# Counting ends on shrinkers

Jia-Yong Wu  Department of Mathematics, Shanghai University, Shanghai 200444,
China<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

In this paper we apply a geometric covering method to study the number of ends
on shrinkers. On one hand, we prove that the number of ends on any complete
non-compact shrinker is at most polynomial growth with fixed degree. On the
other hand, we prove that any complete non-compact shrinker with certain
volume comparison condition has finitely many ends. Some special cases of
shrinkers are also discussed.

###### Key words and phrases:

Gradient shrinking Ricci soliton, end, volume comparison, asymptotic scalar
curvature ratio, asymptotic volume ratio.

###### 2010 Mathematics Subject Classification:

Primary 53C21; Secondary 53C20.

##  1\. Introduction and main results

An $n$-dimensional Riemannian manifold $(M,g)$ is called a gradient shrinking
Ricci soliton or shrinker (see [19]) if there exists a smooth function $f$ on
$(M,g)$ such that the Ricci curvature Ric and the Hessian of $f$ satisfy

${\mathrm{Ric}}+\mathrm{Hess}\,f=\lambda g$

for some constant $\lambda>0$. Function $f$ is often called a potential of the
shrinker. Upon scaling the metric $g$ by a constant, we may assume
$\lambda=1/2$ so that

$\displaystyle{\mathrm{Ric}}+\mathrm{Hess}\,f=\frac{1}{2}g.$ (1.1)

Furthermore, we can normalize $f$ such that (1.1) simultaneously satisfies

$\mathrm{S}+|\nabla f|^{2}-f=0,$ (1.2)

where $\mathrm{S}$ is the scalar curvature of $(M,g)$, and

$\int_{M}(4\pi)^{-\frac{n}{2}}e^{-f}dv=e^{\mu},$ (1.3)

where $dv$ is the volume element with respect to metric $g$, and
$\mu=\mu(g,1)$ is the entropy functional of Perelman [37]. By Lemma 2.5 in
[26], we see that the term $e^{\mu}$ is almost equivalent to the volume of
geodesic ball $B(p,1)$ with radius $1$ and center $p$. Here $p\in M$ is a
infimum point of $f$, which can be always achieved for any complete shrinker;
see [20].

Shrinkers play an important role in the Ricci flow as they correspond to some
self-similar solutions and usually arise as the limit solutions of type I
singularity models of the Ricci flow [15]. They are regarded as a natural
extension of Einstein manifolds with positive scalar curvature, and are
related to the Bakry-Émery Ricci tensor [2]. Nowadays, the understanding of
geometry and topology for shrinkers is an important subject in the Ricci flow
[19]. For dimensions 2 and 3, the classification of shrinkers is complete.
However dimensions equal to or greater than 4, the complete classification
remains open; see [4, 5] and references therein for nice surveys.

It is an interesting phenomenon that many geometric and analytic properties of
shrinkers are similar to manifolds with nonnegative Ricci curvature or
Einstein manifolds with positive scalar curvature. Some interesting results
are exhibited as follows. Wylie [46] proved that any complete shrinker has
finite fundamental group (the compact case due to Derdziński [14]). Fang, Man
and Zhang [16] showed that any non-compact shrinker with bounded scalar
curvature has finite topological type. Chen and Zhou [6] confirmed that any
non-compact shrinker has at most Euclidean volume growth. Munteanu and Wang
[34] proved that any non-compact shrinker has at least linear volume growth.

Haslhofer and Müller [20, 21] proved a Cheeger-Gromov compactness theorem of
shrinkers with a lower bound on their entropy and a local integral Riemann
bound. Li, Li and Wang [26] gave a structure theory for non-collapsed
shrinkers, which was further developed by Huang, Li and Wang [23]. For the
$4$-dimensional case, Li and Wang [31] proved that any nontrivial flat cone
cannot be approximated by smooth shrinkers with bounded scalar curvature and
Harnack inequality under the pointed-Gromov-Hausdorff topology. Huang [22]
applied the strategy of Cheeger-Tian [9] in Einstein manifolds and proved an
$\epsilon$-regularity theorem for $4$-dimensional shrinkers, confirming a
conjecture of Cheeger-Tian [9].

Recently, Li and Wang [32] obtained a sharp logarithmic Sobolev inequality,
the Sobolev inequality, heat kernel estimates, the no-local-collapsing
theorem, the pseudo-locality theorem, etc. on complete shrinkers, which can be
further extended to the other geometric inequalities, such as Nash
inequalities, Faber-Krahn inequalities and Rozenblum-Cwikel-Lieb inequalities
in [43]. For more function theory on shrinkers, the interested readers are
referred to [18, 33, 35, 36, 40, 44, 45] and references therein.

On a manifold $M$, a set $E$ is called an end with respect to a compact set
$\Omega\subset M$, if it is an unbounded connected component of
$M\backslash\Omega$. The number of ends with respect to $\Omega$, denoted by
$N_{\Omega}(M)$, is the number of unbounded connected components of
$M\backslash\Omega$. If $\Omega_{1}\subset\Omega_{2}$, then
$N_{\Omega_{1}}(M)\leq N_{\Omega_{2}}(M)$. Hence if $\Omega_{i}$ is a compact
exhaustion of $M$, then $N_{\Omega_{i}}(M)$ is a nondecreasing sequence. If
this sequence is bounded, then we say that $M$ has finitely many ends. In this
case, the number of ends of $M$ is defined by

$N(M)=\lim_{i\to\infty}N_{\Omega_{i}}(M).$

Obviously, the number of ends is independent of the compact exhaustion
$\\{\Omega_{i}\\}$. Ends of manifolds are related to the geometry and topology
of manifolds; the interested reader may refer to the book [27].

The Cheeger-Gromoll’s splitting theorem [8] indicates that any complete non-
compact manifold with nonnegative Ricci curvature has at most two ends. Later,
Cai [3] and Li-Tam [28] independently proved that any manifold with
nonnegative Ricci curvature outside a compact set has at most finitely many
ends (see also Liu [25]); see [41] for an extension to smooth metric measure
spaces. Cai’s approach is pure geometrical, strongly depending on a local
version of Cheeger-Gromoll’s splitting theorem, while Li-Tam’s proof is
analytic in nature by taking full advantage of the harmonic function theory.
Liu’s proof is also geometrical, not adapting the local splitting theorem but
using various volume comparisons. At present, an interesting question of
whether the Cheeger-Gromoll splitting theorem holds on any complete non-
compact shrinker still remains unresolved. In the next attempt to consider the
number of ends, it is natural to ask

Question. Does any a complete non-compact shrinker have finitely many ends?

For the Kähler case, Munteanu and Wang [36] proved that any Kähler shrinker
has only one end. For the Riemannian case, Munteanu, Schulze and Wang [33]
showed that the number of ends is finite when the scalar curvature satisfies
certain scalar curvature integral at infinity. Their proof depends on the Li-
Tam’s analytic theory [28]. In this paper, we use a geometric covering
argument and prove that

######  Theorem 1.1.

The number of ends on $n$-dimensional complete non-compact shrinker with the
scalar curvature

$\mathrm{S}\geq\delta$

for some constant $\delta\geq 0$ is at most polynomial growth with degree
$2(n-\delta)$.

######  Remark 1.2.

From (2.8) in Section 2, we will see that $\mathrm{S}\geq\delta$ implies
$\delta\leq n/2$ on shrinkers. From Remark 2.6, we have that the point-wise
assumption $\mathrm{S}\geq\delta$ can be replaced by a lower of the average
scalar curvature over the level set $\\{f<r\\}:=\left\\{x\in
M|f(x)<r\right\\}$ for any $r>0$, that is,

$\frac{1}{\int_{\\{f<r\\}}dv}\int_{\\{f<r\\}}\mathrm{S}\,dv\geq\delta$

for any $r>0$. If the scalar curvature also has a uniformly upper bound, then
the degree $2(n-\delta)$ in theorem can be reduced to $n-2\delta$; see Remark
3.4.

The following condition introduced in [29] will play an important role in this
paper.

######  Definition 1.3.

A Riemannian manifold $(M,g)$ has volume comparison condition if there exists
a constant $\eta>0$ such that for all $r\geq r_{0}$ for some $r_{0}>0$, and
all $x\in\partial B(q,r)$,

$\mathrm{Vol}(B(q,r))\leq\eta\,\mathrm{Vol}\left(B(x,\frac{r}{16})\right),$

where $\mathrm{Vol}(B(q,r))$ is the volume of geodesic ball $B(q,r)$ of radius
$r$ with center at a fixed point $q\in M$.

If the shrinker satisfies volume comparison condition, we prove that

######  Theorem 1.4.

Any complete non-compact shrinker with volume comparison condition must have
finitely many ends.

Many special cases of shrinkers satisfy volume comparison condition. The
detailed discussion can be referred to Section 4. Here we summarize some
results as follows:

(I) If a manifold satisfies volume doubling property, then it admits volume
comparison condition; see Proposition 4.2. Recall that $(M,g)$ is said to be
volume doubling property if

$\mathrm{Vol}(B(x,2r))\leq D\,\mathrm{Vol}(B(x,r))$

for any $x\in M$ and $r>0$, where $D$ is a fixed constant. Clearly, any
manifold with nonnegative Ricci curvature satisfies volume doubling property.

