# Quantitative estimate of diameter
for weighted manifolds
under integral curvature bounds and $\varepsilon$-range

Taku Ito  $\ast$ Department of Mathematical Sciences, Tokyo Metropolitan
University, Tokyo 192-0397, Japan<EMAIL_ADDRESS>
###### Abstract.

In this article, we extend the compactness theorems proved by Sprouse [SC] and
Hwang–Lee [HL] to a weighted manifold under the assumption that the weighted
Ricci curvature is bounded below in terms of its weight function. With the
help of the $\varepsilon$-range, we treat the case that the effective
dimension is at most $1$ in addition to the case that the effective dimension
is at least the dimension of the manifold. To show these theorems, we extend
the segment inequality of Cheeger–Colding [CC] to a weighted manifold.

###### Key words and phrases:

Myers theorem, integral curvature, weighted manifold, $\varepsilon$-range

###### 2022 Mathematics Subject Classification:

53C20,58C35

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and segment inequality with $\varepsilon$-range
    1. 2.1 Preliminaries
    2. 2.2 Segment inequality with $\varepsilon$-range
  3. 3 Proof of Theorem 1.7
  4. 4 Proof of Theorem 1.8
  5. 5 Fundamental group under integral curvature bound and $\varepsilon$-range

##  1\. Introduction

One of the most fundamental theorems in Riemannian geometry is the Myers
theorem [SBM], which states that if an $n$-dimensional complete Riemannian
manifold $(M,g)$ satisfies $\mathrm{Ric}_{g}\geq(n-1)K$ with $K>0$ and $n\geq
2,$ then $M$ is compact and its diameter $\mathrm{diam}(M)$ is at most
$\pi/\sqrt{K}$. Moreover, its fundamental group $\pi_{1}(M)$ is a finite
group. Here, we denote by $\mathrm{Ric}_{g}$ the Ricci curvature of $(M,g)$.
For $\kappa\in C^{\infty}(M)$, $\mathrm{Ric}_{g}\geq\kappa$ means that
$\mathrm{Ric}_{g}(v,v)\geq\kappa(p)g(v,v)$ holds for $v\in T_{p}M$. This
theorem has been generalized by several ways. As one of these attempts, we
introduce the theorems by Sprouse [SC] below.

In this article, we always assume that $(M,g)$ is a complete Riemannian
manifold and its dimension is at least $2$. We denote by $d$ and
$\mathrm{vol}_{g}$ the Riemannian distance function and the volume measure on
$(M,g)$, respectively. We write $a_{+}=\max\\{a,0\\}$ for $a\in\mathbb{R}$.
Set $\mathbb{N}_{\geq 2}=\\{\,n\in\mathbb{N}\mid n\geq 2\,\\}$. For $p\in M$
and $R>0$, we denote

$\displaystyle U_{p}M$ $\displaystyle=\\{\,v\in T_{p}M\mid g(v,v)=1\,\\},$
$\displaystyle\mathrm{Ric}_{g-}(p)$
$\displaystyle=\inf\\{\,\mathrm{Ric}_{g}(v,v)\mid v\in U_{p}M\,\\},$
$\displaystyle B(p,R)$ $\displaystyle=\\{\,q\in M\mid d(p,q)<R\,\\}.$

######  Theorem 1.1 ([SC]*Theorem 1.1).

Let $n\in\mathbb{N}_{\geq 2}$ and $\eta>0$. There exists a positive constant
$\delta(n,\eta)$ such that if an $n$-dimensional compact Riemannian manifold
$(M,g)$ satisfies $\mathrm{Ric}_{g}\geq 0$ and

$\frac{1}{\mathrm{vol}_{g}(M)}\int_{M}\big{(}(n-1)-\mathrm{Ric}_{g-}\big{)}_{+}\,d\mathrm{vol}_{g}<\delta(n,\eta)$

then $\mathrm{diam}(M)<\pi+\eta$.

######  Theorem 1.2 ([SC]*Theorem 1.2).

Let $n\in\mathbb{N}_{\geq 2},K\leq 0$ and $R,\eta>0$. There exists a positive
constant $\delta(n,K,R,\eta)$ such that if an $n$-dimensional complete
Riemannian manifold $(M,g)$ satisfies $\mathrm{Ric}_{g}\geq(n-1)K$ and

$\sup_{p\in
M}\frac{1}{\mathrm{vol}_{g}(B(p,R))}\int_{B(p,R)}\big{(}(n-1)-\mathrm{Ric}_{g-}\big{)}_{+}\,d\mathrm{vol}_{g}<\delta(n,K,R,\eta)$

then $M$ is compact and $\mathrm{diam}(M)<\pi+\eta$.

Hwang–Lee [HL] extended Theorems 1.1 and 1.2 to a weighted manifold having a
bounded $\infty$-Ricci curvature. A _weighted manifold_ is a triple
$(M,g,\mu_{f})$, where $(M,g)$ is an $n$-dimensional Riemannian manifold,
endowed with a measure $\mu_{f}=e^{-f}d\mathrm{vol}_{g}$ having smooth
positive density with respect to the volume measure $\mathrm{vol}_{g}$. We
call $f\in C^{\infty}(M)$ the _weight function_ of $\mu_{f}$. For given
$N\in(-\infty,\infty]$, the _$N$ -Ricci curvature_ is defined for $v\in TM$ by

$\displaystyle\mathrm{Ric}_{N}(v,v)=\begin{cases}\mathrm{Ric}_{g}(v,v)+\mathrm{Hess}_{g}f(v,v)-\dfrac{1}{N-n}g(\nabla
f,v)^{2}&\text{if }N\in\mathbb{R}\setminus\\{n\\},\\\
\mathrm{Ric}_{g}(v,v)+\mathrm{Hess}_{g}f(v,v)&\text{if }N=\infty,\end{cases}$

and for $N=n$,

$\displaystyle\mathrm{Ric}_{n}(v,v)=\begin{cases}\mathrm{Ric}_{g}(v,v)&\text{if
}f\text{ is a constant function},\\\ -\infty&\text{otherwise}.\end{cases}$

We call the parameter $N$ the _effective dimension_. We refer to a _weighted
Ricci curvature_ as a generic term for the $N$-Ricci curvature with
$N\in(-\infty,\infty]$. As similar as $\mathrm{Ric}_{-}$, we define
$\mathrm{Ric}_{N-}:M\to\mathbb{R}$ by

$\mathrm{Ric}_{N-}(p)=\inf\\{\,\mathrm{Ric}_{N}(v,v)\mid v\in U_{p}M\,\\}.$

######  Theorem 1.3 ([HL]*Theorem 1.4).

Let $n\in\mathbb{N}_{\geq 2},k\geq 0$ and $\eta>0$. There exists a positive
constant $\delta(n+4k,\eta)$ such that if an $n$-dimensional compact weighted
manifold $(M,g,\mu_{f})$ satisfies $\mathrm{Ric}_{\infty}\geq 0$, $|f|\leq k$
and

$\frac{1}{\mu_{f}(M)}\int_{M}\big{(}(n-1)-\mathrm{Ric}_{\infty-}\big{)}_{+}\,d\mu_{f}<\delta(n+4k,\eta)$

then

$\mathrm{diam}(M)<\left(\pi+\frac{\eta}{2}\right)\sqrt{1+\frac{8k}{(n-1)\pi}}+\frac{\eta}{2}.$

######  Theorem 1.4 ([HL]*Theorem 1.5).

Let $n\in\mathbb{N}_{\geq 2},k\geq 0,K\leq 0$ and $R,\eta>0$. There exists a
positive constant $\delta(n+4k,K,R,\eta)$ such that if an $n$-dimensional
complete weighted manifold $(M,g,\mu_{f})$ satisfies $|f|\leq k$,
$\mathrm{Ric}_{\infty}\geq(n-1)K$ and

$\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)-\mathrm{Ric}_{\infty-}\big{)}_{+}\,d\mu_{f}<\delta(n+4k,K,R,\eta)$

then $M$ is compact and

$\mathrm{diam}(M)<\left(\pi+\frac{\eta}{2}\right)\sqrt{1+\frac{8k}{(n-1)\pi}}+\frac{\eta}{2}.$

The aim of this article is to extend Theorems 1.1–1.4 for a weighted manifold
under the assumption that the weighted curvature is bounded below in terms of
its weight function. The key of the proof is the notion of the following
$\varepsilon$-range introduced by Lu–Minguzzi–Ohta [LMO].

######  Definition 1.5.

For $n\in\mathbb{N}_{\geq 2}$ and $N\in(-\infty,1]\cup[n,\infty]$, we say that
_$\varepsilon\in\mathbb{R}$ is in the $\varepsilon(n,N)$-range_ if
$\varepsilon\in\mathbb{R}$ satisfies the following conditions:

$\varepsilon=0\,\text{ for
}N=1,\qquad|\varepsilon|<\sqrt{\frac{N-1}{N-n}}\,\text{ for }N\neq
1,n,\qquad\varepsilon\in\mathbb{R}\,\text{ for }N=n.$

For $n,N$ as above and $\varepsilon$ in the $\varepsilon(n,N)$-range, we
define the constant $c=c(n,N,\varepsilon)$ by

$c=\frac{1}{n-1}\left(1-\varepsilon^{2}\frac{N-n}{N-1}\right)>0$

for $N\neq 1$. If $\varepsilon=0$, then one can take $N\to 1$ and set
$c(1,0)=1/(n-1)$.

