_{i-1})),\tau\in[\tau_{i-1},\tau_{i}].$

Splicing these flows together, we define the evolving metric as

(3.4) $g(\tau)=\left\\{\begin{array}[]{ll}g_{0}(\tau),&\tau\in[0,\tau_{0}],\\\
g_{i}(\tau),&\tau\in[\tau_{i-1},\tau_{i}].\end{array}\right.$

We need to verify that $g(\tau)$ is a smooth Ricci flow. When $g_{0}(\tau)$
has bounded curvature, this can be easily done by applying the forward and
backward uniqueness of the Ricci flow at time $\tau_{i}$; see [13] and [21].
In our case, we verify the smoothness of $g(\tau)$ by direct computation.

######  Lemma 3.2.

$g(\tau)$ defined in (3.4) is a smooth ancient solution to the Ricci flow on
the manifold $M$.

###### Proof.

By Lemma 3.1, we need only to check the smoothness of $g(\tau)$ at each
$\tau_{i}$. Applying (3.1), we get

$\displaystyle
g_{i}(\tau_{i-1})=\alpha^{-i}(\phi^{i})^{*}g_{0}(0)=\alpha^{-(i-1)}(\phi^{i-1})^{*}g_{0}(1)$
$\displaystyle=$
$\displaystyle\alpha^{-(i-1)}(\phi^{i-1})^{*}g_{0}(\alpha^{i-1}(\tau_{i-1}-\tau_{i-2}))=g_{i-1}(\tau_{i-1}).$

Since by (3.1) and (3.4) we have

$\displaystyle\frac{\partial_{+}}{\partial\tau}g_{i}(\tau_{i-1})=(\phi^{i})^{*}\frac{\partial_{+}}{\partial\tau}g_{0}(0)=(\phi^{i})^{*}Rc(g_{0}(0))$

and

$\displaystyle\frac{\partial_{-}}{\partial\tau}g_{i-1}(\tau_{i-1})=(\phi^{i-1})^{*}\frac{\partial_{-}}{\partial\tau}g_{0}(1)=(\phi^{i-1})^{*}Rc(g_{0}(1))=(\phi^{i})^{*}Rc(g_{0}(0)),$

where $\displaystyle\frac{\partial_{+}}{\partial\tau}$ and
$\displaystyle\frac{\partial_{-}}{\partial\tau}$ stand for the right and the
left derivatives, respectively, it then follows that

$\frac{\partial_{-}}{\partial\tau}g_{i-1}(\tau_{i-1})=\frac{\partial_{+}}{\partial\tau}g_{i-1}(\tau_{i-1}).$

Moreover, because of the smoothness of $g_{0}(\tau)$ up to $\tau=0$ and
$\tau=1$, from the following calculations

$\displaystyle\frac{\partial_{+}^{2}}{\partial\tau^{2}}g_{i}(\tau_{i-1})=\alpha^{i}(\phi^{i})^{*}(\frac{\partial_{+}}{\partial\tau}Rc)(g_{0}(0)),$
$\displaystyle\frac{\partial_{-}^{2}}{\partial\tau^{2}}g_{i-1}(\tau_{i-1})=\alpha^{i-1}(\phi^{i-1})^{*}(\frac{\partial_{-}}{\partial\tau}Rc)(g_{0}(1)),$

and

$\displaystyle(\frac{\partial_{-}}{\partial\tau}Rc)(g_{0}(1))=(\Delta_{L}Rc)(g_{0}(1))=\alpha\phi^{*}((\Delta_{L}Rc)(g_{0}(0)))=\alpha\phi^{*}((\frac{\partial_{+}}{\partial\tau}Rc)(g_{0}(0))),$

we conclude that

$\displaystyle\frac{\partial_{+}^{2}}{\partial\tau^{2}}g_{i}(\tau_{i-1})=\frac{\partial_{-}^{2}}{\partial\tau^{2}}g_{i-1}(\tau_{i-1}).$

It is straightforward to prove
$\frac{\partial_{+}^{k}}{\partial\tau^{k}}g_{i}(\tau_{i-1})=\frac{\partial_{-}^{k}}{\partial\tau^{k}}g_{i-1}(\tau_{i-1})$
for $k\geq 3$ in the similar way. ∎

Chen [6] showed that any ancient solution to the Ricci flow has nonnegative
scalar curvature, and any 3-dimensional ancient solution has nonnegative
sectional curvature. As a direct application of Lemma 3.2, we have the
following corollary.

######  Corollary 3.3.

Any complete shrinking breather has nonnegative scalar curvature. In
particular, any 3-dimensional complete shrinking breather has nonnegative
sectional curvature.

Let us choose a sequence of points on this ancient flow, and estimate the
$l$-function at these points. Indeed, these points will serve as the base
points of the Cheeger-Gromov-Hamilton convergence [8]. Fix an arbitrary point
$p_{0}\in M^{n}$, and for each $i\geq 0$ define

(3.5) $x_{i}=\phi^{-{(i+1)}}(p_{0}).$

Let $\sigma:[0,1]\to M^{n}$ be a smooth curve such that $\sigma(0)=p_{0}$ and
$\sigma(1)=x_{0}$. We define

(3.6)
$\sigma_{i}(\tau)=\phi^{-{(i+1)}}\circ\sigma\big{(}\alpha^{i+1}(\tau-\tau_{i})\big{)},\tau\in[\tau_{i},\tau_{i+1}],$

and $\gamma_{i}:[0,\tau_{i+1}]\to M$ as

(3.7)
$\gamma_{i}(\tau)=\left\\{\begin{array}[]{ll}\sigma(\tau),&\tau\in[0,1],\\\
\sigma_{j}(\tau),&\tau\in[\tau_{j},\tau_{j+1}].\end{array}\right.$

Since

$\sigma_{i}(\tau_{i})=\phi^{-(i+1)}\circ\sigma(0)=\phi^{-i}\circ\sigma(1)=\phi^{-i}\circ\sigma(\alpha^{i}(\tau_{i}-\tau_{i-1}))=\sigma_{i-1}(\tau_{i}),$

we have that $\gamma_{i}(\tau)$ defined in (3.7) is a continuous and piecewise
smooth curve with $\gamma_{i}(0)=p_{0}$ and $\gamma_{i}(\tau_{i})=x_{i+1}$.
Since for $\tau\in[\tau_{j},\tau_{j+1}]$

$\displaystyle R(\sigma_{j}(\tau),\tau)=R_{g_{j+1}(\tau)}(\sigma_{j}(\tau))$
$\displaystyle=$ $\displaystyle
R_{\alpha^{-(j+1)}(\phi^{j+1})^{*}g_{0}(\alpha^{j+1}(\tau-\tau_{j}))}\Big{(}\phi^{-(j+1)}\circ\sigma\big{(}\alpha^{j+1}(\tau-\tau_{j})\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\alpha^{j+1}R_{g_{0}(\alpha^{j+1}(\tau-\tau_{j}))}(\sigma(\alpha^{j+1}(\tau-\tau_{j})))\leq
C_{1}\alpha^{j+1}$ $\displaystyle\leq$
$\displaystyle\tau_{j+1}\alpha^{j+1}\frac{C_{1}}{\tau}\leq\frac{B}{\tau},$

where $\displaystyle
C_{1}=\max\limits_{\tau\in[0,1]}R_{g_{0}(\tau)}(\sigma(\tau))$. We can then
compute the $\mathcal{L}$-energy for $\gamma(\tau)$ defined in (3.7)