(II) If the asymptotic scalar curvature ratio of shrinker is finite, then such
shrinker has volume comparison condition; see Proposition 4.3. Given a point
$q\in(M,g)$, the asymptotic scalar curvature ratio ($\operatorname{ASCR}$) is
defined by

$\operatorname{ASCR}(g):=\underset{r(q,x)\to\infty}{\lim\sup}\,\mathrm{S}(x)\cdot
r(q,x)^{2},$

$r(q,x)$ is the distance function from $q$ to $x$. It is easy to see that
$\operatorname{ASCR}(g)$ is independent of the base point $q$. Chow, Lu and
Yang [12] proved that a non-compact non-flat shrinker has at most quadratic
scalar curvature decay. Therefore, except the flat shrinker, our assumption is
in fact equivalent to $\operatorname{ASCR}(g)=c_{0}$ for some constant
$c_{0}>0$, which takes place at least for the asymptotically conical shrinker
[24].

(III) If a family of average of scalar curvature integral has at least
quadratic decay of radius, precisely, for a infimum point $p\in M$ of $f$,
there exists a constant $c_{1}>0$ such that

$\frac{r^{2}}{\mathrm{Vol}\left(B(x,r)\right)}\int_{B(x,r)}\mathrm{S}\,dv\leq
c_{1}$

for all $r>0$ and all $x\in\partial B(p,r)$, then such shrinker has volume
comparison condition; see Proposition 4.5. The class of average scalar
curvature integral can be regarded as some energy functions of scalar
curvature, which is derived from Li-Wang (logarithmic) Sobolev inequalities;
see Lemma 2.7 or Lemma 2.8.

(IV) If a complete non-compact shrinker $(M,g,f)$ with a infimum point $p\in
M$ of $f$ satisfies

$\mathrm{Vol}\left(B(x,\frac{r}{16})\right)\geq c_{2}\,r^{n}$

for all $r>0$ and all $x\in\partial B(p,r)$, where $c_{2}$ is a positive
constant, then such shrinker satisfies volume comparison condition; see
Corollary 4.8. This condition can be regarded as a family of Euclidean volume
growth, which seems to be stronger than the positive asymptotic volume ratio;
see the end of Section 4 for the detailed discussion.

Besides, Li and Tam [29] proved that if a Riemannian manifold with each end
has asymptotically non-negative sectional curvature, then it satisfies the
volume comparison condition. Recall that $(M,g)$ has asymptotically non-
negative sectional curvature if there exists a point $q\in M$ and a continuous
decreasing function $\tau:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
$\int^{+\infty}_{0}t\tau(t)\,dt<\infty$ and the sectional curvature $K(x)$ at
any point $x\in M$ satisfies $K(x)\geq-\tau(r(q,x))$, where $r(q,x)$ is a
distance function from $q$ to $x$. Li and Tam [29] also proved that if a
Riemannian manifold with finite first Betti number has nonnegative Ricci
curvature outside a compact set, then it satisfies volume comparison
condition. We refer the readers to [29] for further related discussions.

Different from Munteanu-Schulze-Wang’s analytic argument, our proof of Theorem
1.1 is geometrical, which stems from Liu’s approach [25], but we have a major
obstacle due to the lack of volume comparison at different points and radii.
For manifolds with nonnegative Ricci curvature (outside a compact set), such
properties come from classical relative volume comparisons. With these
comparisons, Liu was able to get a ball covering property of manifolds with
nonnegative Ricci curvature (outside a compact set) and hence proved finitely
many ends. But for shrinkers, we only prove relative volume comparisons about
geodesic balls with center at a base point; see Theorem 2.3 in Section 2. We
do not know if they could hold for geodesic balls centered at different
points. To overcome this difficulty, we extend Cao-Zhou upper volume bound [6]
(further development by Munteanu-Wang [34], Zhang [47]) to a more precise
statement; see Lemma 2.5; while we generalize the Li-Wang lower volume bound
[32]; see Lemmas 2.7 and 2.8. Applying these upper and lower volume estimates,
we could get a weak volume comparison condition; see Proposition 3.1 in
Section 3. This proposition is enough to produce a weak ball covering property
(see Theorem 3.2 in Section 3) and finally leads to Theorem 1.1. In
particular, when the shrinker satisfies volume comparison condition, we can
prove Theorem 1.4 in a similar spirit.

The rest of paper is organized as follows. In Section 2, we will prove upper
and lower relative volume comparisons of the shrinker in geodesic balls with
center at a base point. We also give some upper and lower volume estimates. In
Section 3, we will use volume comparisons of Section 2 to prove a weak ball
covering property. Then we apply the weak ball covering property to prove
Theorem 1.1. In Section 4, when the shrinker satisfies volume comparison
condition, we will prove Theorem 1.4 by adapting the argument of Theorem 1.1.
Meanwhile, we will provide various sufficient condition to ensure volume
comparison condition. In Section 5, we will apply the ball covering property
of shrinkers to study the diameter growth of ends.

In the whole of this paper, we let $c(n)$ denote a constant depending only on
dimension $n$ of shrinker $(M,g,f)$ whose value may change from line to line.

Acknowledgements. The author thanks Yu Li for his valuable suggestions and
stimulating discussions, which improves some results in this paper. The author
also thanks Guoqiang Wu for his helpful comments on an earlier version of this
paper. Finally the author sincerely thanks Professor Ovidiu Munteanu for
valuable comments and pointing out a mistake of an earlier version of the
paper.

##  2\. Volume comparison

In this section, we will discuss upper and lower relative volume comparisons
of shrinker about geodesic balls with center at a base point. We will also
discuss upper and lower volume estimates of shrinkers.

Recall that the potential $f$ of shrinker is uniformly equivalent to the
distance function squared. Precisely, the following sharp estimate was
established originally due to Cao-Zhou [6] and later improved by Haslhofer-
Müller [20]; see also Chow et al. [11].

######  Lemma 2.1.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker satisfying
(1.1) and (1.2). For any point $q\in M$, $f$ satisfies

$\frac{1}{4}\left[\left(r(q,x)-2\sqrt{f(q)}-4n+\frac{4}{3}\right)_{+}\right]^{2}\leq
f(x)\leq\frac{1}{4}\left(r(q,x)+2\sqrt{f(q)}\right)^{2}$

for all $x\in M$, where $r(q,x)$ denotes a distance function from $q$ to $x$.

Moreover, there exists a point $p\in M$ where $f$ attains its infimum in $M$
such that $f(p)\leq n/2$; meanwhile $f$ has a simple estimate

$\frac{1}{4}\left[\big{(}r(p,x)-5n\big{)}_{+}\right]^{2}\leq
f(x)\leq\frac{1}{4}\left(r(p,x)+\sqrt{2n}\right)^{2}$

for all $x\in M$. Here $a_{+}=\max\\{a,0\\}$ for $a\in\mathbb{R}$.

Chen [10] proved that the scalar curvature of shrinkers has a lower bound

$\mathrm{S}\geq 0.$

Pigola, Rimoldi and Setti [38] showed that the scalar curvature $\mathrm{S}$
is strictly positive, unless $(M,g,f)$ is the Gaussian shrinking Ricci
soliton. By Lemma 2.1 and (1.2), the scalar curvature naturally has an upper
bound

$\mathrm{S}(x)\leq\frac{1}{4}\left(r(p,x)+\sqrt{2n}\right)^{2}$ (2.1)

for all $x\in M$. This upper bound will be used in this paper.

Recently, Li and Wang [32] applied the monotonicity of Perelman’s functional
along Ricci flow and the invariance of Perelman’s functional under
diffeomorphism actions to obtain (logarithmic) Sobolev inequalities on
complete shrinkers.

######  Lemma 2.2.

Let $(M,g,f)$ be an $n$-dimensional shrinker satisfying (1.1), (1.2) and
(1.3). Then for any $\varphi\in C^{\infty}_{0}(M)$ with
$\int_{M}\varphi^{2}dv=1$ and any $\tau>0$,

$\mu+n+\frac{n}{2}\ln(4\pi)\leq\tau\int_{M}\left(4|\nabla\varphi|^{2}+\mathrm{S}\varphi^{2}\right)dv-\int_{M}\varphi^{2}\ln\varphi^{2}dv-\frac{n}{2}\ln\tau.$
(2.2)

Moreover, for any $u\in C^{\infty}_{0}(M)$,

$\left(\int_{M}u^{\frac{2n}{n-2}}dv\right)^{\frac{n-2}{n}}\leq
c(n)e^{-\frac{2\mu}{n}}\int_{M}\left(4|\nabla
u|^{2}+\mathrm{S}u^{2}\right)dv.$ (2.3)

The above inequalities are useful for understanding the geometry and topology
for shrinkers; see some recent works [32], [33], [42] and [43]. In the
following sections, we will apply them to study the volume growth of
shrinkers.

We start to discuss some applications of the above lemmas. First, applying
Lemma 2.1, we can provide a relative volume comparison with center at any a
base point for large geodesic balls. Similar volume comparison was ever
considered by Carrillo and Ni [7] under some extra assumption.