Notice that $|\varepsilon|=1$ happens only if $N\in[n,\infty)$. For
$N\in(-\infty,1]\cup[n,\infty]$ and $K\in\mathbb{R}$, Lu–Minguzzi–Ohta [LMO2]
established a curvature bound $\mathrm{Ric}_{N}\geq
Ke^{4(\varepsilon-1)f/(n-1)}$ on a weighted manifold $(M,g,\mu_{f})$ by using
the $\varepsilon(n,N)$-range. This curvature bound is a generalization of a
different kind of curvature bounds $\mathrm{Ric}_{1}\geq Ke^{4f/(1-n)}$
introduced by Wylie–Yeroshkin [WY] and for $N\in(-\infty,1]$,
$\mathrm{Ric}_{N}\geq Ke^{4f/(N-n)}$ introduced by Kuwae–Li [KL]. They
presented several comparison theorems with each curvature bounds. Furthermore,
Kuwae–Sakurai [KS] also provided several comparison theorems with the
$\varepsilon(n,N)$-range.

To state our theorems, we prepare the following condition.

######  Definition 1.6.

Let $n\in\mathbb{N}_{\geq 2}$, $N\in(-\infty,1]\cup[n,\infty]$, $\varepsilon$
in the $\varepsilon(n,N)$-range, $K\leq 0$ and $b\geq a>0$.

  1. (1)

We say that an $n$-dimensional weighted manifold $(M,g,\mu_{f})$ satisfies a
_$(N,K,\varepsilon,a,b)$ -condition_ if one has

$\displaystyle\mathrm{Ric}_{N}\geq Ke^{\frac{4(\varepsilon-1)}{n-1}f},\qquad
a\leq e^{\frac{2(1-\varepsilon)}{n-1}f}\leq b\quad\text{on }M.$

  2. (2)

We define the constants $\widetilde{D}=\widetilde{D}(n,N,\varepsilon,a,b)$ by

$\displaystyle\widetilde{D}=\begin{cases}\sqrt{\dfrac{N-1}{n-1}}&\text{if
}\varepsilon=1,\\\\[-2.84544pt] \\\
\sqrt{1+\dfrac{2}{|1-\varepsilon|\pi}\log\dfrac{b}{a}}&\text{if
}\varepsilon\neq 1\text{ and }N\in[n,\infty],\\\\[-2.84544pt] \\\
\sqrt{\left(\dfrac{b}{a}\right)^{\lambda_{0}}+\dfrac{2}{(1-\varepsilon)\pi\lambda_{0}}\left(\left(\dfrac{b}{a}\right)^{\lambda_{0}}-1\right)}&\text{otherwise}.\end{cases}$

Moreover, for $N\in(-\infty,1]$, we define
$\lambda_{0}=\lambda_{0}(n,N,\varepsilon)$ by

$\lambda_{0}=\frac{1}{1-\varepsilon}\left(1-\sqrt{\frac{N-1}{N-n}}\right).$

For a weighted manifold satisfying the $(N,K,\varepsilon,a,b)$-condition, we
estimate its diameter quantitatively.

######  Theorem 1.7.

Let $n\in\mathbb{N}_{\geq 2},N\in(-\infty,1]\cup[n,\infty]$, $\varepsilon$ in
the $\varepsilon(n,N)$-range, $b\geq a>0$ and $H,\eta>0$. There exists a
positive constant $\delta_{1}(n,N,\varepsilon,a,b,H,\eta)$ such that if an
$n$-dimensional compact weighted manifold $(M,g,\mu_{f})$ satisfies the
$(N,0,\varepsilon,a,b)$-condition and

$\frac{1}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta_{1}(n,N,\varepsilon,a,b,H,\eta)$

then

$\mathrm{diam}(M)\leq\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}(n,N,\varepsilon,a,b).$

######  Theorem 1.8.

Let $n\in\mathbb{N}_{\geq 2},N\in(-\infty,1]\cup[n,\infty]$ and $\varepsilon$
in the $\varepsilon(n,N)$-range. Take real numbers $K,a,b,H,R$ and $\eta$ such
that $K\leq 0$ and $a,b,H,R,\eta>0$ with $a\leq b$. There exists a positive
constant $\delta(n,N,K,\varepsilon,a,b,H,R,\eta)$ such that if an
$n$-dimensional complete weighted manifold $(M,g,\mu_{f})$ satisfies the
$(N,K,\varepsilon,a,b)$-condition and

$\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta(n,N,K,\varepsilon,a,b,H,R,\eta)$

then $M$ is compact and

$\mathrm{diam}(M)\leq\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}(n,N,\varepsilon,a,b).$

######  Remark 1.9.

Assume that an $n$-dimensional complete weighted manifold
$(M,g,\mathrm{vol}_{g})$ satisfies $\mathrm{Ric}_{g}\geq(n-1)H$ for $H>0$.
Then $(M,g,\mathrm{vol}_{g})$ satisfies the $(n,0,1,1,1)$-condition and

$\sup_{p\in
M}\frac{1}{\mathrm{vol}_{g}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{n-}\big{)}_{+}\,d\mathrm{vol}_{g}=0$

holds for each $R>0$. Furthermore, we find $\widetilde{D}(n,n,1,1,1)=1$. By
Theorem 1.8, we see that $\mathrm{diam}(M)\leq(\pi+\eta)/\sqrt{H}$ for any
$\eta>0$. Taking $\eta\to 0$, we obtain $\mathrm{diam}(M)\leq\pi/\sqrt{H}$.
This means that Theorem 1.8 recovers the Myers theorem.

This article is organized as follows: In Section 2, we recall a Bishop-type
inequality, the volume comparison theorem on a weighted manifold given in
[LMO2]. Then we extend the segment inequality of Cheeger–Colding [CC] to a
weighted manifold under the assumption that the weighted Ricci curvature is
bounded below in terms of its weight function. In Sections 3 and 4, we prove
Theorems 1.7 and 1.8, respectively. In Sections 5, we analyze the fundamental
group of a weighted manifold satisfying the $(N,K,\varepsilon,a,b)$-condition.

######  Acknowledgements.

The author would like to express his deepest thanks to his supervisor, Asuka
Takatsu, as well as to Manabu Akaho, Shin-ichi Ohta, Takashi Sakai, Homare
Tadano who contributed their support.

##  2\. Preliminaries and segment inequality with $\varepsilon$-range

We recall some results by Lu–Minguzzi–Ohta [LMO2]. Although they discussed a
weighted Finsler manifold, throughout this article, we treat a weighted
manifold.

###  2.1. Preliminaries

Lu–Minguzzi–Ohta [LMO2] introduced the $\varepsilon$-range and proved a volume
comparison theorem for a weighted manifold $(M,g,\mu_{f})$.

For $p\in M$, let $\gamma:[0,l)\to M$ be a unit speed geodesic such that
$\gamma(0)=p$. Set $f_{\gamma}(t)=f\big{(}\gamma(t)\big{)}$. We denote by

$\mu_{f}=e^{-f}d\mathrm{vol}_{g}={\mathcal{A}}^{f}_{\gamma}(t)dtd\theta_{n-1}=e^{-f_{\gamma}(t)}{\mathcal{A}}_{\gamma}(t)dtd\theta_{n-1}$

the weighted measure in the geodesic polar coordinates along geodesics
$\gamma$, where $d\theta_{n-1}$ is the volume measure of the unit sphere in
$T_{p}M$.

First we introduce a Bishop-type inequality.

######  Proposition 2.1 ([LMO2]*Theorem 3.5).

Let $(M,g,\mu_{f})$ be an $n$-dimensional complete weighted manifold,
$N\in(-\infty,1]\cup[n,\infty]$, $\varepsilon$ in the $\varepsilon(n,N)$-range
and $c=c(n,N,\varepsilon)$. For a unit speed geodesic $\gamma:[0,l)\to M$, set

$h(t)=e^{-cf_{\gamma}(t)}{\mathcal{A}}_{\gamma}(t)^{c},\qquad
h_{1}(\tau)=h(\varphi_{\gamma}^{-1}(\tau))$

for $t\in[0,l)$ and $\tau\in[0,\varphi_{\gamma}(l))$, where

$\varphi_{\gamma}(t)=\int_{0}^{t}e^{\frac{2(\varepsilon-1)}{n-1}f_{\gamma}(s)}\,ds.$

Then, for all $\tau\in(0,\varphi_{\gamma}(l))$,

$h_{1}^{\prime\prime}(\tau)\leq-
ch_{1}(\tau)\mathrm{Ric}_{N}\big{(}(\gamma\circ\varphi_{\gamma}^{-1})^{\prime}(\tau)\big{)}.$

We define the comparison function ${\bf sn}_{\kappa}$ by

${\bf
sn}_{\kappa}(t)=\begin{cases}\dfrac{1}{\sqrt{\kappa}}\sin(\sqrt{\kappa}t)&\kappa>0,\\\
t&\kappa=0,\\\
\dfrac{1}{\sqrt{-\kappa}}\sinh(\sqrt{-\kappa}t)&\kappa<0,\end{cases}$

where $t\in[0,\pi/\sqrt{\kappa}]$ for $\kappa>0$ and $t\in\mathbb{R}$ for
$\kappa\leq 0$. Notice that ${\bf sn}_{\kappa}$ satisfies

$\begin{cases}{\bf sn}_{\kappa}^{\prime\prime}(t)+\kappa{\bf
sn}_{\kappa}(t)=0,\\\ {\bf sn}_{\kappa}(0)=0,\\\ {\bf
sn}^{\prime}_{\kappa}(0)=1.\end{cases}$

Let us recall a volume comparison theorem.

######  Proposition 2.2 ([LMO2]*Theorem 3.11).

Let $(M,g,\mu_{f})$ be an $n$-dimensional complete weighted manifold
satisfying the $(N,K,\varepsilon,a,b)$-condition. Then

$\dfrac{\mu_{f}\big{(}B(p,R)\big{)}}{\mu_{f}\big{(}B(p,r)\big{)}}\leq\dfrac{b}{a}\cdot\dfrac{\displaystyle\int_{0}^{R/a}{\bf
sn}_{cK}(\tau)^{1/c}\,d\tau}{\displaystyle\int_{0}^{r/b}{\bf
sn}_{cK}(\tau)^{1/c}\,d\tau}$

holds for all $p\in M$ and $0<r\leq R$.