$\displaystyle\mathcal{L}(\gamma_{i})$
$\displaystyle=\mathcal{L}(\sigma)+\sum\limits_{j=1}^{i}\int\limits_{\tau_{j}}^{\tau_{j+1}}\sqrt{\tau}(R(\sigma_{j}(\tau),\tau)+|\sigma_{j}^{\prime}(\tau)|^{2}_{g(\tau)})d\tau$
$\displaystyle\leq
D+\sum\limits_{j=1}^{i}\int\limits_{\tau_{j}}^{\tau_{j+1}}\sqrt{\tau}(\frac{B}{\tau}+A\alpha^{j+1})d\tau$
$\displaystyle\leq D+C\sum\limits_{j=1}^{i}\alpha^{-\frac{j+1}{2}},$

where $A=\max\limits_{\tau\in[0,1]}|\sigma^{\prime}(\tau)|^{2}_{g_{0}(\tau)}$.
Hence

$\displaystyle l(x_{i+1},\tau_{i+1})$
$\displaystyle\leq\frac{\mathcal{L}(\gamma_{i})}{2\sqrt{\tau_{i+1}}}$
$\displaystyle\leq\frac{1}{2}D\alpha^{\frac{j+1}{2}}+\frac{1}{2}C\sum\limits_{j=1}^{i}\alpha^{\frac{j+1}{2}}\leq
C_{2},$

where $C_{2}$ is a constant independent of $i$.

##  4\. Locally Uniformly Type I Ancient Solutions

In this section, we will consider a certain type of ancient solutions to the
Ricci flow of which (3.4) is a special case. As in [15] and [11], we will
eventually consider the rescaled sequence of backward Ricci flows

(4.1) $\displaystyle(M,\tau_{i}^{-1}g(\tau_{i}\tau),(x_{i},1))\text{\ \ for\ \
}\tau\in[0,\infty),$

where $\tau_{i}$ is as defined in (3.2) and $g(\tau)$ as defined in (3.4). It
turns out that, around the space-time base points $(x_{i},1)$, every member of
this sequence differs the original shrinking breather only by an isometry and
a bounded constant. More specifically, (3.4) implies that for
$\tau\in[1,\frac{\tau_{i+1}}{\tau_{i}}]$, it holds that

$\tau_{i}^{-1}g(\tau_{i}\tau)=\tau_{i}^{-1}\alpha^{-(i+1)}(\phi^{i+1})^{*}g_{0}(\alpha^{i+1}(\tau_{i}\tau-\tau_{i})),$

where the scaling factors $\tau_{i}^{-1}\alpha^{-(i+1)}$ are uniformly bounded
from above and below by constants independent of $i$; indeed, this factor
converges to $1$. This means that around the base points $(x_{i},1)$, the
scaled metrics $\tau_{i}^{-1}g(\tau_{i}\tau)$ have locally uniformly bounded
geometry: the curvature norms are bounded in terms of distance from $x_{i}$
and the injectivity radii at $(x_{i},1)$ are bounded independent of $i$. This
is model on which the following definition is based.

######  Definition 4.1.

(Locally uniformly Type I ancient solution.) Let
$(M,g(\tau))_{\tau\in[0,\infty)}$ be an ancient solution to the Ricci flow,
where $\tau$ is the backward time. Fix $p_{0}\in M$ and let
$\\{(x_{i},\tau_{i})\\}_{i=1}^{\infty}\subset M\times(0,\infty)$ be a sequence
of space-time points with $\tau_{i}\nearrow\infty$. $(M,g(\tau))$ is called
_locally uniformly Type I_ along the space-time sequence
$\\{(x_{i},\tau_{i})\\}_{i=1}^{\infty}$, if the following hold.

  1. (1)

(4.2) $\displaystyle\limsup_{i\rightarrow\infty}l(x_{i},\tau_{i})<\infty,$

where $l$ is the reduced distance function based at $(p_{0},0)$.

  2. (2)

Around the space-time points $(x_{i},\tau_{i})$, the curvature is locally
uniformly Type I. More precisely, there exists a positive function
$C:(0,\infty)\rightarrow(0,\infty)$ with the following property: for all
$A>0$, there exists $i_{0}\in\mathbb{N}$, depending on $A$, such that

(4.3)
$\displaystyle\sup_{B_{g(\tau_{i})}(x_{i},\sqrt{\tau_{i}}r)\times[\tau_{i},2\tau_{i}]}|Rm_{g(\tau)}|\leq\frac{C(r)}{\tau_{i}}\text{
for all }i\geq i_{0}\text{ and }r\leq A.$

  3. (3)

There is time-wise Ricci curvature lower bound for $g(\tau)$. In other words,
there exists a positive function $K:[0,\infty)\rightarrow(0,\infty)$, such
that

(4.4) $\displaystyle Ric_{g(\tau)}\geq-K(\tau)g(\tau)$

for all $\tau\in[0,\infty)$.

  4. (4)

$g(\tau)$ is noncollapsed along $(x_{i},\tau_{i})$. In other words,

(4.5)
$\displaystyle\liminf_{i\rightarrow\infty}\Big{(}(\tau_{i})^{-\frac{1}{2}}\text{inj}(g(\tau_{i}),x_{i})\Big{)}>0,$

where $\text{inj}(g,x)$ stands for the injectivity radius of the Riemannian
metric $g$ at $x$.

According to the argument at the beginning of this section, we have that an
ancient solution constructed with a shrinking breather with Ricci curvature
bounded from below must be locally uniformly Type I.

######  Proposition 4.2.

Let $(M,g_{0}(\tau))_{\tau\in[0,1]}$ be a shrinking breather as defined in
(3.1) with Ricci curvature bounded from below, and
$(M,g(\tau))_{\tau\in[0,\infty)}$ the ancient solution as defined in (3.4).
Let $p_{0}\in M$ be a fixed point, $x_{i}=\phi^{-{(i+1)}}(p_{0})$, and
$\displaystyle\tau_{i}=\sum^{i}_{k=0}\alpha^{-k}$. Then,
$(M,g(\tau))_{\tau\in[0,\infty)}$ is a locally uniformly Type I ancient
solution along $\\{(x_{i},\tau_{i})\\}_{i=1}^{\infty}$.

Furthermore, the scaled flows
$\displaystyle\big{(}M,\tau_{i}^{-1}g(\tau\tau_{i}),(x_{i},1)\big{)}_{\tau\in[1,\frac{\tau_{i+1}}{\tau_{i}}]}$
differ the original shrinking breather
$\displaystyle\big{(}M,g_{0}(\tau-1),(p_{0},1)\big{)}_{\tau\in[1,2]}$ only by
a scaling factor $\tau_{i}^{-1}\alpha^{-(i+1)}$ and a base-point-preserving
diffeomorphism $\phi^{-(i+1)}$.

_Remark_ : Though the ratios $\displaystyle\frac{\tau_{i+1}}{\tau_{i}}$ are
not necessarily equal to $2$, and item (2) of Definition 4.1 requires locally
uniformly bounded curvature on interval $[\tau_{i},2\tau_{i}]$, yet one may
verify from (3.4) that in the special case of an ancient solution constructed
from a shrinking breather, (4.3) is indeed satisfied. Or one may simply take
$\alpha$ to be, say, $\frac{1}{4}$, for simplicity.