######  Theorem 2.3.

Let $(M,g,f)$ be a shrinker satisfying (1.1). For any point $q\in M$,

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,r))}\leq
2\left(\frac{R+c}{r-c}\right)^{n}$

for all $R\geq r\geq 2\sqrt{n}+c$. In particular, for any $0<\alpha<1$,

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,\alpha R))}\leq
2\left(1+\frac{2}{\alpha}\right)^{n}$

for all $R\geq 2\alpha^{-1}(\sqrt{n}+c)$. Here $c:=2\sqrt{f(q)}+4n-4/3$.

###### Proof of Theorem 2.3.

The proof is essentially contained in the argument of Cao and Zhou [6], and we
include it for the completeness. Define

$\rho(x):=2\sqrt{f(x)}.$

By Lemma 2.1,

$r(q,x)-c\leq\rho(x)\leq r(q,x)+c,$

where $c=2\sqrt{f(q)}+4n-4/3$. Denote by

$D(r):=\\{x\in M|\rho(x)<r\\}\quad\mathrm{and}\quad V(r):=\int_{D(r)}dv.$

We trace (1.1) and get

$\mathrm{S}+\Delta f=\frac{n}{2}.$

Integrating this equality and using some properties on shrinkers, Cao and Zhou
[6] established the following interesting equality:

$nV(r)-rV^{\prime}(r)=2\int_{D(r)}\mathrm{S}\,dv-2\int_{\partial
D(r)}\frac{\mathrm{S}}{|\nabla f|}dv.$ (2.4)

Letting

$\chi(r):=\int_{D(r)}\mathrm{S}\,dv,$

then by the co-area formula, (2.4) can be rewritten as

$nV(r)-rV^{\prime}(r)=2\chi(r)-\frac{4}{r}\chi^{\prime}(r),$

that is,

$(r^{-n}V(r))^{\prime}=4r^{-n-2}\chi^{\prime}(r)-2r^{-n-1}\chi(r).$

Integrating this from $r$ to $R$ yields

$\displaystyle R^{-n}V(R)-r^{-n}V(r)$
$\displaystyle=4R^{-n-2}\chi(R)-4r^{-n-2}\chi(r)$
$\displaystyle\quad+2\int^{R}_{r}t^{-n-3}\chi(t)\left(2(n+2)-t^{2}\right)dt.$

For the last term of the above equality, since $\chi(t)$ is positive and
increasing in $t$, then for any $R\geq r\geq\sqrt{2(n+2)}$, we have

$\displaystyle 2\int^{R}_{r}t^{-n-3}\chi(t)\left(2(n+2)-t^{2}\right)dt$
$\displaystyle\leq 2\chi(r)\int^{R}_{r}t^{-n-3}\left(2(n+2)-t^{2}\right)dt$
$\displaystyle=2\chi(r)\left(-2t^{-n-2}+\frac{t^{-n}}{n}\right){\bigg{|}}^{R}_{r}$
$\displaystyle=-4R^{-n-2}\chi(r)+4r^{-n-2}\chi(r)+\frac{2}{n}\chi(r)(R^{-n}-r^{-n}).$

Hence,

$R^{-n}V(R)-r^{-n}V(r)\leq
4R^{-n-2}\left(\chi(R)-\chi(r)\right)+\frac{2}{n}\chi(r)(R^{-n}-r^{-n})$

for $R\geq r\geq\sqrt{2(n+2)}$. Therefore,

$V(R)\leq(r^{-n}V(r))R^{n}+4R^{-2}\chi(R)$ (2.5)

for all $R\geq r\geq\sqrt{2(n+2)}$.

On the other hand, for any $R\geq 2\sqrt{n}$, we have

$4R^{-2}\chi(R)\leq 2nR^{-2}V(R)\leq\frac{1}{2}V(R).$ (2.6)

Substituting (2.6) into (2.5) gives

$\frac{V(R)}{V(r)}\leq 2\left(\frac{R}{r}\right)^{n}$

for any $R\geq r\geq 2\sqrt{n}(\geq\sqrt{2(n+2)})$. This implies

$\frac{V(R+c)}{V(r-c)}\leq 2\left(\frac{R+c}{r-c}\right)^{n}$

for $R\geq r\geq 2\sqrt{n}+c$, where $c:=2\sqrt{f(q)}+4n-4/3$. We also notice

$\mathrm{Vol}(B(q,R))\leq V(R+c)\quad\mathrm{and}\quad\mathrm{Vol}(B(q,r))\geq
V(r-c)$

for any $R\geq 0$ and $r\geq c$. Therefore,

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,r))}\leq
2\left(\frac{R+c}{r-c}\right)^{n}$

for $R\geq r\geq 2\sqrt{n}+c$, which proves the first part of theorem.

In particular, we choose $r=\alpha R$, where $0<\alpha<1$ and the above
estimate becomes

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,\alpha R))}\leq
2\left(\frac{R+c}{\alpha R-c}\right)^{n}$

for $R\geq\alpha^{-1}(2\sqrt{n}+c)$. Furthermore, we let $\alpha
R-c>\frac{\alpha}{2}R$, that is, $R\geq 2\alpha^{-1}c$, then

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,\alpha R))}\leq
2\left(1+\frac{2}{\alpha}\right)^{n}$

for $R\geq 2\alpha^{-1}(\sqrt{n}+c)$. This finishes the second part of
theorem. ∎

Second, following the argument of [6], we can apply Lemma 2.1 to give a
reverse relative volume comparison.

######  Theorem 2.4.

Let $(M,g,f)$ be a shrinker with a base point $q\in M$ satisfying (1.1). If
the scalar curvature $\mathrm{S}\leq\sigma$ for some constant $0<\sigma<n/2$,
then

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,r))}\geq\left(\frac{R-c}{r+c}\right)^{n-2\sigma}$

for all $R\geq r+2c$ and $r>0$, where $c:=2\sqrt{f(q)}+4n-4/3$.

###### Proof of Theorem 2.4.

By (2.4), $\mathrm{S}\geq 0$ and our curvature assumption
$\mathrm{S}\leq\sigma$, we have

$(n-2\sigma)V(t)\leq tV^{\prime}(t)$

for any $t\geq 0$. Integrating this inequality from $r$ to $R$, we get

$\frac{V(R)}{V(r)}\geq\left(\frac{R}{r}\right)^{n-2\sigma}$

for any $R\geq r>0$. We also see that

$\mathrm{Vol}(B(q,r))\leq V(r+c)\quad\mathrm{and}\quad\mathrm{Vol}(B(q,R))\geq
V(R-c)$

for any $r\geq 0$ and $R\geq c$. Therefore,

$\frac{\mathrm{Vol}(B(q,R))}{\mathrm{Vol}(B(q,r))}\geq\frac{V(R-c)}{V(r+c)}\geq\left(\frac{R-c}{r+c}\right)^{n-2\sigma}$

for any $R\geq r+2c$ and $r>0$. ∎

Next we will discuss some volume estimates of geodesic balls on shrinkers. The
sharp upper volume estimate was first proved by Cao-Zhou (see Theorem 1.2 in
[6]), later a explicit coefficient was stated by Munteanu-Wang (see Theorem
1.4 in [35]) by using a delicate generalized Laplace comparison. Furthermore,
Zhang [47] proved a sharp quantitative upper volume of the shrinker with
scalar curvature bounded below; see also [11]. In the following we will
improve previous upper volume estimates when $r$ is not large.

######  Lemma 2.5.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker satisfying
(1.1), (1.2) and (1.3). For any point $q\in M$ and for all $r\geq 0$,

$\mathrm{Vol}(B(q,r))\leq c(n)e^{f(q)}r^{n}.$

Moreover, if the scalar curvature $\mathrm{S}\geq\delta$ for some constant
$\delta\geq 0$, then

$\mathrm{Vol}(B(q,r))\leq
c(n)e^{f(q)}e^{-\frac{\delta}{r^{2}}}\,r^{n-2\delta}$

for all $r\geq 2\sqrt{n+2}+c$, where $c:=2\sqrt{f(q)}+4n-4/3$; in particular,
if $p\in M$ is a infimum point of $f$, then

$\mathrm{Vol}(B(p,r))\leq c(n)e^{-\frac{\delta}{r^{2}}}\,r^{n-2\delta}$

for all $r\geq c(n)$.

###### Proof of Lemma 2.5.

The first estimate is Theorem 1.4 in [35]. So we only need to prove the second
and third estimates. We remark that the second estimate with a rough
coefficient has been proved by Zhang [47] (see also [11]). Here, we need to
figure out the accurate coefficients, which plays a key role in our
application.