It should be mentioned that Theorem 2.2 was originaly proved for all
$K\in\mathbb{R}$ in [LMO2]*Theorem 3.11.

######  Remark 2.3.

If $K=0$, we have ${\bf sn}_{0}(\tau)=\tau$, which gives

$\displaystyle\int_{0}^{r}{\bf
sn}(\tau)^{\frac{1}{c}}\,d\tau=\frac{c}{c+1}r^{\frac{1}{c}+1}$

for $r>0$. In the setting of Theorem 2.2 for $K=0$, we find that

$\dfrac{\mu_{f}\big{(}B(p,R)\big{)}}{\mu_{f}\big{(}B(p,r)\big{)}}\leq\dfrac{b}{a}\cdot\left(\frac{bR}{ar}\right)^{\frac{1}{c}+1}.$

###  2.2. Segment inequality with $\varepsilon$-range

The segment inequality proved by Cheeger–Colding [CC]*Theorem 2.11 plays an
important role in the proof of Theorems 1.1 and 1.2. We extend this theorem as
follows.

######  Theorem 2.4.

Let $(M,g,\mu_{f})$ be an $n$-dimensional complete weighted manifold,
$N\in(-\infty,1]\cup[n,\infty]$, $\varepsilon$ in the $\varepsilon(n,N)$-range
and $K\in\mathbb{R}$. Assume that

$\mathrm{Ric}_{N}\geq Ke^{\frac{4(\varepsilon-1)}{n-1}f}$

holds. For $i=1,2$, let $A_{i}$ be bounded open subsets of $M$ and W be an
open subset of $M$ such that, for each two points $y_{i}\in A_{i}$, any unit
minimal geodesics $\gamma_{y_{1},y_{2}}$ from $y_{1}$ to $y_{2}$ is containd
in W. Then for any non-negative integrable function $F$ on $W$, we have

$\displaystyle\quad\int_{A_{1}\times
A_{2}}\left(\int_{0}^{d(y_{1},y_{2})}F(\gamma_{y_{1},y_{2}})\,ds\right)d\mu_{f\times
f}$ $\displaystyle\leq
C(n,c,K)\left[\mu_{f}(A_{2})\,\mathrm{diam}(A_{1})+\mu_{f}(A_{1})\,\mathrm{diam}(A_{2})\right]\int_{W}F\,d\mu_{f},$

where $\mu_{f\times f}$ is the product measure on $M\times M$ induced by
$\mu_{f}$ and

$C(n,c,K)=\sup_{y_{1}\in A_{1},y_{2}\in A_{2}}\left(\sup_{0<\frac{s}{2}\leq
u<s\leq d(y_{1},y_{2})}\frac{{\bf
sn}_{cK}\big{(}\varphi_{\gamma}(s)\big{)}^{1/c}}{{\bf
sn}_{cK}\big{(}\varphi_{\gamma}(u)\big{)}^{1/c}}\right).$

  * Proof. Set

$B=\\{(y_{1},y_{2})\in A_{1}\times A_{2}\mid\mathrm{there\ exists\ a\ unique\
minimal\ geodesic\ from}\ y_{1}\ \mathrm{to}\ y_{2}\\}.$

Then $\mu_{f\times f}(B)=\mu_{f\times f}(A_{1}\times A_{2})$ holds since the
measure of a cut locus of each $y_{i}\in A_{i}$ with respect to $\mu_{f}$ is
zero for $i=1,2$.

Define maps $E_{1},E_{2},E:A_{1}\times A_{2}\to\mathbb{R}$ by

$\displaystyle E_{1}(y_{1},y_{2})$
$\displaystyle=\int_{d(y_{1},y_{2})/2}^{d(y_{1},y_{2})}F\big{(}\gamma_{y_{1},y_{2}}(u)\big{)}\,du,$
$\displaystyle E_{2}(y_{1},y_{2})$
$\displaystyle=\int_{0}^{d(y_{1},y_{2})/2}F\big{(}\gamma_{y_{1},y_{2}}(u)\big{)}\,du,$
$\displaystyle E(y_{1},y_{2})$
$\displaystyle=E_{1}(y_{1},y_{2})+E_{2}(y_{1},y_{2})=\int_{0}^{d(y_{1},y_{2})}F\big{(}\gamma_{y_{1},y_{2}}(u)\big{)}\,du.$

Fix $y_{i}\in A_{i}$ and $v_{i}\in U_{y_{i}}M$ for $i=1,2$. Set

$I(y_{i},v_{i})=\\{\,s>0\mid\exp_{y_{i}}(sv_{i})\in
A_{i+1},\,d\big{(}y_{i},\exp_{y_{i}}(sv_{i})\big{)}=s\,\\},$

where we put $A_{3}=A_{1}$. We denote by $|I(y_{1},v_{1})|$ the
$1$-dimensional Lebesgue measure of $I(y_{1},v_{1})$. Since
$\\{\,\exp_{y_{i}}(sv_{i})\mid s\in I(y_{i},v_{i})\,\\}$ is contained only in
$A_{i+1}$, we have

$|I(y_{i},v_{i})|\leq\mathrm{diam}(A_{i+1})$

for $i=1,2$.

Fix $s\in I(y_{1},v_{1})$. For $\gamma(u)=\exp_{y_{1}}(uv_{1})$ on
$u\in[0,s)$, by Proposition 2.1, we find that
$\mathcal{A}^{f}_{\gamma}(u)/\big{(}{\bf
sn}_{cK}(\varphi_{\gamma}(u))\big{)}^{1/c}$ is non-increasing, where
$\varphi_{\gamma}(u)<\pi/\sqrt{cK}$ holds if $K>0$ (see [LMO2]*Proof of
Theorem 3.6). Then we find that

$\frac{\mathcal{A}^{f}_{\gamma}(s)}{\mathcal{A}^{f}_{\gamma}(u)}\leq\frac{{\bf
sn}_{cK}\big{(}\varphi_{\gamma}(s)\big{)}^{1/c}}{{\bf
sn}_{cK}\big{(}\varphi_{\gamma}(u)\big{)}^{1/c}}\leq C(n,c,K)$

for $0<s/2\leq u<s\leq T(v_{1})$, where $T(v_{1})$ is the supremum of $s$ such
that $s\in I(y_{1},v_{1})$. This yields

$\displaystyle E_{1}(y_{1},\gamma(s))\mathcal{A}^{f}_{\gamma}(s)$
$\displaystyle=\mathcal{A}^{f}_{\gamma}(s)\int^{s}_{s/2}F\big{(}\gamma(u)\big{)}\,du$
$\displaystyle\leq
C(n,c,K)\int^{s}_{s/2}F\big{(}\gamma(u)\big{)}\mathcal{A}^{f}_{\gamma}(u)\,du$
$\displaystyle\leq
C(n,c,K)\int^{T(v_{1})}_{0}F\big{(}\gamma(u)\big{)}\mathcal{A}^{f}_{\gamma}(u)\,du.$

Integrating this inequality on $I(y_{1},v_{1})$ implies that

$\displaystyle\int_{I(y_{1},v_{1})}E_{1}(y_{1},\gamma(s))\mathcal{A}^{f}_{\gamma}(s)\,ds$
$\displaystyle\leq
C(n,c,K)\int_{I(y_{1},v_{1})}\,ds\int_{0}^{T(v_{1})}F\big{(}\gamma(u)\big{)}\mathcal{A}^{f}_{\gamma}(u)\,du$
$\displaystyle\leq
C(n,c,K)\,\mathrm{diam}(A_{2})\int_{0}^{T(v_{1})}F\big{(}\gamma(u)\big{)}\mathcal{A}^{f}_{\gamma}(u)\,du.$

By integrating this inequality over the unit sphere in $T_{y_{1}}M$, we see
that

$\displaystyle\int_{A_{2}}E_{1}(y_{1},y_{2})d\mu_{f}$
$\displaystyle=\int_{U_{y_{1}}M}\,d\theta_{n-1}\int_{I(y_{1},v_{1})}E_{1}(y_{1},\gamma(s))\mathcal{A}^{f}_{\gamma}(s)\,ds$
$\displaystyle\leq
C(n,c,K)\,\mathrm{diam}(A_{2})\int_{U_{y_{1}}M}\,d\theta_{n-1}\int_{0}^{T(v_{1})}F\big{(}\gamma(u)\big{)}\mathcal{A}^{f}_{\gamma}(u)\,du$
$\displaystyle\leq C(n,c,K)\,\mathrm{diam}(A_{2})\int_{W}F\,d\mu_{f},$

where, in the third inequality, we use the fact that any unit minimal
geodesics from $y_{1}\in A_{1}$ to $y_{2}\in A_{2}$ is containd in $W$.
Therefore we have

(2.1) $\int_{A_{1}\times A_{2}}E_{1}(y_{1},y_{2})d\mu_{f\times f}\leq
C(n,c,K)\,\mathrm{diam}(A_{2})\,\mu_{f}(A_{1})\int_{W}F\,d\mu_{f}.$

On the other hand, substituting $t=d(y_{1},y_{2})-u$ for $E_{2}(y_{1},y_{2})$
yields

(2.2)
$\displaystyle\begin{split}E_{2}(y_{1},y_{2})&=\int_{0}^{d(y_{1},y_{2})/2}F\big{(}\gamma_{y_{1},y_{2}}(u)\big{)}\,du\\\
&=\int_{d(y_{2},y_{1})/2}^{d(y_{2},y_{1})}F\big{(}\sigma_{y_{2},y_{1}}(t)\big{)}\,dt\\\
&=E_{1}(y_{2},y_{1}),\end{split}$

where
$\sigma_{y_{2},y_{1}}(t)=\gamma_{y_{1},y_{2}}\big{(}d(y_{1},y_{2})-t\big{)}$
on $[d(y_{2},y_{1})/2,d(y_{2},y_{1})]$. Interchanging the roles of $A_{1}$ and
$A_{2}$ in (2.1), it turns out that, by (2.2),