##  5\. Estimates on the $l$-distance

In this section and the next, we proceed to prove that a locally uniformly
Type I ancient solution always has an asymptotic shrinking gradient Ricci
soliton. According to Perelman’s [15] idea of asymptotic shrinking gradient
Ricci soliton, we aim at showing that the scaled sequence of backward Ricci
flows,

$\displaystyle\Big{\\{}\big{(}M,g_{i}(\tau),x_{i},l_{i}\big{)}_{\tau\in[1,2]}\Big{\\}}_{i=1}^{\infty},$

where $g_{i}(\tau)=\tau_{i}^{-1}g(\tau\tau_{i})$,
$l_{i}(\tau)=l(\tau\tau_{i})$, and all other notations are as in the
definition in previous section, after passing to a subsequence, converges to a
shrinking gradient Ricci soliton. To this end, we need not only the locally
uniform geometric bound for the scaled sequence (4.1), but also the locally
uniform bound for the $l$-function and its gradient, as in [15] and [11]; this
is the goal of this section.

In our case, we have neither nonnegative curvature operator nor type-I bound.
These two assumptions are made by Perelman and Naber to deal with Hamilton’s
trace Harnack, respectively. Nevertheless, the bound of $l$-function at
$(x_{i},\tau_{i})$ and the locally uniform Type I curvature bound turn out to
be sufficient. For the locally uniform estimate on $l$, the idea of triangle
inequality is very useful: one may use as a test curve the
$\mathcal{L}$-geodesic from the base point to $(x_{i},\tau_{i})$ concatenated
with another curve from $(x_{i},\tau_{i})$ to the target point, where the
bound of the former is a consequence of section 3, and the latter can be
estimated using the local bound on curvature. For the gradient estimate of
$l$, we use the following idea: Let $(x,\tau)$ be a point not too far away
from $(x_{i},\tau_{i})$, and let $\gamma$ be the minimal
$\mathcal{L}$-geodesic from $(p_{0},0)$ to $(x,\tau)$. If $\gamma$ runs too
fast as it approaches $(x,\tau)$, then the latter must be quite far away from
$(x_{i},\tau_{i})$, contradicting our assumption on $(x,\tau)$. In combination
with (2.3) we obtain the gradient estimate.

Let $\displaystyle(M,g(\tau))_{\tau\in[0,\infty)}$, $p_{0}\in M$, and
$\displaystyle\\{(x_{i},\tau_{i})\\}_{i=1}^{\infty}\subset M\times(0,\infty)$
be as in the Definition 4.1. Let us consider the following scaled backward
Ricci flows

(5.1)
$\displaystyle\Big{(}M,g_{i}(\tau),(x_{i},1)\Big{)}_{\tau\in[0,\infty)},$

where $g_{i}(\tau)=\tau_{i}^{-1}g(\tau\tau_{i})$. Then, let

(5.2) $\displaystyle l_{i}(\tau):=l(\tau\tau_{i})\text{ for
}\tau\in(0,\infty)$

where $l$ is the reduced distance from the base point $(p_{0},0)$ with respect
to the ancient flow $g(\tau)$. Indeed, $l_{i}$ is the reduced distance from
$(p_{0},0)$ with respect to the flow $g_{i}(\tau)$.

Next, we interpret (4.2)—(4.5) in terms of the scaled flows (5.1). First of
all, (4.2) becomes

(5.3) $\displaystyle l_{i}(x_{i},1)=l(x_{i},\tau_{i})\leq C\text{ for all
}i\geq 0,$

where $C$ is a constant independent of $i$.

The locally uniform curvature bound (4.3) can be interpreted as follows. There
exists a positive function $C_{0}:(0,\infty)\rightarrow(0,\infty)$, such that
the following holds: for all $r>0$,

(5.4) $\displaystyle|Rm_{g_{i}(\tau)}|\leq C_{0}(r)\text{ on
}B_{g_{i}(1)}(x_{i},r)\times[1,2]\text{ for all }i\text{ large enough.}$

The time-wise Ricci curvature lower bound (4.4) now becomes the following.
There exists a positive sequence $\\{K_{i}\\}_{i=1}^{\infty}$, such that

(5.5) $\displaystyle Ric_{g_{i}(\tau)}\geq-K_{i}g_{i}(\tau)\text{ for all
}i\text{ and for all }\tau\in[1,2]\text{.}$

The injectivity radii condition (4.5) becomes

(5.6)
$\displaystyle\liminf_{i\rightarrow\infty}\Big{(}\text{inj}(g_{i}(1),x_{i})\Big{)}>0.$

Finally, by using (5.4), we may apply a straightforward distance distortion
estimate to obtain the following. For all $r>0$,

(5.7) $\displaystyle B_{g_{i}(\tau)}(x_{i},r)$ $\displaystyle\subset$
$\displaystyle B_{g_{i}(1)}\big{(}x_{i},C_{0}(r)\big{)},$ (5.8) $\displaystyle
B_{g_{i}(1)}(x_{i},r)$ $\displaystyle\subset$ $\displaystyle
B_{g_{i}(\tau)}\big{(}x_{i},C_{0}(r)\big{)},$

for all $\tau\in[1,2]$ and for all $i$ large enough, where $C_{0}(r)$ is a
positive function of $r$. The function $C_{0}$ in (5.7), (5.8), and (5.4) are
not necessarily the same. Nevertheless, we may always use the same notation
for the sake of convenience.

The above setting is what we work with in the current section. Indeed, (5.5)
and (5.6) are not needed until the next section. The following is the main
result of this section

######  Proposition 5.1.

Consider the sequence of backward Ricci flows in (5.1) with locally uniformly
curvature bound (5.4). Let $l_{i}$ be the functions defined in (5.2),
satisfying (5.3). Then the following hold. For any
$\displaystyle\varepsilon\in\left(0,\frac{1}{4}\right)$ there exists a
positive function $C(\epsilon,\cdot):(0,\infty)\rightarrow(0,\infty)$, such
that for any $r>0$ it holds that

  1. (1)

$\displaystyle 0\leq l_{i}(x,\tau)\leq C(\epsilon,r)$ for $(x,\tau)\in
B_{g_{i}(1)}(x_{i},r)\times[1+\varepsilon,2]$ for all $i$ large enough

  2. (2)

$\displaystyle\left|\frac{\partial l_{i}}{\partial\tau}(x,\tau)\right|+|\nabla
l_{i}(x,\tau)|\leq C(\epsilon,r)$ for
$(x,\tau)\in\big{(}B_{g_{i}(1)}(x_{i},r)\times[1+\varepsilon,2-\varepsilon]\big{)}\cap\big{(}\cup_{\tau\in[1+\varepsilon,2-\varepsilon]}\Omega(\tau)\times\\{\tau\\}\big{)}$
and for all $i$ large enough.

Here $\Omega(\tau)$ is as defined in section 2.

###### Proof.

In the following proof of both (1) and (2), we assume that
$\displaystyle\varepsilon\in\left(0,\frac{1}{4}\right)$ is a fixed number.
Furthermore, whenever we fix an $r>0$, we always assume $i$ is taken large
enough so that the formulas (5.4), (5.7), and (5.8) can be applied.

(1) The positivity of $l$-function follows from the positivity of the scalar
curvature on ancient solutions; see [6].