For convenience of our computation, we adapt the notations of [47] (see also
[11]), which are sight different from those in [6]. For any $t\in\mathbb{R}$,
let

$\\{f<t\\}:=\left\\{x\in M|f(x)<t\right\\}$

and define

$\mathscr{V}(t):=\int_{\\{f<t\\}}dv\quad\mathrm{and}\quad\mathscr{R}(t):=\int_{\\{f<t\\}}\mathrm{S}dv.$

Notice that for any $q\in M$, $f(x)$ satisfies

$\frac{1}{4}\left[(r(x,q)-c)_{+}\right]^{2}\leq
f(x)\leq\frac{1}{4}\left(r(x,q)+c\right)^{2}$

where $c:=2\sqrt{f(q)}+4n-4/3$. Therefore, if $r\geq c$, then

$\left\\{f<\frac{1}{4}(r-c)^{2}\right\\}\subset
B(q,r)\subset\left\\{f<\frac{1}{4}(r+c)^{2}\right\\}$

and hence

$\mathscr{V}\left(\frac{1}{4}(r-c)^{2}\right)\leq\mathrm{Vol}(B(q,r))\leq\mathscr{V}\left(\frac{1}{4}(r+c)^{2}\right).$
(2.7)

Using present notations, (2.4) can be rewritten as

$0\leq\frac{n}{2}\mathscr{V}(t)-\mathscr{R}(t)=t\mathscr{V}^{\prime}(t)-\mathscr{R}^{\prime}(t).$
(2.8)

For any $t>0$, let

$\operatorname{P}(t):=\frac{\mathscr{V}(t)}{t^{\frac{n}{2}}}-\frac{\mathscr{R}(t)}{t^{\frac{n}{2}+1}}\quad\mathrm{and}\quad\operatorname{N}(t):=\frac{\mathscr{R}(t)}{t\mathscr{V}(t)}.$

Then (2.8) implies

$\displaystyle\operatorname{P}^{\prime}(t)$
$\displaystyle=-\left(1-\frac{n+2}{2t}\right)\frac{\mathscr{R}(t)}{t^{\frac{n}{2}+1}}$
(2.9)
$\displaystyle=-\frac{\left(1-\frac{n+2}{2t}\right)\operatorname{N}(t)}{1-\operatorname{N}(t)}\operatorname{P}(t).$

This implies $\operatorname{P}(t)$ is decreasing and

$\left(1-\frac{n}{2t}\right)\frac{\mathscr{V}(t)}{t^{\frac{n}{2}}}\leq\operatorname{P}(t)\leq\frac{\mathscr{V}(t)}{t^{\frac{n}{2}}}$
(2.10)

for $t\geq n/2+1$, where we used $\frac{\mathscr{R}(t)}{\mathscr{V}(t)}\leq
n/2$. Integrating equality (2.9) gives

$\operatorname{P}(t)=\operatorname{P}(n+2)e^{-\int_{n+2}^{t}\frac{\left(1-\frac{n+2}{2\tau}\right)\operatorname{N}(\tau)}{1-\operatorname{N}(\tau)}d\tau}$

for all $t\geq n+2$. Since $\mathrm{S}\geq\delta$, then
$\operatorname{N}(\tau)\geq\delta/\tau$. Also noticing that
$\frac{\operatorname{N}(\tau)}{1-\operatorname{N}(\tau)}$ is increasing in
$\operatorname{N}(\tau)$, hence the above equality can be estimated by

$\displaystyle\operatorname{P}(t)$
$\displaystyle\leq\operatorname{P}(n+2)e^{-\int_{n+2}^{t}\left(1-\frac{n+2}{2\tau}\right)\frac{\delta}{\tau-\delta}d\tau}$
$\displaystyle\leq\operatorname{P}(n+2)e^{-\int_{n+2}^{t}\left(1-\frac{n+2}{2\tau}\right)\frac{\delta}{\tau}d\tau}$
$\displaystyle=\operatorname{P}(n+2)(n+2)^{\delta}e^{\frac{\delta}{2}}e^{-\frac{n+2}{2t}\delta}\,t^{-\delta}$

for all $t\geq n+2$. Combining this with (2.10),

$\mathscr{V}(t)\leq
c(n)\operatorname{P}(n+2)e^{-\frac{n+2}{2t}\delta}\,t^{\frac{n}{2}-\delta}$
(2.11)

for all $t\geq n+2$, where we used $\delta<n/2$. By Lemma 2.1, since
$B(q,2\sqrt{t}-c)\subset\\{f<t\\}$, where $c:=2\sqrt{f(q)}+4n-4/3$, combining
(2.7), it follows that

$\mathrm{Vol}\left(B(q,2\sqrt{t}-c)\right)\leq\mathscr{V}(t)$

for $t\geq c^{2}/4$. Combining this with (2.11) yields

$\mathrm{Vol}\left(B(q,2\sqrt{t}-c)\right)\leq
c(n)\operatorname{P}(n+2)e^{-\frac{n+2}{2t}\delta}\,t^{\frac{n}{2}-\delta}$

for all $t\geq n+2+c^{2}/4$, so that

$\mathrm{Vol}\left(B(q,r)\right)\leq
c(n)\operatorname{P}(n+2)e^{-\frac{2(n+2)\delta}{(r+c)^{2}}}\left(\frac{r+c}{2}\right)^{n-2\delta}$

for all $r\geq 2\sqrt{n+2}$. Noticing that

$\operatorname{P}(n+2)\leq\frac{\mathscr{V}(n+2)}{(n+2)^{\frac{n}{2}}}\quad\mathrm{and}\quad\mathscr{V}(n+2)\leq\mathrm{Vol}\left(B(q,2\sqrt{n+2}+c)\right),$

then

$\mathrm{Vol}\left(B(q,r)\right)\leq
c(n)\mathrm{Vol}\left(B(q,2\sqrt{n+2}+c)\right)e^{-\frac{\delta}{r^{2}}}\,r^{n-2\delta}$

for all $r\geq 2\sqrt{n+2}+c$. Therefore the second estimate follows by
applying the first estimate of Lemma 2.5

$\displaystyle\mathrm{Vol}\left(B(q,2\sqrt{n+2}+c)\right)$ $\displaystyle\leq
c(n)e^{f(q)}(2\sqrt{n+2}+c)^{n}$ $\displaystyle\leq c(n)e^{f(q)},$

where we used a fact that

$e^{f(q)}(2\sqrt{n+2}+c)^{n}\leq
c(n)e^{f(q)}f(q)^{n/2}\leq\widetilde{c}(n)e^{f(q)}.$

Finally, the third estimate of the lemma follows by the second estimate and a
basic fact $f(p)\leq n/2$. ∎

######  Remark 2.6.

The above argument shows that the point-wise condition of scalar curvature in
Lemma 2.5 can be replaced by a condition of the average scalar curvature over
the level set $\\{f<r\\}$, that is,

$\frac{1}{\int_{\\{f<r\\}}dv}\int_{\\{f<r\\}}\mathrm{S}\,dv\geq\delta$

for any $r>0$. This is because we only used
$\frac{\mathscr{R}(t)}{\mathscr{V}(t)}\geq\delta$ in the proof of Lemma 2.5.

For a lower volume estimate, a sharp version was proved by Munteanu-Wang (see
Theorem 1.6 in [34] or Theorem 1.4 in [35]). But coefficients of these
estimates all depend on a base point, which will be trouble in dealing with
our issue. So in the following we shall adopt a Li-Wang’s local lower volume
estimate for any base point, which comes from the Sobolev inequality (see
Theorem 23 in [32]). This estimate is more useful when $r$ is sufficiently
large.

######  Lemma 2.7.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker satisfying
(1.1), (1.2) and (1.3). For any point $q\in M$ and for any $r>0$,

$\frac{\mathrm{Vol}(B(q,r))}{r^{n}}\left[1+\sup_{s\in[0,r]}\frac{s^{2}\int_{B(q,s)}\mathrm{S}\,dv}{\mathrm{Vol}(B(q,s))}\right]^{n/2}\geq
c(n)e^{\mu}.$

In particular, if the scalar curvature $\mathrm{S}\leq\Lambda$ for some
constant $\Lambda\geq 0$ in $B(q,r)\subset M$, then

$\frac{\mathrm{Vol}(B(q,r))}{r^{n}}(1+\Lambda r^{2})^{n/2}\geq c(n)e^{\mu}.$

###### Proof of Lemma 2.7.

The argument is essentially the same as the proof of Theorem 23 in [32]. For
the reader’s convience, we provide the detailed proof. For a base point $q\in
M$, we choose $r_{0}\in[0,r]$ such that

$\inf_{s\in[0,r]}\frac{\mathrm{Vol}(B(q,s))}{s^{n}}$

is attained at $r_{0}$. Below we discuss two cases $r_{0}=0$ and $r_{0}>0$
separately.

Case one: $r_{0}=0$. We have

$\mathrm{Vol}(B(q,r))\geq\omega_{n}r^{n},$

where $\omega_{n}$ is the volume of the unit Euclidean $n$-ball. Now we claim
that $\mu\leq 0$. Indeed, for $\tau\to 0+$, we have that
$(M^{n},p,\tau^{-1}g)$ converges to Euclidean space $(\mathbb{R}^{n},0,g_{E})$
smoothly in the Cheeger-Gromov sense. By Lemma 3.2 of [30], we know

$\underset{\tau\to 0+}{\lim\sup}\,\mu(g,\tau)=\underset{\tau\to
0+}{\lim\sup}\,\mu(\tau^{-1}g,1)\leq\mu(g_{E},1)=0.$

Also, since $\mu(g,\tau)\geq\mu(g,1)=\mu$ for each $\tau\in(0,1)$ by Lemma 15
in [32], then the claim $\mu\leq 0$ follows. Hence the estimate of Case one
follows.