$\displaystyle\int_{A_{1}\times A_{2}}E_{2}(y_{1},y_{2})\,d\mu_{f\times f}$
$\displaystyle=\int_{A_{1}\times A_{2}}E_{1}(y_{2},y_{1})\,d\mu_{f\times f}$
$\displaystyle\leq
C(n,c,K)\,\mathrm{diam}(A_{1})\,\mu_{f}(A_{2})\int_{W}F\,d\mu_{f}.$

Thus we obtain

$\displaystyle\quad\int_{A_{1}\times A_{2}}E(y_{1},y_{2})\,d\mu_{f\times f}$
$\displaystyle=\int_{A_{1}\times A_{2}}E_{1}(y_{1},y_{2})\,d\mu_{f\times
f}+\int_{A_{1}\times A_{2}}E_{2}(y_{1},y_{2})\,d\mu_{f\times f}$
$\displaystyle\leq
C(n,c,K)\big{[}\mathrm{diam}(A_{2})\,\mu_{f}(A_{1})+\mathrm{diam}(A_{1})\,\mu_{f}(A_{2})\big{]}\int_{W}F\,d\mu_{f}.$

This completes the proof of the theorem. $\square$

######  Remark 2.5.

Set $S=\sup_{y_{1}\in A_{1},y_{2}\in A_{2}}d(y_{1},y_{2})$. If we take $N=n$,
$f\equiv 0$, $\varepsilon=0$ and $K=(n-1)H$ with $H>0$, then we have

$C(n,c,K)=\sup_{0<\frac{s}{2}\leq u<s\leq S}\left(\frac{{\bf sn}_{H}(s)}{{\bf
sn}_{H}(u)}\right)^{n-1}$

and the estimate in Theorem 2.4 coincides with that of [CC]*Theorem 2.11.
Whereas, for $N=\infty$, since we impose the different condition compared with
[JM]*Proposition 2.3, the estimate in Theorem 2.4 differs from that of
[JM]*Proposition 2.3

Finally, we provide the following lemma towards the proof of Theorems 1.7 and
1.8.

######  Lemma 2.6.

Let $(M,g,\mu_{f})$ be an $n$-dimensional complete weighted manifold
satisfying the $(N,K,\varepsilon,a,b)$-condition. Take $A_{1}$, $A_{2}$ and
$C(n,c,K)$ as in Theorem 2.4. Then

$C(n,c,K)\leq\frac{{\bf sn}_{cK}(S/a)^{1/c}}{{\bf sn}_{cK}(S/2b)^{1/c}}$

holds, where $S=\sup_{y_{1}\in A_{1},y_{2}\in A_{2}}d(y_{1},y_{2})$.

  * Proof. The boundedness of the weight function $f$ yields $s/b\leq\varphi_{\gamma}(s)\leq s/a$ for any unit speed minimal geodesic $\gamma$ from $y_{1}\in A_{1}$ to $y_{2}\in A_{2}$. Then for $K\leq 0$, the monotonicity of ${\bf sn}_{cK}(\tau)$ in $\tau>0$ gives

(2.3) $C(n,c,K)\leq\sup_{0<s\leq S}\frac{{\bf sn}_{cK}(s/a)^{1/c}}{{\bf
sn}_{cK}(s/2b)^{1/c}}.$

For $K=0$, ${\bf sn}_{0}(s)=s$ implies

$C(n,c,K)\leq\sup_{0<s\leq S}\frac{{\bf sn}_{0}(s/a)^{1/c}}{{\bf
sn}_{0}(s/2b)^{1/c}}=\left(\frac{2b}{a}\right)^{\frac{1}{c}}.$

When $K<0$, we will see the following claim:

######  Claim 2.7.

Let $F:\mathbb{R}\to\mathbb{R}$ be

$F(s)=\frac{\sinh(As)}{\sinh(Bs)}$

for $A>B>0$. Then $F(s)$ is strictly increasing on $(0,\infty)$.

  * Proof. Differentiating $F(s)$ gives

$F^{\prime}(s)=\frac{1}{\sinh^{2}(Bs)}\left(A\cosh(As)\sinh(Bs)-B\sinh(As)\cosh(Bs)\right).$

We use the following hyperbolic function identities:

$\displaystyle
2\cosh(As)\sinh(Bs)=\sinh\left(A+B\right)s+\sinh\left(-A+B\right)s,$
$\displaystyle
2\sinh(As)\cosh(Bs)=\sinh\left(A+B\right)s-\sinh\left(-A+B\right)s.$

These yield

$F^{\prime}(s)=\frac{1}{2\sinh^{2}(Bs)}\left\\{\left(A-B\right)\sinh\left(A+B\right)s-\left(A+B\right)\sinh\left(A-B\right)s\right\\}.$

Setting

$F_{1}(s)=\left(A-B\right)\sinh\left(A+B\right)s-\left(A+B\right)\sinh\left(A-B\right)s,$

we have $F_{1}(0)=0$ and

$F^{\prime}_{1}(s)=AB\left\\{\cosh\left(A+B\right)s-\cosh\left(A-B\right)s\right\\}>0$

on $(0,\infty)$. Therefore we get $F_{1}(s)>0$ on $(0,\infty)$. Hence we
obtain $F^{\prime}(s)>0$ on $(0,\infty)$. This completes the proof of the
claim. $\Diamond$

When $K<0$, applying this claim to (2.3) gives

$C(n,c,K)\leq\frac{{\bf sn}_{cK}(S/a)^{1/c}}{{\bf sn}_{cK}(S/2b)^{1/c}}.$

$\square$

##  3\. Proof of Theorem 1.7

We extend Theorem 1.1 to the case that the weighted manifold has the non-
negative weighted curvature.

  * Proof of Theorem 1.7. Set $D=\mathrm{diam}(M)$ and take $p_{1},p_{2}\in M$ such that $D=d(p_{1},p_{2})$. We put $W=M$ and $A_{i}=B(p_{i},r)$ for $i=1,2$, where $r>0$ is later determined. Then Theorem 2.4 gives

(3.1) $\displaystyle\begin{split}&\quad\int_{A_{1}\times
A_{2}}\left(\inf_{(y_{1},y_{2})\in\overline{A}_{1}\times\overline{A}_{2}}\int_{0}^{d(y_{1},y_{2})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}(\gamma_{y_{1},y_{2}})\,ds\right)d\mu_{f\times
f}\\\ &\leq\int_{A_{1}\times
A_{2}}\left(\int_{0}^{d(y_{1},y_{2})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}(\gamma_{y_{1},y_{2}})\,ds\right)d\mu_{f\times
f}\\\ &\leq
2rC(n,c,K)\left[\mu_{f}(A_{1})+\mu_{f}(A_{2})\right]\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.\end{split}$

We observe from Theorem 2.2 with $K=0$ that

$\dfrac{\mu_{f}(M)}{\mu_{f}(A_{i})}\leq\frac{b}{a}\left(\frac{bD}{ar}\right)^{\frac{1}{c}+1}$

for $i=1,2$ and from Lemma 2.6 that

$C(n,c,K)\leq\left(\frac{2b}{a}\right)^{\frac{1}{c}}.$

Dividing (3.1) by $\mu_{f}(A_{1})\mu_{f}(A_{2})$ yields

(3.2)
$\displaystyle\begin{split}&\quad\inf_{(y_{1},y_{2})\in\overline{A}_{1}\times\overline{A}_{2}}\int_{0}^{d(y_{1},y_{2})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}(\gamma_{y_{1},y_{2}})\,ds\\\
&\leq
2rC(n,c,K)\left(\frac{1}{\mu_{f}(A_{1})}+\frac{1}{\mu_{f}(A_{2})}\right)\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\\\
&\leq
2r\,\left(\frac{2b^{2}D}{a^{2}r}\right)^{\frac{1}{c}+1}\frac{1}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.\end{split}$

There exists a unit speed minimal geodesic $\gamma$ from
$y_{1}\in\overline{A}_{1}$ to $y_{2}\in\overline{A}_{2}$ that attains the
infimum of (LABEL:L-com). Let $L=d(y_{1},y_{2})$ and
$\\{E_{1},\ldots,E_{n}=\dot{\gamma}\\}$ be a parallel orthonormal frame along
$\gamma$. For a smooth function $\alpha\in C^{\infty}([0,L])$ such that
$\alpha(0)=\alpha(L)=0$, we set $Y_{i}(t)=\alpha(t)E_{i}(t)$,
$i=1,\ldots,n-1$. We denote by $L_{i}(s)$ the length functional of a fixed-
endpoint variation of a curves $c(s,t):(-\epsilon,\epsilon)\times[0,L]\to M$
such that

$c(0,t)=\gamma(t),\qquad\left.\frac{\partial}{\partial
s}c(s,t)\right|_{s=0}=Y_{i}(t).$

Then the second variation formula for $L_{i}(s)$ (see [ST]*Chapter III Theorem
2.5) provides

(3.3)
$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle=\sum_{i=1}^{n-1}\int_{0}^{L}\left\\{g(\nabla_{\dot{\gamma}}Y_{i},\nabla_{\dot{\gamma}}Y_{i})-R_{g}\big{(}Y_{i}(t),\gamma^{\prime}(t),\gamma^{\prime}(t),Y_{i}(t)\big{)}\right\\}\,dt$
$\displaystyle=\int_{0}^{L}\left\\{(n-1)\alpha^{\prime}(t)^{2}-\alpha(t)^{2}\mathrm{Ric}_{g}(\gamma^{\prime}(t),\gamma^{\prime}(t))\right\\}\,dt$
$\displaystyle=\int_{0}^{L}\big{[}-(n-1)H\alpha(t)^{2}+(n-1)\alpha^{\prime}(t)^{2}+\alpha(t)^{2}\mathrm{Hess}_{g}\,f\big{(}\gamma^{\prime}(t),\gamma^{\prime}(t)\big{)}$
$\displaystyle\quad-\frac{\alpha(t)^{2}}{N-n}f^{\prime}_{\gamma}(t)^{2}+\alpha(t)^{2}\left\\{(n-1)H-\mathrm{Ric}_{N}\big{(}\gamma^{\prime}(t),\gamma^{\prime}(t)\big{)}\right\\}\big{]}\,dt,$

where $R_{g}$ is the Riemannian curvature tensor of $(M,g)$. Note that if
$N=n$, the forth term on the right-hand side of (3.3) always vanishes since we
set $f^{\prime}_{\gamma}(t)^{2}/(N-n)=0$.