Fix a $\tau\in[1+\varepsilon,2]$ and an $r>0$. By (5.3) we have that there
exists a curve $\alpha_{i}:[0,1]\to M$ from $(p_{0},0)$ to $(x_{i},1)$ such
that

(5.9) $\mathcal{L}_{i}(\alpha_{i})\leq C.$

Let $\zeta_{i}(s):[0,\tau-1]$ be the minimal geodesic from $x_{i}$ to $x$ with
respect to $g_{i}(\tau)$, where $\displaystyle x\in
B_{g_{i}(\tau)}(x_{i},r)\subset B_{g_{i}(1)}\big{(}x_{i},C_{0}(r)\big{)}$; the
latter inclusion is by (5.7). Set

(5.10) $\eta_{i}(s)=\left\\{\begin{array}[]{ll}\alpha_{i}(s),&s\in[0,1],\\\
\zeta_{i}(s-1),&s\in(1,\tau].\end{array}\right.$

By (5.4), we have $\displaystyle|\eta^{\prime}_{i}(s)|^{2}_{g_{i}(s)}\leq
e^{2nC_{1}(r)\tau}|\zeta^{\prime}_{i}(s-1)|^{2}_{g_{i}(\tau)}$ for
$s\in(1,\tau]$. This is because

$\displaystyle\zeta_{i}\subset B_{g_{i}(\tau)}(x_{i},r)\subset
B_{g_{i}(1)}\big{(}x_{i},C_{0}(r)\big{)},$
$\displaystyle\max_{{\zeta_{i}}\times[1,2]}|Rm_{g_{i}}|\leq C_{1}(r),$

where $C_{1}(r)$ is a constant depending on $r$. Hence

$\displaystyle L_{i}(x,\tau)$ $\displaystyle\leq$
$\displaystyle\mathcal{L}_{i}(\eta_{i}(s))$ $\displaystyle=$
$\displaystyle\mathcal{L}_{i}(\alpha_{i})+\int^{\tau}_{1}\sqrt{s}\Big{(}R_{g_{i}(s)}(\eta_{i}(s))+|\eta_{i}^{\prime}(s)|^{2}_{g_{i}(s)}\Big{)}ds$
$\displaystyle\leq$
$\displaystyle\mathcal{L}_{i}(\alpha_{i})+\int^{\tau}_{1}\sqrt{s}\Big{(}R_{g_{i}(s)}(\zeta_{i}(s-1))+e^{2nC_{1}(r)\tau}|\zeta^{\prime}_{i}(s-1)|^{2}_{g_{i}(\tau)}\Big{)}ds$
$\displaystyle\leq$
$\displaystyle\mathcal{L}_{i}(\alpha_{i})+2C_{1}(r)+\frac{2}{3}e^{2nC_{1}(r)\tau}\frac{r^{2}}{(\tau-1)^{2}}(\tau^{\frac{3}{2}}-1)$
$\displaystyle\leq$ $\displaystyle
2C+2C_{1}(r)+4e^{2nC_{1}(r)\tau}\varepsilon^{-2}.$

Taking (5.8) into account, this finishes the proof of part (1).

(2) Let $\gamma_{i}$ be the minimal $\mathcal{L}-$geodesic from $(p_{0},0)$ to
$(x,\tau)$ with respect to the backward Ricci flow $g_{i}$, where
$x\in(B_{g_{i}(\tau)}(x_{i},r)\cap\Omega(\tau))$ and
$\tau\in[1+\varepsilon,2-\varepsilon]$. Since

$B_{g_{i}(\tau)}(x,r)\subset B_{g_{i}(\tau)}(x_{i},2r)\subset
B_{g_{i}(1)}\big{(}x_{i},C_{0}(2r)\big{)},$

we have by (5.4) that

(5.11) $\displaystyle\max_{B_{g_{i}(\tau)}(x,r)\times[1,2]}|Rm_{g_{i}}|\leq
C_{2}(r).$

Let $q_{i}=\gamma_{i}(\tau_{q_{i}})$ be defined as

$\displaystyle\tau_{q_{i}}=\inf\Big{\\{}s\in[0,\tau]\ \Big{|}\
\gamma_{i}|_{[s,\tau]}\subset B_{g_{i}(\tau)}(x,r)\Big{\\}}.$

Claim: There exists a positive number $c_{1}$ depending on $r$ and
$\varepsilon$, but independent of $i$, such that
$\sqrt{\tau}-\sqrt{\tau_{q_{i}}}\geq c_{1}$ for all $i$ large enough.

###### Proof of the claim.

We argue by contradiction. Let $x_{i}^{\prime}\in B_{g_{i}(\tau)}(x_{i},r)$ be
a sequence of counterexamples. In other words, let $\gamma_{i}$ be minimal
$\mathcal{L}$-geodesic from $(p_{0},0)$ to $(x_{i}^{\prime},\tau)$ with
respect to the flow $g_{i}$ and

$\displaystyle\tau_{q_{i}}=\inf\Big{\\{}s\in[0,\tau]\ \Big{|}\
\gamma_{i}|_{[s,\tau]}\subset B_{g_{i}(\tau)}(x_{i}^{\prime},r)\Big{\\}}.$

But $\sqrt{\tau}-\sqrt{\tau_{q_{i}}}\to 0$ as $i\to\infty$. We can then assume
that $\tau_{q_{i}}\geq 1$ when $i$ is large. As before, since

$\gamma_{i}|_{[\tau_{q_{i}},\tau]}\subset
B_{g_{i}(\tau)}(x_{i}^{\prime},r)\subset B_{g_{i}(\tau)}(x_{i},2r)\subset
B_{g_{i}(1)}\big{(}x_{i},C_{0}(2r)\big{)},$

we have

(5.12)
$\displaystyle\max_{\gamma_{i}|_{[\tau_{q_{i}},\tau]}\times[1,2]}|Rm_{g_{i}}|\leq\max_{B_{g_{i}(\tau)}(x_{i}^{\prime},r)\times[1,2]}|Rm_{g_{i}}|\leq
C_{2}(r).$

Let us change the variable so that $\beta_{i}(\sigma)=\gamma_{i}(\sigma^{2})$.
We can then write the $\mathcal{L}$-energy of $\gamma_{i}$ as

$\mathcal{L}_{i}(\gamma_{i})=\int^{\sqrt{\tau}}_{0}\left(2\sigma^{2}R_{g_{i}(\sigma^{2})}(\beta_{i}(\sigma))+\frac{1}{2}|\beta_{i}^{\prime}(\sigma)|^{2}_{g_{i}(\sigma^{2})}\right)d\sigma.$

Then we have

$\displaystyle\int^{\sqrt{\tau}}_{\sqrt{\tau_{q_{i}}}}\frac{1}{2}|\beta_{i}^{\prime}(\sigma)|^{2}_{g_{i}(\sigma^{2})}d\sigma$
$\displaystyle=\mathcal{L}_{i}(\gamma_{i}|_{[\tau_{q_{i}},\tau]})-\int^{\sqrt{\tau}}_{\sqrt{\tau_{q_{i}}}}2\sigma^{2}R_{g_{i}(\sigma^{2})}(\beta_{i}(\sigma))d\sigma$
$\displaystyle\leq\mathcal{L}_{i}(\gamma_{i})=2\sqrt{\tau}l_{i}(x_{i}^{\prime},\tau)$
$\displaystyle\leq C_{3}(r),$

where we have also used part (1) as well as the positivity of scalar curvature
on ancient Ricci flows. It follows from the definition of $q_{i}$ and (5.12)
that