Case two: $r_{0}>0$. Let $\phi:\mathbb{R}\to[0,1]$ be a smooth function such
that $\phi(t)=1$ on $(-\infty,1/2]$, $\phi(t)=0$ on $[1,+\infty)$ and
$|\phi^{\prime}|\leq 2$ on $[0,\infty)$. For any point $q\in M$, let

$u(x):=\phi\left(\frac{r(q,x)}{r_{0}}\right).$

Clearly, $u$ is supported in $B(q,r_{0})$ and it satisfies $|\nabla u|\leq
2r_{0}^{-2}$. We substitute the above special function $u$ into (2.3) of Lemma
2.2 and get

$\displaystyle\mathrm{Vol}\left(B(q,\frac{r_{0}}{2})\right)^{\frac{n-2}{n}}$
$\displaystyle\leq c(n)e^{-\frac{2\mu}{n}}\int_{B(q,r_{0})}\left(4|\nabla
u|^{2}+\mathrm{S}u^{2}\right)dv$ $\displaystyle\leq
c(n)e^{-\frac{2\mu}{n}}\frac{\mathrm{Vol}(B(q,r_{0}))}{r^{2}_{0}}\left[1+\frac{r^{2}_{0}\int_{B(q,r_{0})}\mathrm{S}\,dv}{\mathrm{Vol}(B(q,r_{0})}\right].$

From the choice of $r_{0}$, we see that

$\mathrm{Vol}\left(B(q,\frac{r_{0}}{2})\right)\geq\frac{\mathrm{Vol}(B(q,r_{0}))}{2^{n}}.$

Combining the above two inequalities yields

$\frac{\mathrm{Vol}(B(q,r_{0}))}{r^{n}_{0}}\left[1+\frac{r^{2}_{0}\int_{B(q,r_{0})}\mathrm{S}\,dv}{\mathrm{Vol}(B(q,r_{0}))}\right]^{n/2}\geq
c(n)e^{\mu}.$

According to the definition of $r_{0}$, we have

$\frac{\mathrm{Vol}(B(q,r))}{r^{n}}\geq\frac{\mathrm{Vol}(B(q,r_{0}))}{r^{n}_{0}}.$

Combining the above two inequalities gives the conclusion of Case two. ∎

At the end of this section, we give another version of lower volume estimate
by using the logarithmic Sobolev inequality (2.2), which is sharper than Lemma
2.7 when $r$ is not sufficiently large.

######  Lemma 2.8.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker satisfying
(1.1), (1.2) and (1.3). For any point $q\in M$,

$\mu+n+\frac{n}{2}\ln(4\pi)+16(1-2\cdot 5^{n})\leq+2\cdot
5^{n}r^{2}\frac{\int_{B(q,r)}\mathrm{S}\,dv}{\mathrm{Vol}(B(q,r))}+\ln\frac{\mathrm{Vol}(B(q,r))}{r^{n}}$
(2.12)

for any $r\geq 4(\sqrt{n}+c)$, where $c:=2\sqrt{f(q)}+4n-4/3$.

###### Proof of Lemma 2.8.

Let $\phi:[0,\infty)\to[0,1]$ be a smooth cut-off function supported in
$[0,1]$ such that $\phi(t)=1$ on $[0,1/2]$ and $|\phi^{\prime}|\leq 2$ on
$[0,\infty)$. For any $q\in M$ and any $r>0$, let

$\varphi(x):=e^{-\theta/2}\phi\left(\frac{r(q,x)}{r}\right),$

where $\theta$ is some constant determined by condition
$\int_{M}\varphi^{2}dv=1$. Clearly, $\varphi$ is supported in $B(p,r)$ and it
satisfies $|\nabla\varphi|\leq 2r^{-1}\cdot e^{-\theta/2}$. Moreover, $\theta$
satisfies

$\mathrm{Vol}\left(B(q,\frac{r}{2})\right)\leq
e^{\theta}\int_{M}\varphi^{2}dv=e^{\theta}$

and

$e^{\theta}=e^{\theta}\int_{M}\varphi^{2}dv=\int_{M}\phi^{2}\left(\frac{r(q,x)}{r}\right)dv\leq\mathrm{Vol}(B(q,r)).$

Now we shall substitute the above cut-off function $\varphi$ into Lemma 2.2 to
simplify the inequality (2.2).

First, by the definition of $\varphi$ and lower bound of $e^{\theta}$, we have

$\displaystyle 4\tau\int_{M}|\nabla\varphi|^{2}dv$
$\displaystyle=4\tau\int_{B(q,r)\backslash
B(q,\frac{r}{2})}|\nabla\varphi|^{2}dv$ (2.13)
$\displaystyle\leq\frac{16\tau}{r^{2}}\left[\mathrm{Vol}(B(q,r))-\mathrm{Vol}\left(B(q,\frac{r}{2})\right)\right]e^{-\theta}$
$\displaystyle\leq\frac{16\tau}{r^{2}}\left[\frac{\mathrm{Vol}(B(q,r))}{\mathrm{Vol}\left(B(q,\frac{r}{2})\right)}-1\right]$
$\displaystyle\leq 16(2\cdot 5^{n}-1)\frac{\tau}{r^{2}}$

for all $r\geq 4(\sqrt{n}+c)$, where $c:=2\sqrt{f(q)}+4n-4/3$. In the last
inequality, we used Theorem 2.3 in the following form:

$\frac{\mathrm{Vol}(B(q,r))}{\mathrm{Vol}(B(q,\frac{r}{2}))}\leq 2\cdot 5^{n}$

for any $r\geq 4(\sqrt{n}+c)$.

Second, by the definition of $\varphi$ and the lower bound of $e^{\theta}$, we
have the estimate

$\displaystyle\tau\int_{M}\mathrm{S}\varphi^{2}dv$ $\displaystyle\leq\tau
e^{-\theta}\int_{B(q,r)}\mathrm{S}\,dv$ (2.14)
$\displaystyle\leq\frac{\tau}{\mathrm{Vol}\left(B(q,\frac{r}{2})\right)}\int_{B(q,r)}\mathrm{S}\,dv$
$\displaystyle\leq 2\cdot
5^{n}\frac{\tau\int_{B(q,r)}\mathrm{S}\,dv}{\mathrm{Vol}(B(q,r))}$

for all $r\geq 4(\sqrt{n}+c)$, where $c:=2\sqrt{f(q)}+4n-4/3$. Here we still
used Theorem 2.3 in the above last inequality.

Third, we will apply the Jensen’s inequality to estimate the term:
$-\int_{M}\varphi^{2}\ln\varphi^{2}dv$. Since smooth function $H(t):=-t\ln t$
is concave in $t>0$ and the Riemannian measure $dv$ is supported in $B(q,r)$,
by the following Jensen’s inequality

$\frac{\int H(\varphi^{2})dv}{\int dv}\leq
H\left(\frac{\int\varphi^{2}dv}{\int dv}\right)$

and the definition of $H(t)$, we obtain

$-\frac{\int_{B(q,r)}\varphi^{2}\ln\varphi^{2}dv}{\int_{B(q,r)}dv}\leq-\frac{\int_{B(q,r)}\varphi^{2}dv}{\int_{B(q,r)}dv}\ln\left(\frac{\int_{B(q,r)}\varphi^{2}dv}{\int_{B(q,r)}dv}\right).$

Since $\int_{B(q,r)}\varphi^{2}dv=1$, we further have a simple form

$-\int_{B(q,r)}\varphi^{2}\ln\varphi^{2}dv\leq\ln\mathrm{Vol}(B(q,r)).$

Therefore,

$\displaystyle-\int_{M}\varphi^{2}\ln\varphi^{2}dv$
$\displaystyle=-\int_{B(q,r)}\varphi^{2}\ln\varphi^{2}dv$ (2.15)
$\displaystyle\leq\ln\mathrm{Vol}(B(q,r)).$

Now we substitute (2.13), (2.14) and (2.15) into (2.2) and get that

$\mu+n+\frac{n}{2}\ln(4\pi)\leq 16(2\cdot 5^{n}-1)\frac{\tau}{r^{2}}+2\cdot
5^{n}\frac{\tau\int_{B(q,r)}\mathrm{S}\,dv}{\mathrm{Vol}(B(q,r))}+\ln\frac{\mathrm{Vol}(B(q,r))}{\tau^{\frac{n}{2}}}$

for any $\tau>0$ and for any $r\geq 4(\sqrt{n}+c)$, where
$c:=2\sqrt{f(q)}+4n-4/3$. Finally we let $\tau=r^{2}$ and the result follows.
∎

##  3\. Ends on a general shrinker

In this section, we will give a weak ball covering property depending on the
radius of a general shrinker without any assumption. Then we will apply the
weak ball covering to prove Theorem 1.1. With the help of Lemmas 2.5 and 2.7,
we first establish a weak volume comparison condition on shrinkers.