We take a parameter $\lambda$ satisfying

(3.4)
$\displaystyle\begin{cases}|(1-\varepsilon)\lambda-1|\leq\sqrt{\dfrac{N-1}{N-n}}\qquad&\text{
if }N\in(-\infty,1]\cup(n,\infty]\text{ and }\varepsilon\neq
1,\\\\[-2.84544pt] \\\ \lambda\in\mathbb{R}\qquad&\text{ if either }N=n\text{
or }\varepsilon=1.\end{cases}$

We define $\Phi(\lambda)$, $\Psi(\lambda)$ by

$\Phi(\lambda)=\begin{cases}b^{\lambda}\,&\text{if }\lambda>0,\\\ 1\,&\text{if
}\lambda=0,\\\ a^{\lambda}\,&\text{if
}\lambda<0,\end{cases}\qquad\Psi(\lambda)=\begin{cases}a^{\lambda}\,&\text{if
}\lambda>0,\\\ 1\,&\text{if }\lambda=0,\\\ b^{\lambda}\,&\text{if
}\lambda<0.\end{cases}$

If we choose

$\alpha(t)=e^{\frac{(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\sin\left(\frac{\pi
t}{L}\right),$

then we have

$\alpha^{\prime}(t)=\frac{(1-\varepsilon)\lambda}{n-1}f^{\prime}_{\gamma}(t)\alpha(t)+\frac{\pi
t}{L}e^{\frac{(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\cos\left(\frac{\pi
t}{L}\right).$

We estimate the right-hand side of (3.3). The first term constructs to

(3.5) $\displaystyle-(n-1)H\int_{0}^{L}\alpha(t)^{2}\,dt$
$\displaystyle\leq-(n-1)H\int_{0}^{L}\Psi(\lambda)\sin^{2}\left(\frac{\pi
t}{L}\right)\,dt$ $\displaystyle\leq-\frac{(n-1)HL}{2}\Psi(\lambda).$

The second term is estimated as

(3.6) $\displaystyle\quad\int_{0}^{L}(n-1)\alpha^{\prime}(t)^{2}\,dt$
$\displaystyle=\frac{(n-1)\pi^{2}}{L^{2}}\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\cos^{2}\left(\frac{\pi
t}{L}\right)\,dt+\frac{(1-\varepsilon)\pi\lambda}{L}\int_{0}^{L}f^{\prime}_{\gamma}(t)e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\sin\left(\frac{2\pi
t}{L}\right)\,dt$
$\displaystyle\quad+\frac{(1-\varepsilon)^{2}\lambda^{2}}{n-1}\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt$
$\displaystyle=\frac{(n-1)\pi^{2}}{L^{2}}\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\cos^{2}\left(\frac{\pi
t}{L}\right)\,dt-\frac{(n-1)\pi^{2}}{L^{2}}\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\cos\left(\frac{2\pi
t}{L}\right)\,dt$
$\displaystyle\quad+\frac{(1-\varepsilon)^{2}\lambda^{2}}{n-1}\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt$
$\displaystyle=\frac{(n-1)\pi^{2}}{2L^{2}}\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\left(1-\cos\left(\frac{\pi
t}{L}\right)\right)\,dt+\frac{(1-\varepsilon)^{2}\lambda^{2}}{n-1}\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt$
$\displaystyle\leq\frac{(n-1)\pi^{2}}{2L}\Phi(\lambda)+\frac{(1-\varepsilon)^{2}\lambda^{2}}{n-1}\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt.$

We calculate the third term as

(3.7)
$\displaystyle\begin{split}&\quad\int_{0}^{L}\alpha(t)^{2}f_{\gamma}^{\prime\prime}(t)\,dt\\\
&=-\int_{0}^{L}2\alpha(t)\alpha^{\prime}(t)f^{\prime}_{\gamma}(t)\,dt\\\
&=-\frac{\pi}{L}\int_{0}^{L}f^{\prime}_{\gamma}(t)e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\sin\left(\frac{2\pi
t}{L}\right)\,dt-\frac{2(1-\varepsilon)\lambda}{n-1}\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt.\end{split}$

The last term constructs to

(3.8)
$\displaystyle\begin{split}&\quad\int_{0}^{L}\alpha(t)^{2}\left\\{(n-1)H-\mathrm{Ric}_{N-}\big{(}\gamma^{\prime}(t),\gamma^{\prime}(t)\big{)}\right\\}\,dt\\\
&\leq\int_{0}^{L}\alpha(t)^{2}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,dt\\\
&\leq\Phi(\lambda)\int_{0}^{L}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,dt.\end{split}$

Combining (3.5), (3.6), (3.7), (3.8) gives

(3.9)
$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle\leq-\frac{(n-1)HL}{2}\Psi(\lambda)+\frac{(n-1)\pi^{2}}{2L}\Phi(\lambda)$
$\displaystyle\quad+\widetilde{D}_{1}(\lambda)+\Phi(\lambda)\int_{0}^{L}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,dt,$

where

$\displaystyle\widetilde{D}_{1}(\lambda)$
$\displaystyle=-\frac{\pi}{L}\int_{0}^{L}f^{\prime}_{\gamma}(t)e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\sin\left(\frac{2\pi
t}{L}\right)\,dt$
$\displaystyle\quad+\left(\frac{(1-\varepsilon)^{2}\lambda^{2}}{(n-1)^{2}}-\frac{2(1-\varepsilon)\lambda}{n-1}-\frac{1}{N-n}\right)\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt.$

First we consider the case of $\varepsilon=1$. We see that

$\displaystyle\widetilde{D}_{1}(\lambda)$
$\displaystyle=-\frac{\pi}{L}\int_{0}^{L}f^{\prime}_{\gamma}(t)e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\sin\left(\frac{2\pi
t}{L}\right)\,dt-\frac{1}{N-n}\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt$
$\displaystyle=(N-n)\frac{\pi^{2}}{L^{2}}\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\cos^{2}\left(\frac{\pi
t}{L}\right)\,dt$
$\displaystyle\quad-(N-n)\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\left(\frac{\pi}{L}\cos\left(\frac{\pi
t}{L}\right)+\frac{f^{\prime}_{\gamma}(t)}{N-n}\sin\left(\frac{\pi
t}{L}\right)\right)^{2}\,dt$
$\displaystyle\leq\frac{(N-n)\pi^{2}}{2L}\Phi(\lambda).$

If $\varepsilon\neq 1$, then $\lambda\neq 0$ and it follows from (3.4) that

$\displaystyle\widetilde{D}_{1}(\lambda)$
$\displaystyle=-\frac{\pi}{L}\int_{0}^{L}f^{\prime}_{\gamma}(t)e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\sin\left(\frac{2\pi
t}{L}\right)\,dt$
$\displaystyle\quad+\left(\frac{(1-\varepsilon)^{2}\lambda^{2}}{(n-1)^{2}}-\frac{2(1-\varepsilon)\lambda}{n-1}-\frac{1}{N-n}\right)\int_{0}^{L}\alpha(t)^{2}f^{\prime}_{\gamma}(t)^{2}\,dt$
$\displaystyle\leq\frac{(n-1)\pi^{2}}{(1-\varepsilon)L^{2}\lambda}\int_{0}^{L}e^{\frac{2(1-\varepsilon)\lambda}{n-1}f_{\gamma}(t)}\cos\left(\frac{2\pi
t}{L}\right)\,dt$
$\displaystyle\leq\frac{(n-1)\pi}{|1-\varepsilon|L\lambda}\big{(}\Phi(\lambda)-\Psi(\lambda)\big{)}.$

Moreover, if we define

$\displaystyle\widetilde{D}(\lambda)=\begin{cases}\sqrt{\dfrac{N-1}{n-1}\left(\dfrac{\Phi(\lambda)}{\Psi(\lambda)}\right)}&\text{if
}\varepsilon=1,\\\\[1.0pt] \\\
\sqrt{\dfrac{\Phi(\lambda)}{\Psi(\lambda)}+\dfrac{2}{|1-\varepsilon|\pi\lambda}\left(\dfrac{\Phi(\lambda)}{\Psi(\lambda)}-1\right)}&\text{if
}\varepsilon\neq 1,\end{cases}$

then we find that

(3.10)
$\displaystyle\begin{split}\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}&\leq-\frac{(n-1)HL\Psi(\lambda)}{2}\left(1-\frac{\pi^{2}}{HL^{2}}\widetilde{D}(\lambda)^{2}\right)\\\
&\quad+\Phi(\lambda)\int_{0}^{L}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,dt.\end{split}$

Substituting (LABEL:L-com) for (3.10) gives

$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle\leq-\frac{(n-1)HL\Psi(\lambda)}{2}\left(1-\frac{\pi^{2}}{HL^{2}}\widetilde{D}(\lambda)^{2}\right)$
$\displaystyle\quad+2r\,\left(\frac{2b^{2}D}{a^{2}r}\right)^{\frac{1}{c}+1}\frac{\Phi(\lambda)}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

Choose $T=T(\eta)>2$ such that

(3.11) $\frac{1}{1-\frac{2}{T}}\leq\frac{\pi+\eta}{\pi+\frac{\eta}{2}}$

and let $r=D/T$. By the assumption $y_{i}\in\overline{B}(p_{i},r)$ for $i=1,2$
together with the triangle inequality, we find that