$\displaystyle r^{2}=d^{2}_{g_{i}(\tau)}(x_{i}^{\prime},q_{i})$
$\displaystyle\leq\left(\int^{\sqrt{\tau}}_{\sqrt{\tau_{q_{i}}}}|\beta_{i}^{\prime}(\sigma)|_{g_{i}(\tau)}d\sigma\right)^{2}$
$\displaystyle\leq
e^{2nC_{2}(r)\tau}\left(\int^{\sqrt{\tau}}_{\sqrt{\tau_{q_{i}}}}|\beta_{i}^{\prime}(\sigma)|_{g_{i}(\sigma^{2})}d\sigma\right)^{2}$
$\displaystyle\leq(\sqrt{\tau}-\sqrt{\tau_{q_{i}}})e^{2nC_{2}(r)\tau}\int^{\sqrt{\tau}}_{\sqrt{\tau_{q_{i}}}}|\beta_{i}^{\prime}(\sigma)|^{2}_{g_{i}(\sigma^{2})}d\sigma$
$\displaystyle\leq C_{4}(r)(\sqrt{\tau}-\sqrt{\tau_{q_{i}}});$

a contradiction against $\sqrt{\tau}-\sqrt{\tau_{q_{i}}}\to 0$. This proves
the claim. ∎

Let us continue working with the minimal $\mathcal{L}$-geodesic connecting
$(p_{0},0)$ and $(x,\tau)$, where $x\in B_{g_{i}(\tau)}(x_{i},r)$. After the
change of the variable $\beta_{i}(\sigma)=\gamma_{i}(\sigma^{2})$, the
$\mathcal{L}-$geodesic equation (2.3) becomes

$\nabla_{\beta_{i}^{\prime}(\sigma)}\beta_{i}^{\prime}(\sigma)-2\sigma^{2}\nabla
R+4\sigma Rc(\beta_{i}^{\prime}(\sigma))=0,$

So we have

$\displaystyle\frac{d}{d\sigma}|\beta_{i}^{\prime}(\sigma)|_{g_{i}(\sigma^{2})}^{2}$
$\displaystyle=$ $\displaystyle 4\sigma^{2}\nabla
R\cdot\beta_{i}^{\prime}-4\sigma
Ric_{g_{i}}(\beta_{i}^{\prime},\beta_{i}^{\prime})$ $\displaystyle\leq$
$\displaystyle
C_{5}(r)\left(|\beta_{i}^{\prime}(\sigma)|_{g_{i}(\sigma^{2})}+|\beta_{i}^{\prime}(\sigma)|_{g_{i}(\sigma^{2})}^{2}\right)$
$\displaystyle\leq$ $\displaystyle
C_{5}(r)\left(1+2|\beta_{i}^{\prime}(\sigma)|_{g_{i}(\sigma^{2})}^{2}\right),$

for all
$\displaystyle\sigma\in\left[\sqrt{\tau_{q_{i}}^{\prime}},\sqrt{\tau}\right]$,
where $\tau_{q_{i}}^{\prime}=\max\\{\tau_{q_{i}},1\\}$; we have made use of
the fact
$\displaystyle\beta_{i}|_{[\sqrt{\tau_{q_{i}}^{\prime}},\sqrt{\tau}]}\subset
B_{g_{i}(\tau)}(x,r)$, (5.11), and Shi’s local gradient estimates [16].
Indeed, (5.11) is still true if we replace $B_{g_{i}(\tau)}(x,r)$ by
$B_{g_{i}(\tau)}(x,2r)$, with possibly a different $C_{2}$. Therefore Shi’s
estimates can be implemented. Integrating the above inequality, we obtain

(5.13) $\displaystyle|\beta_{i}^{\prime}(\sigma)|^{2}_{g_{i}(\sigma^{2})}\geq
c_{6}(r)|\beta_{i}^{\prime}(\sqrt{\tau})|^{2}_{g_{i}(\tau)}-C_{6}(r)$

for $\sigma\in[\sqrt{\tau_{q_{i}}^{\prime}},\sqrt{\tau}]$ . Then, by part (1),
the claim, and (5.13), we have

$\displaystyle
C(r)\geq\mathcal{L}_{i}(\gamma_{i}|_{[\tau_{q}^{\prime},\tau]})$
$\displaystyle=\int^{\sqrt{\tau}}_{\sqrt{\tau_{q}^{\prime}}}\frac{1}{2}|\beta_{i}^{\prime}(\sigma)|^{2}_{g_{i}(\sigma^{2})}d\sigma+\int^{\sqrt{\tau}}_{\sqrt{\tau_{q}^{\prime}}}2\sigma^{2}R_{g_{i}}(\beta_{i}(\sigma),\sigma^{2})d\sigma$
$\displaystyle\geq
c_{7}(r)|\beta_{i}^{\prime}(\sqrt{\tau})|_{g_{i}(\tau)}^{2}-C_{7}(r)$

Using Lemma 2.3, we obtain

$\displaystyle|\nabla
l_{i}(x,\tau)|=|\gamma_{i}^{\prime}(\tau)|=\frac{|\beta_{i}^{\prime}(\sqrt{\tau})|}{2\sqrt{\tau}}\leq
C_{8}(r).$

Finally, by (2.6), (5.4), and part (1), we obtain

$\displaystyle\left|\frac{\partial l_{i}}{\partial\tau}(x,\tau)\right|\leq
C_{9}(r);$

this completes the proof of the theorem. ∎

##  6\. Asymptotic Shrinking Gradient Ricci Soliton

After the preparation in the previous section, we are ready to show that a
locally uniformly Type I ancient solution has an asymptotic shrinking gradient
Ricci soliton.

######  Theorem 6.1.

Let $(M,g(\tau))_{\tau\in[0,\infty)}$ be an ancient solution to the Ricci
flow, where $\tau$ is the backward time. Let $p_{0}\in M$ and
$\\{(x_{i},\tau_{i})\\}_{i=1}^{\infty}$ be a space-time sequence such that
$\tau_{i}\nearrow\infty$. Assume that $(M,g(\tau))_{\tau\in[0,\infty)}$ is
locally uniformly Type I along $\\{(x_{i},\tau_{i})\\}_{i=1}^{\infty}$. Then
the sequence of tuples

(6.1)
$\displaystyle\Big{\\{}\big{(}M,g_{i}(\tau),x_{i},l_{i}\big{)}_{\tau\in[1,2]}\Big{\\}}_{i=1}^{\infty},$

where $g_{i}(\tau)=\tau_{i}^{-1}g(\tau\tau_{i})$,
$l_{i}(\tau)=l(\tau\tau_{i})$, and $l$ is the reduced distance function based
at $(p_{0},0)$, after passing to a subsequence, converges to the canonical
form of a shrinking gradient Ricci soliton

$\displaystyle\big{(}M_{\infty},g_{\infty}(\tau),x_{\infty},l_{\infty}\big{)}_{\tau\in(1,2)}$

with $l_{\infty}$ being the potential function, that is,

$\displaystyle
Ric_{g_{\infty}(\tau)}+\nabla^{2}l_{\infty}-\frac{1}{2\tau}g_{\infty}(\tau)=0,$

for all $\tau\in(1,2)$. Here the Ricci flows
$\big{(}M,g_{i}(\tau),x_{i}\big{)}_{\tau\in[1,2]}$ converge in the smooth
Cheeger-Gromov-Hamilton sense [8], and the functions $l_{i}$ converge in the
$C_{\text{loc}}^{0,\alpha}$ sense as well as in the weak
$*W_{\text{loc}}^{1,2}$ sense.