######  Proposition 3.1.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker with a
infimum point $p\in M$ of $f$ satisfying (1.1), (1.2) and (1.3). If the scalar
curvature $\mathrm{S}\geq\delta$ for some constant $\delta\geq 0$, then for
any $r\geq c(n)$ and for any $x\in\overline{B(p,2r)}$,

$\frac{\mathrm{Vol}(B(p,2r))}{\mathrm{Vol}\left(B(x,\tfrac{r}{8})\right)}\leq
c(n)e^{-\mu}r^{2(n-\delta)}.$

In addition, if the scalar curvature $\mathrm{S}\leq\sigma$ for some constant
$\sigma\geq\delta$ in $M$, then

$\frac{\mathrm{Vol}(B(p,2r))}{\mathrm{Vol}\left(B(x,\tfrac{r}{8})\right)}\leq
c(n)e^{-\mu}\sigma^{n/2}r^{n-2\delta}$

for any point $x\in M$ and for any $r\geq c(n)$.

###### Proof of Proposition 3.1.

By Lemma 2.5, we have

$\mathrm{Vol}(B(p,2r))\leq c(n)r^{n-2\delta}$ (3.1)

for any $r\geq c(n)$. On the other hand, the second estimate of Lemma 2.7
shows that

$\mathrm{Vol}(B(x,r))\geq c(n)e^{\mu}r^{n}(1+\Lambda r^{2})^{-n/2}$ (3.2)

with $\mathrm{S}\leq\Lambda$ in $B(x,r)\subset M$. Now we want to find an
upper bound of scalar curvature $\mathrm{S}$ in $B(x,r)$. From (2.1), we know

$\mathrm{S}(y)\leq\frac{1}{4}\left(r(y,p)+\sqrt{2n}\right)^{2}$

for all $y\in B(x,r)$. Since $x\in\overline{B(p,2r)}$, by the triangle
inequality, we further have

$\displaystyle\mathrm{S}(y)$
$\displaystyle\leq\frac{1}{4}\left(r(y,x)+r(x,p)+\sqrt{2n}\right)^{2}$
$\displaystyle\leq\frac{1}{4}\left(r+2r+\sqrt{2n}\right)^{2}$
$\displaystyle\leq\frac{1}{4}(3+\sqrt{2})^{2}r^{2}$

for all $y\in B(x,r)$ and for all $r\geq\sqrt{n}$. Substituting this into
(3.2) yields

$\mathrm{Vol}(B(x,r))\geq c(n)e^{\mu}r^{-n}$

for all $r\geq\sqrt{n}$. Combining this with (3.1) immediately yields the
first estimate of theorem.

Next we will prove the second part of theorem. Since we also assume that
$\mathrm{S}\leq\sigma$ for some constant $\sigma\geq\delta$ in $M$,
substituting this into (3.2), we have

$\mathrm{Vol}(B(x,r))\geq c(n)e^{\mu}\sigma^{-n/2}$

for any point $x\in M$ and any $r\geq 1$. Combining this with (3.1) gives the
second estimate. ∎

Inspired by Liu’s argument [25], we shall apply Proposition 3.1 to give a weak
ball covering property for sufficiently large balls in a shrinker without any
assumption. Our argument will be focused on a sufficiently large fixed radius.

######  Theorem 3.2.

Let $(M,g,f)$ be a complete non-compact shrinker with a infimum point $p\in M$
of $f$ satisfying (1.1), (1.2) and (1.3). If the scalar curvature
$\mathrm{S}\geq\delta$ for some constant $\delta\geq 0$, then for sufficiently
large $r\geq c(n)$, there exists

$N=c(n)e^{-\mu}r^{2(n-\delta)}$

such that we can find points $p_{1},\ldots,p_{k}\in
B(p,2r)\backslash\overline{B(p,r)}$, where $k=k(r)\leq N$, with

$\bigcup^{k}_{i=1}B\left(p_{i},\frac{r}{4}\right)\supset
B(p,2r)\backslash\overline{B(p,r)}.$

###### Proof of Theorem 3.2.

For a sufficiently large fixed $r\geq c(n)$, we let $k:=k(r)$ denote the
maximum number of disjoint geodesic balls of radius $r/8$ with centers
$p_{1},\ldots,p_{k}$ in $B(p,2r)\backslash\overline{B(p,r)}$. Obviously, in
this case,

$\bigcup^{k}_{i=1}B\left(p_{i},\frac{r}{4}\right)\supset
B(p,2r)\backslash\overline{B(p,r)}.$

See Figure 1 for a detailed description.

Figure 1.  Annulus is covered by small balls

Since $p_{i}\in B(p,2r)\backslash\overline{B(p,r)}$, we may let
$p_{i}\in\partial B(p,\beta_{i}r)$ for some $1<\beta_{i}<2$, where
$i=1,\ldots,k$. By the first estimate of Proposition 3.1, we have

$\displaystyle\mathrm{Vol}(B(p,\beta_{i}r))$
$\displaystyle\leq\mathrm{Vol}(B(p,2r))$ $\displaystyle\leq
c(n)e^{-\mu}r^{2(n-\delta)}\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)$

for $r\geq c(n)$. By Theorem 2.3, we also have

$\mathrm{Vol}(B(p,3r))\leq
2\left(1+\frac{6}{\beta_{i}}\right)^{n}\mathrm{Vol}\left(B(p,\beta_{i}r)\right)$

for $r\geq 2(\sqrt{n}+c)$, where $c:=2\sqrt{n/2}+4n-4/3$ and $i=1,\ldots,k$.
Combining the above two estimates, for each $i$,

$\mathrm{Vol}(B(p,3r))\leq
c(n)e^{-\mu}r^{2(n-\delta)}\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)$

for $r\geq c(n)$, where we used $1<\beta_{i}<2$. Summing the above $k$
inequalities, we get

$k(r)\mathrm{Vol}(B(p,3r))\leq
c(n)e^{-\mu}r^{2(n-\delta)}\sum^{k}_{i=1}\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)$

for $r\geq c(n)$. On the other hand, we easily see that

$\sum^{k}_{i=1}\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)\leq\mathrm{Vol}(B(p,3r)).$

Combining the above two estimates gives

$k(r)\leq c(n)e^{-\mu}r^{2(n-\delta)}$

for $r\geq c(n)$, which completes the proof. ∎

######  Remark 3.3.

In Theorem 3.2, if the scalar curvature also satisfies $\mathrm{S}\leq\sigma$
for some constant $\sigma\geq\delta$ in $M$, then for a sufficiently large
$r$, we can choose an $(n-2\delta)$-degree as follows:

$N=c(n)e^{-\mu}\sigma^{n/2}r^{n-2\delta}.$

The above weak ball covering property immediately implies Theorem 1.1.

###### Proof of Theorem 1.1.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker satisfying
(1.1), (1.2) and (1.3). Since the number of ends on the shrinker is
independent of the choice of the base point, we can choose a infimum point $p$
of $f$ as a base point in $M$.

Given a sufficiently large fixed number $r$, let

$N_{1}=c(n)e^{-\mu}r^{2(n-\delta)}$

as in Theorem 3.2. That is we can find points $p_{1},\ldots,p_{k}\in
B(p,2r)\backslash\overline{B(p,r)}$, where $k=k(r)\leq N_{1}$, with

$\bigcup^{k}_{i=1}B\left(p_{i},\frac{r}{4}\right)\supset
B(p,2r)\backslash\overline{B(p,r)}.$

Next we will prove Theorem 1.1 by a contradiction argument.

If Theorem 1.1 is not true, that is, the number of ends grows faster than
polynomial growth with degree $2(n-\delta)$, then for the above mentioned
sufficiently large $r$, there exists more than

$\widetilde{N}_{1}=c(n)e^{-\mu}r^{2(n-\delta)+\epsilon},$

where $\epsilon>0$ is any small constant, unbounded ends $E_{j}$ with respect
to $\overline{B(p,r)}$.

It is obvious that geodesic balls of radius $r/4$ with centers in different
components $E_{j}\cap B(p,2r)$ do not intersect. Thus we need at least
$\widetilde{N}_{1}$ geodesic balls of radius $r/4$ to cover the sets
$E_{j}\cap B(p,2r)\subset B(p,2r)\backslash\overline{B(p,r)}$, which
contradicts Theorem 3.2. ∎

######  Remark 3.4.

For Theorem 1.1, if the scalar curvature $\mathrm{S}\leq\sigma$ for some
constant $\sigma\geq\delta$ in $M$, then we can apply Remark 3.3 to the above
argument and get the same conclusion whereas the degree $2(n-\delta)$ of
polynomial growth can be reduced to $n-2\delta$.

##  4\. Ends with volume comparison condition

In this section we will discuss the finite number of ends when the shrinker
satisfies volume comparison condition. In this case we first give a ball
covering property, which is similar to the manifold case of nonnegative Ricci
curvature.