(3.12) $L=d(y_{1},y_{2})\geq d(p_{1},p_{2})-2r=D\left(1-\frac{2}{T}\right).$

Therefore we have

$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle\leq-\frac{(n-1)HL\Psi(\lambda)}{2}\left(1-\frac{\pi^{2}}{HL^{2}}\widetilde{D}(\lambda)^{2}\right)$
$\displaystyle\quad+2\,\left(\frac{2b^{2}}{a^{2}}\right)^{\frac{1}{c}+1}T^{\frac{1}{c}}\frac{L}{1-\frac{2}{T}}\frac{\Phi(\lambda)}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

We consider the limit as

(3.13) $\displaystyle\lambda\to\lambda_{0}=\begin{cases}0&\text{if
}N\in[n,\infty],\\\
\dfrac{1}{1-\varepsilon}\left(1-\sqrt{\dfrac{N-1}{N-n}}\right)&\text{if
}N\in(-\infty,1].\end{cases}$

Then we have $\widetilde{D}(\lambda)\to\widetilde{D}(n,N,\varepsilon,a,b)$ as
$\lambda\to\lambda_{0}$. We set

$\displaystyle\delta_{1}(n,N,\varepsilon,a,b,H,\eta)$
$\displaystyle=\frac{H(n-1)(1-\frac{2}{T})}{2^{\frac{1}{c}+3}T^{\frac{1}{c}}}\left(1-\frac{\pi^{2}}{\left(\pi+\frac{\eta}{2}\right)^{2}}\right)\left(\frac{a}{b}\right)^{\frac{2}{c}+2+\lambda_{0}}.$

If we assume that

$\frac{1}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta_{1}(n,N,\varepsilon,a,b,H,\eta),$

then we get

$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle\leq-\frac{(n-1)HLa^{\lambda_{0}}}{2}\left(1-\frac{\pi^{2}}{HL^{2}}\widetilde{D}^{2}\right)+\frac{(n-1)HLa^{\lambda_{0}}}{2}\left(1-\frac{\pi^{2}}{\left(\pi+\frac{\eta}{2}\right)^{2}}\right)$
$\displaystyle=-\frac{(n-1)H\pi^{2}a^{\lambda_{0}}}{2L\left(\pi+\frac{\eta}{2}\right)^{2}}\left(L^{2}-\frac{1}{H}\left(\pi+\frac{\eta}{2}\right)^{2}\widetilde{D}^{2}\right).$

Since $\gamma$ is a minimal geodesic, it follows

$\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}\geq 0.$

Therefore we obtain

$L\leq\frac{1}{\sqrt{H}}\left(\pi+\frac{\eta}{2}\right)\widetilde{D}(n,N,\varepsilon,a,b).$

We observe from (3.11), (3.12) that

$D\leq\frac{L}{1-\frac{2}{T}}\leq\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}(n,N,\varepsilon,a,b).$

This completes the proof of the theorem. $\square$

######  Remark 3.1.

In Theorem 1.7, when $\varepsilon\neq 1$, if we take $a^{\prime},b^{\prime}$
such that

$\frac{b^{\prime}}{a^{\prime}}=\exp\left(\frac{2|1-\varepsilon|}{n-1}(\sup
f-\inf f)\right),$

then
$\delta_{1}(n,N,\varepsilon,a,b,H,\eta)\leq\delta_{1}(n,N,\varepsilon,a^{\prime},b^{\prime},H,\eta)$
holds. This implies that the assumption of Theorem 1.7 also holds when we
replace $a,b$ by $a^{\prime},b^{\prime}$. We find that

$\displaystyle\widetilde{D}(n,N,\varepsilon,a^{\prime},b^{\prime})=\begin{cases}\sqrt{1+\dfrac{4(\sup
f-\inf f)}{(n-1)\pi}}&\quad\text{if }N\in[n,\infty],\\\\[1.0pt] \\\
\sqrt{\lambda_{1}+\dfrac{2(n-N)(\lambda_{1}-1)}{(n-1)\pi}\left(1+\sqrt{\dfrac{N-1}{N-n}}\right)}&\quad\text{if
}N\in(-\infty,1],\end{cases}$

where

$\lambda_{1}=\exp\left(\dfrac{2(\sup f-\inf
f)}{n-1}\left(1-\sqrt{\frac{N-1}{N-n}}\right)\right).$

##  4\. Proof of Theorem 1.8

We provide the following lemma to prove Theorem 1.8.

######  Lemma 4.1.

Let $n,N,\varepsilon,K,a,b,H,R$ and $\eta$ as in Theorem 1.8. Assume $R,\eta$
satisfy

(4.1)
$R>\frac{\pi}{\sqrt{H}}\widetilde{D}(n,N,\varepsilon,a,b)\quad\text{and}\quad
0<\eta<\eta_{*}(H,R,\widetilde{D})=\frac{4}{7}\left(\frac{R\sqrt{H}}{\widetilde{D}}-\pi\right).$

There exists a positive constant $\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta)$
such that if an $n$-dimensional complete weighted manifold $(M,g,\mu_{f})$
satisfies the $(N,K,\varepsilon,a,b)$-condition and

(4.2) $\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta)$

then $M$ is compact and

$\mathrm{diam}(M)\leq\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}(n,N,\varepsilon,a,b).$

  * Proof. The proof goes by contradiction, that is, there exist points $p_{1},q\in M$ such that the distance from $p_{1}$ to $q$ is greater than $(\pi+\eta)\widetilde{D}/\sqrt{H}$. Then there exists $p_{2}\in M$ such that $p_{2}$ lies in a unit minimal geodesic from $p_{1}$ to $q$ and

$\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}<d(p_{1},p_{2})<R-\frac{3\eta\widetilde{D}}{4\sqrt{H}}.$

First we set $W=B\left(p_{1},R\right)$ for $p_{1}\in M$ and
$r=\eta\widetilde{D}/4\sqrt{H}$ for $\eta<\eta_{*}(H,R,\widetilde{D})$. We put
$A_{i}=B(p_{i},r)\subset W$ for $i=1,2$. The triangle inequality that for
$y_{i}\in A_{i}$ with $i=1,2$ yields

$d(y_{1},y_{2})\leq d(y_{1},p_{1})+d(p_{1},p_{2})+d(p_{2},y_{2})<R-r.$

On the other hand, the distance from $y_{1}\in A_{1}$ to the boundary of $W$
is greater than $R-r$. This means that all unit minimal geodesics from
$y_{1}\in A_{1}$ to $y_{2}\in A_{2}$ lie in $W$. Using Theorem 2.4, we see
that

$\displaystyle\quad\inf_{(y_{1},y_{2})\in\overline{A}_{1}\times\overline{A}_{2}}\int_{0}^{d(y_{1},y_{2})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}(\gamma_{y_{1},y_{2}})\,ds$
$\displaystyle\leq
2rC(n,c,K)\left(\frac{1}{\mu_{f}(A_{1})}+\frac{1}{\mu_{f}(A_{2})}\right)\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

We set

$v_{cK,a}(R)=\int_{0}^{R/a}{\bf sn}_{cK}(\tau)^{1/c}\,d\tau.$

Theorem 2.2 implies

$\dfrac{\mu_{f}\big{(}B(p_{1},R)\big{)}}{\mu_{f}\big{(}B(p_{1},r)\big{)}}\leq\frac{b}{a}\cdot\frac{v_{cK,a}(R)}{v_{cK,b}(r)}$

and Lemma 2.6 gives

$C(n,c,K)\leq\widetilde{C}=\frac{{\bf
sn}_{cK}\big{(}(R-r)/a\big{)}^{1/c}}{{\bf
sn}_{cK}\big{(}(R-r)/b\big{)}^{1/c}}.$

We observe that

(4.3)
$\displaystyle\begin{split}&\quad\inf_{(y_{1},y_{2})\in\overline{A}_{1}\times\overline{A}_{2}}\int_{0}^{d(y_{1},y_{2})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}(\gamma_{y_{1},y_{2}})\,ds\\\
&\leq
2rC(n,c,K)\frac{b}{a}\left(\frac{v_{cK,a}\left(R\right)}{v_{cK,b}(r)}\frac{1}{\mu_{f}\big{(}B(p_{1},R)\big{)}}+\frac{v_{cK,a}\left(2R\right)}{v_{cK,b}(r)}\frac{1}{\mu_{f}\big{(}B(p_{2},2R)\big{)}}\right)\\\
&\quad\times\int_{B(p_{1},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\\\
&\leq
2r\widetilde{C}\frac{b}{a}\left(\frac{v_{cK,a}\left(R\right)+v_{cK,a}\left(2R\right)}{v_{cK,b}(r)}\right)\frac{1}{\mu_{f}\big{(}B(p_{1},R)\big{)}}\int_{B(p_{1},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f},\end{split}$

where we use $B(p_{1},R)\subset B(p_{2},2R)$ in the second inequality.