Let us consider the sequence of tuples (6.1). Because the locally uniform Type
I assumption, (5.4)—(5.8) are still valid for $g_{i}$ and $l_{i}$.
Furthermore, Proposition 5.1 also implies a locally uniform bound for $l_{i}$
as well as its gradient around $(x_{i},1)$. This is the setting which we work
with subsequently in this section.

The following convergence of underlying Ricci flows is an immediate
consequence of (5.4) and (5.6).

######  Lemma 6.2.

After passing to a subsequence, the sequence of backward Ricci flows

$\displaystyle\Big{\\{}\big{(}M,g_{i}(\tau),(x_{i},1)\big{)}_{\tau\in[1,2]}\Big{\\}}_{i=1}^{\infty}$

converges in the smooth Cheeger-Gromov-Hamilton sense [8] to a smooth backward
Ricci flow

$\displaystyle\big{(}M_{\infty},g_{\infty}(\tau),(x_{\infty},1)\big{)}_{\tau\in[1,2)}.$

_Remark_ : It is not known whether $g_{\infty}$ is defined at $\tau=2$,
because Shi’s estimates [16] cannot be applied at this instance. Nevertheless,
in the case of shrinking breather, that is, if the ancient solution is defined
as in (3.4), $g_{\infty}$ exists and is smooth up to $\tau=2$.

In the definition of Cheeger-Gromov-Hamilton convergence, there are
diffeomorphism involved:

(6.2) $\displaystyle\Phi_{i}:M_{\infty}\times[1,2)\supset U_{i}\rightarrow
V_{i}\subset M\times[1,2],$

where $\\{U_{i}\\}_{i=1}^{\infty}$ is a sequence of open sets exhausting the
space-time $M_{\infty}\times[1,2)$, with $(x_{\infty},1)\in U_{i}$ and
$\Phi_{i}(x_{\infty},1)=(x_{i},1)$ for all $i$ . The composite functions

$\displaystyle l_{i}\circ\Phi_{i}$

are therefore functions on $M_{\infty}\times[1,2)$. In the following argument,
when we are referring to $l_{i}$ as a function on $M_{\infty}\times[1,2)$,
this composition is implicitly understood, though we still write it as $l_{i}$
for simplicity.

Let us then recall some basic properties of $l_{i}$. We define

$\displaystyle\mathcal{V}_{i}(\tau):=\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau),\
\tau\in[1,2].$

By the monotonicity property of reduced volume (Theorem 2.4(2)), we have

(6.3)
$\displaystyle\lim_{\i\rightarrow\infty}\mathcal{V}_{i}(\tau)=V_{\infty}\text{
uniformly for }\tau\in[1,2],$

where $V_{\infty}\in[0,1]$ is a constant. We will see in section 7 that
$V_{\infty}>0$, and $V_{\infty}=1$ implies that the Ricci flow is a static
Euclidean space; see [18]. For the moment, we only need the fact that
$V_{\infty}$ is a constant. Furthermore, $l_{i}$ also satisfies the formulas
(2.6)—(2.10).

######  Lemma 6.3.

After passing to a subsequence, $\\{l_{i}\\}_{i=1}^{\infty}$ converges in the
$C_{\text{loc}}^{0,\alpha}\big{(}M_{\infty}\times(1,2)\big{)}$ sense as well
as in the weak $*W_{\text{loc}}^{1,2}\big{(}M_{\infty}\times(1,2)\big{)}$
sense to a locally Lipschitz function $l_{\infty}$; here $\alpha\in(0,1)$ is
an arbitrarily fixed number. By passing to a further subsequence, we may also
assume that
$\displaystyle(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\rightarrow(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}$
in the same sense. In particular, $(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}$
is a positive function.

###### Proof.

This is a direct consequence of Proposition 5.1. ∎

Subsequently, our main goal for this section is to prove the following.

######  Proposition 6.4.

$(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}$ is a smooth solution to the
conjugate heat equation $\displaystyle\frac{\partial}{\partial\tau}u-\Delta
u+Ru=0$ on $M_{\infty}\times(1,2)$.

We break down the proof of Proposition 6.4 into several steps. First of all,
the existence of a time-independent cut-off function with small gradient is
merely a consequence of (5.5).

######  Lemma 6.5.

For any positive number $A$, there exists a smooth function
$\varphi_{A,i}:M\rightarrow[0,1]$ satisfying the following conditions:

  1. (1)

$\varphi_{A,i}$ is supported in $B_{g_{i}(2)}(x_{i},2A)$,

  2. (2)

$\varphi_{A,i}\equiv 1$ on $B_{g_{i}(2)}(x_{i},A)$,

  3. (3)

$|\nabla\varphi_{A,i}|_{g_{i}(\tau)}\leq\frac{C_{K_{i}}}{A}\ \text{ for all }\
\tau\in[1,2]$,

where the constant $C_{K_{i}}$ depending only on the Ricci curvature lower
bound $K_{i}$. In particular, $C_{K_{i}}$ is independent of $A$.

###### Proof.

Let $\chi(t):[0,+\infty)\to[0,1]$ be a smooth function satisfying $\chi(t)=1$
for $t\in[0,1]$, $\chi(t)=0$ for $t\geq 2$ and $|\chi^{\prime}(t)|\leq 2$.
Then one may easily verify that
$\displaystyle\varphi_{A,i}(x)=\chi\left(\frac{d_{g_{i}(2)}(x_{i},x)}{A}\right)$
satisfies the lemma. ∎

This cut-off function will be used repeatedly in the following arguments.

######  Lemma 6.6.

There exists a constant $C_{0}$ independent of $i$, such that

(6.4) $\displaystyle\int_{M}|\nabla
l_{i}|^{2}_{g_{i}(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)\leq
C_{0},$

for all $i$ and for all $\tau\in[1,2]$.

###### Proof.

Rewriting (2.10) for $l_{i}$, we have

$\displaystyle\int_{M}\Big{(}-2\nabla l_{i}\cdot\nabla\phi-|\nabla
l_{i}|_{g_{i}(\tau)}^{2}\phi\Big{)}dg_{i}(\tau)$ $\displaystyle\leq$
$\displaystyle-\int_{M}\Big{(}R_{g_{i}}+\frac{l_{i}-n}{\tau}\Big{)}\phi
dg_{i}(\tau)$ $\displaystyle\leq$ $\displaystyle\frac{n}{\tau}\int_{M}\phi
dg_{i}(\tau)$

for all $\tau\in[1,2]$, where in the second inequality we have used the facts
that $R_{g_{i}}\geq 0$ and that $l_{i}\geq 0$; see Proposition 5.1(1).