######  Theorem 4.1.

Let $(M,g,f)$ be an $n$-dimensional complete non-compact shrinker with a base
point $q\in M$ satisfying volume comparison condition. There exists a constant

$N=N(n,\eta)$

depending only on $n$ and $\eta$ such that for any $r\geq
2(\sqrt{n}+c)+r_{0}$, where $c:=2\sqrt{f(q)}+4n-4/3$, we can find
$p_{1},\ldots,p_{k}\in B(q,2r)\backslash\overline{B(q,r)}$, $k\leq N$, with

$\bigcup^{k}_{i=1}B\left(p_{i},\frac{r}{4}\right)\supset
B(q,2r)\backslash\overline{B(q,r)}.$

###### Proof of Theorem 4.1.

Let $k$ be the maximum number of disjoint geodesic balls of radius $r/8$ with
centers $p_{1},\ldots,p_{k}$ in $B(q,2r)\backslash\overline{B(q,r)}$. Here we
choose $r$ sufficiently large such that $r\geq 2(\sqrt{n}+c)+r_{0}$. Clearly,

$\bigcup^{k}_{i=1}B\left(p_{i},\frac{r}{4}\right)\supset
B(q,2r)\backslash\overline{B(q,r)}.$

Since $p_{i}\in B(q,2r)\backslash\overline{B(q,r)}$, we may let
$p_{i}\in\partial B(q,\beta_{i}r)$ for some constant $1<\beta_{i}<2$, where
$i=1,\ldots,k$. By the volume comparison condition, we have

$\displaystyle\mathrm{Vol}(B(q,\beta_{i}r))$
$\displaystyle\leq\eta\mathrm{Vol}\left(B(p_{i},\frac{\beta_{i}r}{16})\right)$
$\displaystyle\leq\eta\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)$

for all $r\geq r_{0}$. By Theorem 2.3, we see that

$\mathrm{Vol}(B(q,3r))\leq
2\left(1+\frac{6}{\beta_{i}}\right)^{n}\mathrm{Vol}(B(q,\beta_{i}r))$

for $r\geq 2\beta_{i}^{-1}(\sqrt{n}+c)$, where $i=1,\ldots,k$. Combining the
above two estimates, for each $i$, there exists a constant $C(n,\eta)$
depending only on $n$ and $\eta$ such that

$\mathrm{Vol}(B(q,3r))\leq
C(n,\eta)\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)$

for $r\geq 2(\sqrt{n}+c)+r_{0}$, where $c:=2\sqrt{f(q)}+4n-4/3$ and we used
$1<\beta_{i}<2$, where $i=1,\ldots,k$. This implies

$k\mathrm{Vol}(B(q,3r))\leq
C(n,\eta)\sum^{k}_{i=1}\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)$

for $r\geq 2(\sqrt{n}+c)+r_{0}$. On the other hand,

$\sum^{k}_{i=1}\mathrm{Vol}\left(B(p_{i},\frac{r}{8})\right)\leq\mathrm{Vol}(B(q,3r))$

Combining the above two inequalities yields $k\leq C(n,\eta)$ and the result
follows. ∎

Similar to the preceding discussion in Section 4, we can apply Theorem 4.1 to
prove Theorem 1.4. Here we include it for the completeness.

###### Proof Theorem 1.4.

Under the assumption of Theorem 1.4, we let $N_{2}=N(n,\eta)$ as in Theorem
4.1. If Theorem 1.4 is not true, we can take $r$ large enough such that there
exist more than $N_{2}$ unbounded ends $E_{j}$ with respect to
$\overline{B(q,r)}$.

Because $E_{j}\cap B(q,2r)$ lie in $B(q,2r)\backslash\overline{B(q,r)}$ and
geodesic balls of radius $r/4$ with centers in different components $E_{j}\cap
B(q,2r)$ do not intersect. That is, we need more than $N_{2}$ geodesic balls
of radius $r/4$ to cover $E_{j}\cap B(q,2r)$, which contradicts Theorem 4.1. ∎

In the rest of this section, we will discuss four sufficient assumptions such
that a class of shrinkers satisfies volume comparison condition. As we all
know, if $(M,g)$ has nonnegative Ricci curvature everywhere, then it satisfies
the volume comparison condition. Indeed the volume doubling property
sufficiently leads to volume comparison condition.

######  Proposition 4.2.

Let $(M,g)$ be an $n$-dimensional complete manifold satisfying the volume
doubling property. Then for all $0<r<R<\infty$ and all $x\in M$ and
$y\in\overline{B(x,R)}$,

$\frac{\mathrm{Vol}(B(x,R))}{\mathrm{Vol}(B(y,r))}\leq
D^{2}\left(\frac{R}{r}\right)^{\kappa},$

where $\kappa=\log_{2}D$. In particular, $(M,g)$ satisfies volume comparison
condition.

###### Proof of Proposition 4.2.

Assume $(M,g)$ satisfies the volume doubling property, that is

$\mathrm{Vol}(B(x,2r))\leq D\,\mathrm{Vol}(B(x,r))$

for any $x\in M$ and $r>0$, where $D$ is a fixed constant. Let $m$ be a
positive integer such that $2^{m}<R/r\leq 2^{m+1}$. Since

$B(x,R)\subset B(y,2R)\subset B(y,2^{m+2}r)$

and thus

$\mathrm{Vol}(B(x,R))\leq\mathrm{Vol}(B(y,2^{m+2}r)),$

then we have

$\displaystyle\mathrm{Vol}(B(x,R))$ $\displaystyle\leq
D^{m+2}\mathrm{Vol}(B(y,r))$ $\displaystyle\leq
D^{2}\left(\frac{R}{r}\right)^{\kappa}\mathrm{Vol}(B(y,r)),$

where $\kappa=\log_{2}D$. This proves the first estimate.

In particular, when $y\in\partial B(x,R)$, we let $r=R/16$ in the first
estimate and immediately get volume comparison condition. ∎

Second, we observe that the shrinker with at least quadratic decay of scalar
curvature implies some non-collapsed property and hence satisfies volume
comparison condition.

######  Proposition 4.3.

Let $(M,g,f)$ be a complete non-compact shrinker with a infimum point $p\in M$
of $f$ satisfying (1.1), (1.2) and (1.3). If the scalar curvature satisfies

$\mathrm{S}(x)\cdot r^{2}(p,x)\leq c_{0}$

for any $r(p,x)>0$, where $c_{0}>0$ is a constant and $r(p,x)$ is the distance
function from $p$ to $x$, then the shrinker satisfies volume comparison
condition. In particular, any shrinker with finite asymptotic scalar curvature
ratio satisfies volume comparison condition.

###### Proof of Proposition 4.3.

For any $1/32\leq\alpha\leq 1/2$, for any $r>0$ and for any point
$q\in\partial B(p,r)$, by the second estimate of Lemma 2.7, we have

$(\alpha r)^{-n}\mathrm{Vol}(B(q,\alpha r))\geq
c(n)e^{\mu}\left[1+\frac{c_{0}}{(1-\alpha)^{2}r^{2}}\cdot(\alpha
r)^{2}\right]^{-\frac{n}{2}}$

where we used

$\mathrm{S}\leq\frac{c_{0}}{r^{2}(p,x)}\leq\frac{c_{0}}{(1-\alpha)^{2}r^{2}}.$

Namely, for any $r>0$ and for any point $q\in\partial B(p,r)$,

$\displaystyle\mathrm{Vol}(B(q,\alpha r))$ $\displaystyle\geq
c(n)e^{\mu}\left[1+\frac{c_{0}\alpha^{2}}{(1-\alpha)^{2}}\right]^{-\frac{n}{2}}\alpha^{n}\cdot
r^{n}$ $\displaystyle\geq c(n,c_{0})e^{\mu}r^{n}$

for some constant $c(n,c_{0})$ depending only on $n$ and $c_{0}$, where used
$1/32\leq\alpha\leq 1/2$.

On the other hand, by Lemma 2.5,

$\mathrm{Vol}(B(p,r))\leq c(n)r^{n}$

for any $r>0$. Thus, for any $r>0$ and for any point $q\in\partial B(p,r)$,
the lower and upper volume estimates give

$\frac{\mathrm{Vol}(B(p,r))}{\mathrm{Vol}(B(q,\alpha r))}\leq
c(n,c_{0})e^{-\mu}.$

Letting $\alpha=1/16$ shows that such shrinker satisfies volume comparison
condition. ∎

The proof of Proposition 4.3 indicates that the finite asymptotic scalar
curvature ratio implies the positive asymptotic volume ratio. Moreover,
combining Proposition 4.3 and Theorem 1.4, we easily get the following result
due to Munteanu, Schulze and Wang [33].

######  Corollary 4.4.

Any complete non-compact shrinker with finite asymptotic scalar curvature
ratio must have finitely many ends.

Third, we see that if a family of the average of scalar curvature integral has
at least quadratic decay of radius, then such shrinker also satisfies volume
comparison condition.

######  Proposition 4.5.

Let $(M,g,f)$ be a complete non-compact shrinker with a infimum point $p\in M$
of $f$ satisfying (1.1), (1.2) and (1.3). If there exists a constant $c_{1}>0$
such that

$\frac{r^{2}}{\mathrm{Vol}\left(B(x,r)\right)}\int_{B(x,r)}\mathrm{S}\,dv\leq
c_{1}$ (4.1)

for all $r>0$ and all $x\in\partial B(p,r)$, then the shrinker satisfies
volume comparison condition.