We find a unit speed minimal geodesic $\gamma$ from $y_{1}\in\overline{A}_{1}$
to $y_{2}\in\overline{A}_{2}$ that attains the infimum of (4.3). We put
$L=d(y_{1},y_{2})$. With the same argument of Theorem 1.7, we utilize (3.10)
and take the limit as $\lambda\to\lambda_{0}$ again, we see that

$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle\leq-\frac{(n-1)HLa^{\lambda_{0}}}{2}\left(1-\frac{\pi^{2}}{HL^{2}}\widetilde{D}^{2}\right)+\frac{\eta\widetilde{D}}{2\sqrt{H}}\frac{b^{\lambda_{0}+1}\widetilde{C}}{a}\left(\frac{v_{cK,a}\left(R\right)+v_{cK,a}\left(2R\right)}{v_{cK,b}\left(\frac{\eta\widetilde{D}}{4\sqrt{H}}\right)}\right)$
$\displaystyle\quad\times\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

We set

(4.4)
$\displaystyle\begin{split}&\quad\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta)\\\
&=\frac{H(n-1)(\pi+\frac{\eta}{2})}{\eta\widetilde{C}}\frac{v_{cK,b}\left(\frac{\eta\widetilde{D}}{4\sqrt{H}}\right)}{v_{cK,a}\left(R\right)+v_{cK,a}\left(2R\right)}\left(1-\frac{\pi^{2}}{\left(\pi+\frac{\eta}{2}\right)^{2}}\right)\left(\frac{a}{b}\right)^{\lambda_{0}+1}.\end{split}$

If we assume that

$\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta),$

then we have

$\displaystyle\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}$
$\displaystyle\leq-\frac{(n-1)HLa^{\lambda_{0}}}{2}\left(1-\frac{\pi^{2}}{HL^{2}}\widetilde{D}^{2}\right)$
$\displaystyle\quad+\frac{(n-1)(\pi+\frac{\eta}{2})\sqrt{H}a^{\lambda_{0}}\widetilde{D}}{2}\left(1-\frac{\pi^{2}}{(\pi+\frac{\eta}{2})^{2}}\right)$
$\displaystyle=-\frac{n-1}{2}\left(L-\frac{1}{\sqrt{H}}\left(\pi+\frac{\eta}{2}\right)\widetilde{D}\right)\left(Ha^{\lambda_{0}}+\frac{(n-1)\pi^{2}\widetilde{D}a^{\lambda_{0}}\sqrt{H}}{2L(\pi+\frac{\eta}{2})}\right).$

Since $\gamma$ is a minimal geodesic, it follows

$\left.\sum_{i=1}^{n-1}\frac{d^{2}L_{i}}{ds^{2}}\right|_{s=0}\geq 0.$

Therefore we obtain

$L\leq\frac{1}{\sqrt{H}}\left(\pi+\frac{\eta}{2}\right)\widetilde{D}.$

By the triangle inequality, we have

(4.5) $d(p_{1},p_{2})\leq L+2r\leq\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}.$

On the other hand, we assumed that

$\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}<d(p_{1},p_{2})<R-\frac{3\eta\widetilde{D}}{4\sqrt{H}},$

but by (4.5), no geodesic starting from $p_{1}\in M$ of a length greater than
$(\pi+\eta)\widetilde{D}/\sqrt{H}$ can be minimal. This completes the proof of
the lemma. $\square$

Finally, we show Theorem 1.8.

  * Proof of Theorem 1.8. We divide the cases into three parts:

    1. (1)

$R\leq\pi\widetilde{D}(n,N,\varepsilon,a,b)/\sqrt{H}$,

    2. (2)

$R>\pi\widetilde{D}(n,N,\varepsilon,a,b)/\sqrt{H}$ and
$\eta\geq\eta_{*}(H,R,\widetilde{D})$,

    3. (3)

$R>\pi\widetilde{D}(n,N,\varepsilon,a,b)/\sqrt{H}$ and
$\eta<\eta_{*}(H,R,\widetilde{D})$.

The case (3) is already discussed in Lemma 4.1. Thus it is enough to consider
the cases (1) and (2).

Let $R^{\prime}=R^{\prime}(\eta)>\pi\widetilde{D}/\sqrt{H}$ be fixed such that
$\eta<\eta_{*}(H,R^{\prime},\widetilde{D})$. For each $p\in M$, we consider
the discrete subset $\\{x_{i}\\}\subset B(p,R^{\prime})$ for
$i=1,\ldots,T_{2}$ such that

$B(p,R^{\prime})\subset\bigcup_{i=1}^{T_{2}}B(x_{i},R)$

and $d(x_{i},x_{j})>R$ for $i\neq j$, where $T_{2}$ is the maximal number of
the $R$-discrete net of $B(p,R^{\prime})$ (see [SHT]*Definition 3.1 for the
definition of the $R$-discrete net). We now claim:

######  Claim 4.2.

Let $(M,g,\mu_{f})$ be an $n$-dimensional complete weighted manifold
satisfying the $(N,K,\varepsilon,a,b)$-condition. Set $T_{2}>0$ as above. Then
it follows

$T_{2}\leq\frac{b}{a}\cdot\frac{v_{cK,b}(2R^{\prime}+R)}{v_{cK,a}\left(R/2\right)}.$

  * Proof. Take $i_{0}\in\\{\,1,\ldots,T_{2}\,\\}$ such that

$\mu_{f}\big{(}B(x_{i_{0}},R/2)\big{)}=\min_{1\leq i\leq
T_{2}}\mu_{f}\big{(}B(x_{i},R/2)\big{)}.$

Since $B(x_{i},R/2)\cap B(x_{j},R/2)=\emptyset$ for any $i\neq j$ and
$B(p,R^{\prime}+R)\subset B(x_{i_{0}},2R^{\prime}+R)$ hold, we have

$\displaystyle 1$
$\displaystyle=\frac{\mu_{f}\big{(}B(p,R^{\prime}+R)\big{)}}{\mu_{f}\big{(}B(p,R^{\prime}+R)\big{)}}$
$\displaystyle\leq\frac{\mu_{f}\big{(}B(p,R^{\prime}+R)\big{)}}{\mu_{f}\big{(}B(p,R^{\prime}+R)\cap\bigcup_{i=1}^{T_{2}}B(x_{i},R/2)\big{)}}$
$\displaystyle\leq\frac{\mu_{f}\big{(}B(p,R^{\prime}+R)\big{)}}{T_{2}\cdot\mu_{f}\big{(}B(x_{i_{0}},R/2)\big{)}}$
$\displaystyle\leq\frac{\mu_{f}\big{(}B(x_{i_{0}},2R^{\prime}+R)\big{)}}{T_{2}\cdot\mu_{f}\big{(}B(x_{i_{0}},R/2)\big{)}}.$

By using Theorem 2.2, we obtain

$T_{2}\leq\frac{b}{a}\cdot\frac{v_{cK,b}(2R^{\prime}+R)}{v_{cK,a}(R/2)}.$

$\Diamond$

Then we find that, for any $z\in M$,

$\displaystyle\quad\frac{1}{\mu_{f}\big{(}B(z,R^{\prime})\big{)}}\int_{B(z,R^{\prime})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq\frac{T_{2}}{\mu_{f}\big{(}B(z,R^{\prime})\big{)}}\sup_{x_{i}\in
B(z,R^{\prime})}\int_{B(x_{i},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq
T_{2}\frac{b}{a}\frac{1}{\mu_{f}\big{(}B(z,R^{\prime}+R)\big{)}}\frac{v_{cK,b}(R^{\prime}+R)}{v_{cK,a}(R^{\prime})}$
$\displaystyle\quad\times\sup_{x_{i}\in
B(z,R^{\prime})}\int_{B(x_{i},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq\frac{b^{2}}{a^{2}}\frac{v_{cK,b}(2R^{\prime}+R)}{v_{cK,a}(R/2)}\frac{v_{cK,b}(R^{\prime}+R)}{v_{cK,a}(R^{\prime})}$
$\displaystyle\quad\times\sup_{x_{i}\in
B(z,R^{\prime})}\frac{1}{\mu_{f}\big{(}B(x_{i},R)\big{)}}\int_{B(x_{i},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f},$

where we use the estimate of $T_{2}$ and $B(x_{i},R)\subset B(z,R^{\prime}+R)$
for $1\leq i\leq T_{2}$ in the third inequality. This provides that

$\displaystyle\quad\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R^{\prime})\big{)}}\int_{B(x,R^{\prime})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq\frac{b^{2}}{a^{2}}\frac{v_{cK,b}(2R^{\prime}+R)}{v_{cK,a}\left(R/2\right)}\frac{v_{cK,b}(R^{\prime}+R)}{v_{cK,a}(R^{\prime})}\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R)\big{)}}\int_{B(x,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

We set

$\displaystyle\quad\delta^{\prime}_{2}(n,N,K,\varepsilon,a,b,H,R,\eta)$
$\displaystyle=\frac{a^{2}}{b^{2}}\frac{v_{cK,a}(R/2)}{v_{cK,b}(2R^{\prime}(\eta)+R)}\frac{v_{cK,a}\big{(}R^{\prime}(\eta)\big{)}}{v_{cK,b}\big{(}R^{\prime}(\eta)+R\big{)}}\delta_{2}(n,N,K,\varepsilon,a,b,H,R^{\prime}(\eta),\eta).$

If we assume that

$\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R)\big{)}}\int_{B(x,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta^{\prime}_{2}(n,N,K,\varepsilon,a,b,H,R,\eta),$

we obtain

$\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R^{\prime})\big{)}}\int_{B(x,R^{\prime})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}\leq\delta_{2}(n,N,K,\varepsilon,a,b,H,R^{\prime}(\eta),\eta).$

By Lemma 4.1, this completes the proof of the cases (1), (2) and the theorem.
$\square$

We prepare the following lemma to provide a slightly weak result compared with
Lemma 4.1.

######  Lemma 4.3.

Let $\delta_{2}=\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta)$ be given as in
(4.4). Then $\delta_{2}$ is strictly increasing in $\eta\in(0,\eta_{*})$.