Let us take

$\displaystyle\phi:=\varphi_{A,i}^{2}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}},$

where $\varphi_{A,i}$ is as defined in Lemma 6.5. This is valid, since $l_{i}$
is locally Lipschitz, and $\varphi_{A,i}$ is smooth as well as compactly
supported. Then the above inequality becomes

$\displaystyle\int_{M}\Big{(}\varphi_{A,i}^{2}|\nabla
l_{i}|^{2}_{g_{i}(\tau)}-4\langle\nabla
l_{i},\nabla\varphi_{A,i}\rangle\varphi_{A,i}\Big{)}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)\leq\frac{n}{\tau}\int_{M}\varphi_{A,i}^{2}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau).$

Hence we have

$\displaystyle\int_{M}\varphi_{A,i}^{2}|\nabla
l_{i}|^{2}_{g_{i}(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle\leq$
$\displaystyle\frac{n}{\tau}\int_{M}\varphi_{A,i}^{2}(4\pi\tau)^{\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle+4\int_{M}\Big{(}\langle\nabla
l_{i},\nabla\varphi_{A,i}\rangle\varphi_{A,i}\Big{)}(4\pi\tau)^{\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle\leq$
$\displaystyle\frac{n}{\tau}\int_{M}\varphi_{A,i}^{2}(4\pi\tau)^{\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle+\frac{1}{2}\int_{M}\varphi_{A,i}^{2}|\nabla
l_{i}|^{2}_{g_{i}(\tau)}(4\pi\tau)^{\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle+8\int_{M}|\nabla\varphi_{A_{i}}|^{2}_{g_{i}(\tau)}(4\pi\tau)^{\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau),$

and subsequently

$\displaystyle\int_{M}\varphi_{A,i}^{2}|\nabla
l_{i}|^{2}_{g_{i}(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle\leq$
$\displaystyle\left(\frac{16C_{K_{i}}^{2}}{A^{2}}+\frac{2n}{\tau}\right)\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle=$
$\displaystyle\left(\frac{16C_{K_{i}}^{2}}{A^{2}}+\frac{2n}{\tau}\right)\mathcal{V}_{i}(\tau).$

Taking $A\rightarrow\infty$ completes the proof; note that
$\mathcal{V}_{i}(\tau)\leq 1$ by Theorem 2.4. ∎

Next, we consider the other distributional inequality (2.9) satisfied by
$l_{i}$. Replacing $\phi$ by $\phi(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}$, (2.9)
becomes

(6.5)
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\Bigg{(}\phi\frac{\partial}{\partial\tau}\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}+\big{\langle}\nabla\phi,\nabla\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}\big{\rangle}$
$\displaystyle+\phi
R_{g_{i}}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\Bigg{)}dg_{i}(\tau)d\tau\leq 0,$

where $1<\tau_{1}<\tau_{2}<2$ and $\phi$ is a smooth nonnegative function
compactly supported on $M\times[\tau_{1},\tau_{2}]$. Combining this and the
definition of the reduced volume, we have

######  Lemma 6.7.

For any $1<\tau_{1}<\tau_{2}<2$ and any smooth nonnegative function $\phi$
compactly supported on $M\times[\tau_{1},\tau_{2}]$ with $\sup\phi\leq 1$, it
holds that

$\displaystyle\mathcal{V}_{i}(\tau_{2})-\mathcal{V}_{i}(\tau_{1})$
$\displaystyle\leq$
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\Bigg{(}\phi\frac{\partial}{\partial\tau}\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}$
$\displaystyle+\big{\langle}\nabla\phi,\nabla\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}\big{\rangle}+\phi
R_{g_{i}}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\Bigg{)}dg_{i}(\tau)d\tau\leq 0.$

###### Proof.

The second inequality of (6.7) follows from (6.5).

For the first inequality, let us fix an arbitrary function $\phi$ satisfying
the properties indicated in the lemma. Then we may find a positive number
$A_{0}$ large enough, such that

$\displaystyle\text{spt}\ \phi\subset
B_{g_{i}(2)}(x_{0},A_{0})\times[\tau_{1},\tau_{2}].$

Then, for any $A\geq A_{0}$, we may apply (6.5) to $\varphi_{A,i}-\phi$, where
$\varphi_{A,i}$ is as defined in Lemma 6.5. This yields

(6.7)
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\Bigg{(}\phi\frac{\partial}{\partial\tau}\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}+\big{\langle}\nabla\phi,\nabla\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}\big{\rangle}$
$\displaystyle+\phi
R(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\Bigg{)}dg_{i}(\tau)d\tau$
$\displaystyle\geq$
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\Bigg{(}\varphi_{A,i}\frac{\partial}{\partial\tau}\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}+\big{\langle}\nabla\varphi_{A,i},\nabla\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}\big{\rangle}$
$\displaystyle+\varphi_{A,i}R_{g_{i}}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\Bigg{)}dg_{i}(\tau)d\tau.$

We therefore consider the right-hand-side of the above inequality.

Since $\varphi_{A,i}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}$ is a Lipschitz
function compactly supported on $M\times[\tau_{1},\tau_{2}]$, it is then also
absolutely continuous. Hence

$\displaystyle\int_{M}\varphi_{A,i}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)\
\Bigg{|}_{\tau_{1}}^{\tau_{2}}$ $\displaystyle=$
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\frac{d}{d\tau}\left(\int_{M}\varphi_{A,i}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)\right)$
$\displaystyle=$
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\Bigg{(}\varphi_{A,i}\frac{\partial}{\partial\tau}\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}$
$\displaystyle+\varphi_{A,i}R_{g_{i}}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\Bigg{)}dg_{i}(\tau)d\tau,$

where we have used the time-independence of $\varphi_{A,i}$. Then the right-
hand-sides of (6.7) and (6) differ only by the following term.

(6.9)
$\displaystyle\left|\int_{M}\big{\langle}\nabla\varphi_{A,i},\nabla\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\big{)}\big{\rangle}dg_{i}(\tau)\right|$
$\displaystyle\leq\int_{M}|\nabla\varphi_{A,i}|_{g_{i}(\tau)}|\nabla
l_{i}|_{g_{i}(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle\leq\sup|\nabla\varphi_{A,i}|_{g_{i}(\tau)}\left(\int_{M}|\nabla
l_{i}|^{2}_{g_{i}(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg(\tau)\right)^{\frac{1}{2}}\left(\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{C_{K_{i}}}{A}C_{0}^{\frac{1}{2}}\big{(}\mathcal{V}_{i}(\tau)\big{)}^{\frac{1}{2}},$

where we have used Lemma 6.5, Lemma 6.6, and the Cauchy-Schwarz inequality.

Finally, combining (6.7), (6), (6.9), and taking $A\rightarrow\infty$, we
prove the first inequality of (6.7). ∎

We are now ready to prove Proposition 6.4.

###### Proof of Proposition 6.4.

Let us fix $1<\tau_{1}<\tau_{2}<2$ and an arbitrary smooth nonnegative
function $\phi$ compactly supported on $M_{\infty}\times[\tau_{1},\tau_{2}]$
with $\sup\phi\leq 1$. Hence, whenever $i$ is large enough, we have that
$\phi\circ\Phi_{i}^{-1}$ satisfies (6.7), where $\Phi_{i}$ is defined in
(6.2). Since
$\displaystyle(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}\rightarrow(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}$
locally uniformly and in weak $*W^{1,2}\big{(}\text{spt}\ \phi\big{)}$ sense,
we may take a limit for (6.7), and use (6.3) to obtain

$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M_{\infty}}\Bigg{(}\phi\frac{\partial}{\partial\tau}\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}\big{)}+\big{\langle}\nabla\phi,\nabla\big{(}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}\big{)}\big{\rangle}$
$\displaystyle+\phi
R_{g_{\infty}}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}\Bigg{)}dg_{\infty}(\tau)d\tau=0.$

Since $\tau_{1}$, $\tau_{2}$, and $\phi$ are arbitrary, this shows that
$(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}$ is a weak solution to the conjugate
heat equation $\displaystyle\frac{\partial}{\partial\tau}u-\Delta u+Ru=0$ on
$M_{\infty}\times(1,2)$. By the standard local regularity theory of linear
parabolic equations, we also have that $l_{\infty}$ is smooth and that
$(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}$ is a classical solution to the
conjugate heat equation. ∎

Before the proof of the our main theorem, we need one final lemma.