###### Proof of Proposition 4.5.

For any $r>0$, we let point $q$ be $x\in\partial B(p,r)$ in the first estimate
of Lemma 2.7, and get

$\frac{\mathrm{Vol}\left(B(x,\frac{r}{16})\right)}{(\tfrac{r}{16})^{n}}\left[1+\sup_{s\in\left[0,\tfrac{r}{16}\right]}\frac{s^{2}\int_{B(x,s)}\mathrm{S}\,dv}{\mathrm{Vol}(B(x,s))}\right]^{n/2}\geq
c(n)e^{\mu}.$

By the assumption (4.1), the above inequality becomes

$\mathrm{Vol}\left(B(x,\frac{r}{16})\right)\geq c(n,c_{1})e^{\mu}r^{n}$

for all $r>0$ and all $x\in\partial B(p,r)$. Combining this with the volume
upper growth $\mathrm{Vol}(B(p,r))\leq c(n)r^{n}$ immediately yields

$\frac{\mathrm{Vol}(B(p,r))}{\mathrm{Vol}\left(B(x,\frac{r}{16})\right)}\leq
c(n,c_{1})e^{-\mu}$

for any $r>0$ and all $x\in\partial B(p,r)$. ∎

######  Remark 4.6.

Similar to the above argument, Proposition 4.5 can be also proved by Lemma
2.8. Moreover, when $n\geq 3$, the assumption (4.1) in Proposition 4.5 can be
replaced by the bound of the following maximal function of scalar curvature
introduced by Topping [39]:

$\sup_{s\in\left(0,\tfrac{r}{16}\right]}s^{-1}\left[\mathrm{Vol}(B(x,s))\right]^{-\frac{n-3}{2}}\left(\int_{B(x,s)}\mathrm{S}\,dv\right)^{\frac{n-1}{2}}\leq\delta,$

for all $r>0$ and all $x\in\partial B(p,r)$, where
$\delta:=\min\\{w_{n},\,(4\pi)^{\frac{n}{2}}e^{\mu+n-2^{n}\cdot 17}\\}$ and
$\omega_{n}$ is the volume of the unit Euclidean $n$-ball. This bound
assumption also enables us to get that

$\mathrm{Vol}\left(B(x,\frac{r}{16})\right)>\delta\,r^{n}$

for all $r>0$ and all $x\in\partial B(p,r)$, the interested readers are
referred to Theorem 3.1 of [42] for detailed proof.

Combining Proposition 4.5 and Theorem 1.4 leads to

######  Corollary 4.7.

Any complete non-compact shrinker satisfying (4.1) must have finitely many
ends.

In the proof of Corollaries 4.4 and 4.7, we observe that these curvature
assumptions both imply a family of Euclidean volume growth. These proof indeed
shows that any shrinker with a family of Euclidean volume growth must have
volume comparison condition.

######  Corollary 4.8.

If a complete non-compact shrinker $(M,g,f)$ with a infimum point $p\in M$ of
$f$ satisfies

$\mathrm{Vol}\left(B(x,\frac{r}{16})\right)\geq c\,r^{n}$ (4.2)

for all $r\geq r_{0}$ for some $r_{0}>0$, and all $x\in\partial B(p,r)$, where
$c$ is a positive constant independent of $x$ and $r$, then such shrinker
satisfies volume comparison condition and hence has finitely many ends.

In the end of this section, we give some comments on the relation between
Corollary 4.8 and asymptotic volume ratio on shrinkers. Recall that the
asymptotic volume ratio ($\operatorname{AVR}$) of a complete Riemannian
manifold $(M,g)$ is defined by

$\operatorname{AVR}(g):=\lim_{r\rightarrow\infty}\frac{\operatorname{Vol}B(q,r)}{\omega_{n}r^{n}}$

if the limit exists. Whenever the $\operatorname{AVR}(g)$ exists, it is
independent of point $q$. If $(M,g)$ has nonnegative Ricci curvature, then the
limit always exists by the Bishop-Gromov volume comparison. For any shrinker,
Chow, Lu and Yang [13] proved that $\operatorname{AVR}(g)$ always exists and
is finite. The assumption (4.2) naturally implies positive asymptotic volume
ratio; but the reverse problem is not clear to the author at present. Notice
that Feldman, Ilmanen and Knopf [17] described examples of complete non-
compact Kähler shrinkers, which have $\operatorname{AVR}(g)>0$ and the Ricci
curvature changes sign. We see that positive asymptotic volume ratio provides
the Euclidean volume growth based on a fixed point, which does not seem to
yield a family of Euclidean volume growth (4.2). On the other hand, Carrillo
and Ni [7] proved that any shrinker with Ricci curvature $\mathrm{Ric}(g)\geq
0$ must have $\operatorname{AVR}(g)=0$. Here we may reverse the process and
naively ask that if $\operatorname{AVR}(g)=0$ implies $\mathrm{Ric}(g)\geq 0$?

##  5\. Diameter growth of ends

In the last section, we will apply the ball covering property to study the
diameter growth of ends in the shrinker. The manifold case can be referred to
[1], where Abresch and Gromoll proved that every end of manifolds with
nonnegative Ricci curvature has most linear diameter growth. Later this result
can be generalized by Liu [25] to manifolds with nonnegative Ricci curvature
outside a compact set. Let us first recall the definition diameter of ends on
manifolds; see also [25].

######  Definition 5.1.

Let $q$ be a fixed point in a Riemannian manifold $(M,g)$. For any $r>0$, any
connected component $\Sigma$ of the annulus

$A_{q}(2r,\tfrac{3}{4}r):=B(q,2r)\backslash\overline{B(q,\tfrac{3}{4}r)},$

and any two points $x,y\in\Sigma\cap\partial B(q,r)$, we let

$d_{r}(x,y):=\inf\left\\{\mathrm{length}(\gamma)\right\\},$

where the infimum is taken over all piecewise smooth curves $\gamma$ from $x$
to $y$ in $M\backslash\overline{B(q,r/2)}$. Then we set

$\mathrm{diam}\left(\Sigma\cap\partial
B(q,r)\right):=\sup_{x,y\in\Sigma\cap\partial B(q,r)}d_{r}(x,y).$

Using the above notations, the diameter of ends at $r$ from $q$ is defined by

$\mathrm{diam}_{q}(r):=\sup_{\Sigma\subset
A_{q}(2r,\tfrac{3}{4}r)}\mathrm{diam}\left(\Sigma\cap\partial B(q,r)\right).$

See Figure 2 for a simple description.

Figure 2.  Definition of the diameter of ends

We now apply the above definition to Theorem 5.2 and obtain a diameter growth
for ends in the shrinker without any assumption.

######  Theorem 5.2.

On any $n$-dimensional complete non-compact shrinker with the scalar curvature

$\mathrm{S}\geq\delta$

for some constant $\delta\geq 0$, the diameter growth of ends is at most
polynomial growth with degree $2(n-\delta)+1$.

###### Proof of Theorem 5.2.

Without loss of generality, we choose a infimum point $p\in M$ of $f$ as a
base point. By Theorem 3.2, for a fixed sufficiently large $r$, and for any
connected component $\Sigma$ of the annulus $A_{q}(2r,\tfrac{3}{4}r)$, we can
find no more than

$N:=c(n)e^{-\mu}r^{2(n-\delta)}$

geodesic balls $B_{i}:=B\left(p_{i},\frac{r}{4}\right)$, where $p_{i}\in
A_{q}(2r,\tfrac{3}{4}r)$ and $i\leq N$ such that

$\bigcup_{i=1}B\left(p_{i},\frac{r}{4}\right)\supset\Sigma.$

For any two points $x$ and $y$ in $\Sigma\cap\partial B(q,r)$, since $\Sigma$
is connected, we can find a subsequence of geodesic balls $\\{B_{i}\\}$:
$B_{i_{1}},\ldots,B_{i_{k}}$, where $k\leq N$ such that

$x\in B_{i_{1}},\quad B_{i_{j}}\cap
B_{i_{j+1}}\neq\emptyset\,\,(j=1,\ldots,k-1),\quad y\in B_{i_{k}}.$

Now we choose fixed points $z_{j}\in B_{i_{j}}\cap B_{i_{j+1}}$ and
consecutively connect the above mentioned points

$x,p_{i_{1}},z_{1},p_{i_{2}},z_{2},p_{i_{3}},\ldots,p_{i_{k-1}},z_{k-1},p_{i_{k}},y,$

which forms a piecewise smooth curve $\gamma$. Obviously, the curve $\gamma$
lies in $M\backslash\overline{B(q,r/2)}$ and has the length of $\gamma$

$\mathrm{length}(\gamma)\leq 2k\cdot\frac{r}{4}\leq\frac{N}{2}r\leq
c(n)e^{-\mu}r^{2(n-\delta)+1}.$

This completes the proof. ∎

######  Remark 5.3.

If the scalar curvature of shrinker is uniformly bounded, by Remark 3.3, the
above argument indicates that the degree $2(n-\delta)+1$ in Theorem 5.2 can be
reduced to $n-2\delta+1$.

If the shrinker satisfies volume comparison condition, by the same argument as
above, Theorem 4.1 immediately implies

######  Theorem 5.4.

On any complete non-compact shrinker with volume comparison condition, the
diameter growth of ends is at most linear.

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