  * Proof. Set

(4.6)
$G(\eta)=\left(\frac{1}{2}+\frac{\pi}{\eta}\right)\int_{0}^{\eta\widetilde{D}/4b\sqrt{H}}{\bf
sn}_{cK}(t)^{\frac{1}{c}}\,dt.$

Note that the monotonicity of ${\bf sn}_{cK}$. We observe that

$\displaystyle G^{\prime}(\eta)$
$\displaystyle=\frac{\widetilde{D}}{4b\sqrt{H}}\left(\frac{1}{2}+\frac{\pi}{\eta}\right)\left({\bf
sn}_{cK}\left(\frac{\eta\widetilde{D}}{4b\sqrt{H}}\right)\right)^{\frac{1}{c}}-\frac{\pi}{\eta^{2}}\int_{0}^{\eta\widetilde{D}/4b\sqrt{H}}{\bf
sn}_{cK}(t)^{\frac{1}{c}}\,dt$
$\displaystyle>\frac{\widetilde{D}}{4b\sqrt{H}}\left(\frac{1}{2}+\frac{\pi}{\eta}\right)\left({\bf
sn}_{cK}\left(\frac{\eta\widetilde{D}}{4b\sqrt{H}}\right)\right)^{\frac{1}{c}}-\frac{\pi}{\eta^{2}}\frac{\eta\widetilde{D}}{4b\sqrt{H}}\left({\bf
sn}_{cK}\left(\frac{\eta\widetilde{D}}{4b\sqrt{H}}\right)\right)^{\frac{1}{c}}$
$\displaystyle=\frac{\widetilde{D}}{8b\sqrt{H}}\left({\bf
sn}_{cK}\left(\frac{\eta\widetilde{D}}{4b\sqrt{H}}\right)\right)^{\frac{1}{c}}$
$\displaystyle>0.$

Thus $G(\eta)$ is strictly increasing on $(0,\eta_{*})$. Moreover

(4.7) $1-\frac{\pi^{2}}{\left(\pi+\frac{\eta}{2}\right)^{2}}$

and, by Claim 2.7,

(4.8) $\frac{{\bf
sn}_{cK}\Big{(}\frac{1}{b}\left(R-\frac{\eta\widetilde{D}}{4\sqrt{H}}\right)\Big{)}^{\frac{1}{c}}}{{\bf
sn}_{cK}\Big{(}\frac{1}{a}\left(R-\frac{\eta\widetilde{D}}{4\sqrt{H}}\right)\Big{)}^{\frac{1}{c}}}$

are also strictly increasing with respect to $\eta$. Since $\delta_{2}$ was
given by multiplication of (4.6), (4.7), (4.8) and a positive constant
independent of $\eta$, it follows that $\delta_{2}$ is strictly increasing in
$\eta\in(0,\eta_{*})$. $\square$

We take the limit of (4.4) as $\eta\to\eta_{*}$, where $\eta_{*}$ is given in
(4.1). Then we have a diameter estimate of a weighted manifold.

######  Corollary 4.4.

Let $n,N,\varepsilon,K,a,b,H,R$ and $\eta_{*}$ as in Lemma 4.1. There exists a
positive constant $\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta_{*})$ such that
if an $n$-dimensional complete weighted manifold $(M,g,\mu_{f})$ satisfies the
$(N,K,\varepsilon,a,b)$-condition and

(4.9) $\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}<\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta_{*})$

then $M$ is compact and $\mathrm{diam}(M)<R$.

  * Proof. By Lemma 4.3, if (4.9) holds, then there exists $\eta>0$ such that $\eta<\eta_{*}(H,R,\widetilde{D})$ satisfying (4.2). Hence Lemma 4.1 implies

$\mathrm{diam}(M)\leq\frac{\pi+\eta}{\sqrt{H}}\widetilde{D}(n,N,\varepsilon,a,b)<R.$

$\square$

##  5\. Fundamental group under integral curvature bound and
$\varepsilon$-range

We shall show a finiteness of the fundamental group of $M$. Compared with
Corollary 4.4, we need to assume a slightly strong condition about integral
curvature bound.

######  Corollary 5.1.

Let $(M,g,\mu_{f})$ be an $n$-dimensional complete weighted manifold
satisfying the $(N,K,\varepsilon,a,b)$-condition. If there exists $H>0$,
$R>\pi\tilde{D}(n,N,\varepsilon,a,b)/\sqrt{H}$ such that

$\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}<\frac{a}{b}\frac{v_{cK,b}(R)}{v_{cK,a}(3R)}\delta_{2},$

where $\delta_{2}=\delta_{2}(n,N,K,\varepsilon,a,b,H,R,\eta_{*})$ is given in
(4.4), then the universal cover of $M$ is compact, and hence $\pi_{1}(M)$ is a
finite group.

  * Proof. We find that, by $3R/a\geq R/b$,

$\frac{a}{b}\frac{v_{cK,b}(R)}{v_{cK,a}(3R)}<1.$

Since $(M,g,\mu_{f})$ satisfies the assumption of Corollary 4.4, we have
$\mathrm{diam}(M)<R$.

We denote by $k:\widetilde{M}\to M$ the universal Riemannian covering. Set
$\tilde{g}=k^{*}g$, $\tilde{f}=f\circ k$. Fix $\tilde{x}\in\widetilde{M}$. Let
$F$ be the fundamental domain of $k$ that contains
$\tilde{x}\in\widetilde{M}$. We find that

$\mu_{f}(M)=\mu_{\tilde{f}}(F),\qquad\widetilde{M}=\bigcup_{\alpha\in\Gamma}\alpha
F,$

where $\Gamma$ is the deck transformation group of $\widetilde{M}$. We set

$T=\inf\left\\{\\#\Gamma_{0}\,\middle|\,B(\tilde{x},R)\subset\bigcup_{\alpha\in\Gamma_{0}}\alpha
F\right\\}.$

Fix $w\in B(\tilde{x},R)\cap\alpha F$ and $z\in\alpha F$ for each
$\alpha\in\Gamma_{0}$. Since $d(y,z)<2R$ for all $y\in\alpha F$, we see that

$d(\tilde{x},z)\leq d(\tilde{x},w)+d(w,z)<3R.$

Hence this implies

$\bigcup_{i=1}^{T}\alpha_{i}F\subset B(\tilde{x},3R)$

and we find that

(5.1)
$T\cdot\mu_{f}(M)=T\cdot\mu_{\tilde{f}}(F)\leq\mu_{\tilde{f}}\big{(}B(\tilde{x},3R)\big{)}.$

Then it follows from (5.1) that

$\displaystyle\quad\frac{1}{\mu_{\tilde{f}}\big{(}B(\tilde{x},R)\big{)}}\int_{B(\tilde{x},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{\tilde{f}}$
$\displaystyle\leq\frac{T}{\mu_{\tilde{f}}\big{(}B(\tilde{x},R)\big{)}}\int_{F}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{\tilde{f}}$
$\displaystyle=\frac{T}{\mu_{\tilde{f}}\big{(}B(\tilde{x},R)\big{)}}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq\frac{b}{a}\frac{v_{cK,b}(3R)}{v_{cK,a}(R)}\frac{T}{\mu_{\tilde{f}}\big{(}B(\tilde{x},3R)\big{)}}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq\frac{b}{a}\frac{v_{cK,b}(3R)}{v_{cK,a}(R)}\frac{1}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

Thus we observe that

$\displaystyle\quad\sup_{\tilde{x}\in\widetilde{M}}\frac{1}{\mu_{\tilde{f}}\big{(}B(\tilde{x},R)\big{)}}\int_{B(\tilde{x},R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{\tilde{f}}$
$\displaystyle\leq\frac{b}{a}\frac{v_{cK,b}(3R)}{v_{cK,a}(R)}\frac{1}{\mu_{f}(M)}\int_{M}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle<\delta_{2}.$

Since $(\widetilde{M},\tilde{g},\mu_{\tilde{f}})$ satisfies the assumption of
Corollary 4.4, $\widetilde{M}$ is compact and $\pi_{1}(M)$ is a finite group.
$\square$

Finally, we extend Corollary 5.1 to the cases that
$R\leq\pi\widetilde{D}/\sqrt{H}$.

######  Corollary 5.2.

Let $n,N,\varepsilon,K,a,b,H$ and $R$ as in Theorem 1.8. Set $R^{\prime}>0$
satisfies $R^{\prime}>\pi\widetilde{D}(n,N,\varepsilon,a,b)/\sqrt{H}$. There
exists a positive constant
$\tilde{\delta}(n,N,K,\varepsilon,a,b,H,R,R^{\prime})$ such that if an
$n$-dimensional complete weighted manifold $(M,g,\mu_{f})$ satisfies the
$(\varepsilon,a,b,N,K)$-condition and

$\sup_{p\in
M}\frac{1}{\mu_{f}\big{(}B(p,R)\big{)}}\int_{B(p,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}<\tilde{\delta}(n,N,K,\varepsilon,a,b,H,R,R^{\prime})$

then the universal cover of $M$ is compact, and hence $\pi_{1}(M)$ is a finite
group.

  * Proof. Let $R\leq\pi\widetilde{D}/\sqrt{H}$. With the same argument in the proof of Theorem 1.8, we find that

$\displaystyle\quad\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R^{\prime})\big{)}}\int_{B(x,R^{\prime})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle\leq\frac{b^{2}}{a^{2}}\frac{v_{cK,b}(2R^{\prime}+R)}{v_{cK,a}(R/2)}\frac{v_{cK,b}(R^{\prime}+R)}{v_{cK,a}(R^{\prime})}\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R)\big{)}}\int_{B(x,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}.$

If we assume that

$\displaystyle\quad\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R)\big{)}}\int_{B(x,R)}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle<\frac{a^{3}}{b^{3}}\frac{v_{cK,a}(R/2)}{v_{cK,b}(2R^{\prime}+R)}\frac{v_{cK,a}(R^{\prime})}{v_{cK,b}(R^{\prime}+R)}\frac{v_{cK,b}(R^{\prime})}{v_{cK,a}(3R^{\prime})}\delta_{2}(n,N,K,\varepsilon,a,b,H,R^{\prime},\eta_{*})$

holds, we obtain

$\displaystyle\quad\sup_{x\in
M}\frac{1}{\mu_{f}\big{(}B(x,R^{\prime})\big{)}}\int_{B(x,R^{\prime})}\big{(}(n-1)H-\mathrm{Ric}_{N-}\big{)}_{+}\,d\mu_{f}$
$\displaystyle<\frac{a}{b}\frac{v_{cK,b}(R^{\prime})}{v_{cK,a}(3R^{\prime})}\delta_{2}(n,N,K,\varepsilon,a,b,H,R^{\prime},\eta_{*}).$

Therefore, by Corollary 5.1, this completes the proof of the corollary.
$\square$

## References