######  Lemma 6.8.

$l_{\infty}$ satisfies the following equations on $M_{\infty}\times(1,2)$.

(6.10) $\displaystyle 2\frac{\partial l_{\infty}}{\partial\tau}+|\nabla
l_{\infty}|^{2}-R_{g_{\infty}}+\frac{l_{\infty}}{\tau}$ $\displaystyle=$
$\displaystyle 0,$ (6.11) $\displaystyle 2\Delta l_{\infty}-|\nabla
l_{\infty}|^{2}+R_{g_{\infty}}+\frac{l_{\infty}-n}{\tau}$ $\displaystyle=$
$\displaystyle 0.$

###### Proof.

By Theorem 2.2, we have that $l_{i}$ satisfies (6.10) almost everywhere on
$M\times[1,2]$. Since $l_{i}$ is locally Lipschitz, we also have that $l_{i}$
satisfies (6.10) in the sense of distribution.

On the other hand, $l_{i}\rightarrow l_{\infty}$ in the
$C_{\text{loc}}^{0,\alpha}\big{(}M_{\infty}\times(1,2)\big{)}$ sense as well
as in the weak $*W_{\text{loc}}^{1,2}\big{(}M_{\infty}\times(1,2)\big{)}$
sense, therefore (6.10) holds for $l_{\infty}$ in the sense of distribution;
see Lemma 9.21 in [9], for instance, for the distributional convergence
$|\nabla l_{i}|^{2}\rightarrow|\nabla l_{\infty}|^{2}$, and the distributional
convergence of all other terms in (6.10) is obvious. By Proposition 6.4,
$l_{\infty}$ is smooth, hence (6.10) holds in the classical sense on
$M\times(1,2)$.

Finally, (6.11) follows from (6.10) and the conjugate heat equation. ∎

###### Proof of Theorem 6.1.

Let us define

$\displaystyle u$ $\displaystyle:=$
$\displaystyle(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}},$ $\displaystyle v$
$\displaystyle:=$ $\displaystyle\left(\tau(2\Delta l_{\infty}-|\nabla
l_{\infty}|^{2}+R_{g_{\infty}})+l_{\infty}-n\right)u.$

Since $u$ satisfies the conjugate heat equation, and according to Perelman’s
computation (Proposition 9.1 in [15]), we have

$\displaystyle\left(\frac{\partial}{\partial\tau}-\Delta+R_{g_{\infty}}\right)v=-2\tau\left|Ric_{g_{\infty}}+\nabla^{2}l_{\infty}-\frac{1}{2\tau}g_{\infty}(\tau)\right|^{2}u,$

on $M_{\infty}\times(1,2)$. By (6.11) $v\equiv 0$ and $u>0$, it then follows
that

$\displaystyle
Ric_{g_{\infty}(\tau)}+\nabla^{2}l_{\infty}-\frac{1}{2\tau}g_{\infty}(\tau)=0$

on $M_{\infty}\times(1,2)$; this finishes the proof. ∎

Then, the no shrinking breather theorem follows immediately.

###### Proof of Theorem 1.3.

Let us consider the locally uniformly Type I ancient solution as mentioned in
Proposition 4.2. On the one hand, Theorem 6.1 implies

$\displaystyle\big{(}M,g_{i}(\tau),x_{i},l_{i}\big{)}_{\tau\in[1,2]}\rightarrow\big{(}M_{\infty},g_{\infty}(\tau),x_{\infty},l_{\infty}\big{)}_{\tau\in(1,2)},$

where the latter is the canonical form of a shrinking gradient Ricci soliton.
Note that $l_{\infty}$ is not defined at $\tau=1$, but $g_{\infty}$ is. On the
other hand, by the final statement of Proposition 4.2, and since
$\tau_{i}^{-1}\alpha^{-(i+1)}\rightarrow 1$ and
$\displaystyle\frac{\tau_{i+1}}{\tau_{i}}\rightarrow\alpha^{-1}$, we have

$\displaystyle(M,g_{0}(\tau-1),(p_{0},1))_{\tau\in[1,\alpha^{*})}\cong(M_{\infty},g_{\infty}(\tau),(x_{\infty},1))_{\tau\in[1,\alpha^{*})},$

where $\alpha^{*}=\max\\{2,\alpha^{-1}\\}>1$; this proves the existence of
soliton structure on the original shrinking breather. ∎

##  7\. Further Remarks

In this section, we continue our consideration of a locally uniformly Type I
ancient solution as well as its asymptotic shrinking gradient soliton. All the
notations bear the same meaning as in section 6.

###  7.1. The Possible Loss of Reduced Volume

According to Fatou’s lemma, we have

$\displaystyle\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{\infty}}dg_{\infty}(\tau)$
$\displaystyle\leq$
$\displaystyle\lim_{i\rightarrow\infty}\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-l_{i}}dg_{i}(\tau)$
$\displaystyle=$
$\displaystyle\lim_{s\rightarrow\infty}\mathcal{V}(s):=V_{\infty}.$

Since the left-hand-side is the Gaussian density of a shrinking gradient Ricci
soliton, it must always be positive. On the other hand, if $V_{\infty}=1$,
then $(M,g(\tau))$ must be the static Euclidean space; see [18]. Hence we have

$\displaystyle\lim_{s\rightarrow\infty}\mathcal{V}(s):=V_{\infty}\in(0,1).$

Now, one may naturally ask: is the inequality in (7.1) always an equality? We
do not know the answer yet, though we highly doubt whether the opposite could
happen. Recall that in the classical cases: the Type I case and the case of
“curvature positivity”, this equality always holds, and this is because of the
following estimate which shows that that the integrand of $\mathcal{V}_{i}$ is
negligible outside compact sets

$\displaystyle l_{i}\sim d_{g_{i}}^{2}(x_{i},\cdot).$

###  7.2. Noncollapsedness of Locally Uniformly Type I Ancient Solutions

Next, we observe that a locally uniformly Type I ancient solution is
$\kappa$-noncollapsed on all scaled. According to the definition, we only know
that such an ancient solution is $\kappa$-noncollpased along a sequence of
space-time points $(x_{i},\tau_{i})$ on scales $\tau_{i}^{\frac{1}{2}}$.
However, the existence of the asymptotic shrinking gradient Ricci soliton
enables us to extend the noncollapsedness to everywhere at all scales. We can
write formula (7.1) as

$\displaystyle\lim_{s\rightarrow\infty}\mathcal{V}_{(p_{0},0)}(s)=V_{\infty}\in(0,1),$

where we use the subindex to signify the base point. It then follows from
Lemma 3.1 in [18] that

$\displaystyle\lim_{s\rightarrow\infty}\mathcal{V}_{(p,\tau)}(s)\geq
V_{\infty}>0,$

for all $(p,\tau)\in M\times(0,\infty)$. Note that the proof of Lemma 3.1 in
[18] requires only time-wise Ricci curvature lower bound. A standard argument
as in [15] implies that $(M,g(\tau))_{\tau\in(0,\infty)}$ is
$\kappa$-noncollaposed on all scales, where $\kappa$ depends on the dimension
and $V_{\infty}\in(0,1]$.

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