# Compactness of sequences of warped product circles over spheres with
nonnegative scalar curvature

Wenchuan Tian  Department of Mathematics, University of California, Santa
Barbara, CA93106-3080<EMAIL_ADDRESS>and  Changliang Wang  School of
Mathematical Sciences and Institute for Advanced Study, Tongji University,
Shanghai 200092, China<EMAIL_ADDRESS>
###### Abstract.

Gromov and Sormani conjectured that a sequence of three dimensional Riemannian
manifolds with nonnegative scalar curvature and some additional uniform
geometric bounds should have a subsequence which converges in some sense to a
limit space with some generalized notion of nonnegative scalar curvature. In
this paper, we study the pre-compactness of a sequence of three dimensional
warped product manifolds with warped circles over standard ${\mathbb{S}}^{2}$
that have nonnegative scalar curvature, a uniform upper bound on the volume,
and a positive uniform lower bound on the $\operatorname{MinA}$, which is the
minimum area of closed minimal surfaces in the manifold. We prove that such a
sequence has a subsequence converging to a $W^{1,p}$ Riemannian metric for all
$p<2$, and that the limit metric has nonnegative scalar curvature in the
distributional sense as defined by Lee-LeFloch.

###### Contents

  1. 1  Introduction
  2. 2  Consequences of the geometric hypotheses on ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$
    1. 2.1 Basic consequences of the hypotheses
    2. 2.2 Spherical mean inequality
    3. 2.3 Ball average monotonicity
  3. 3 $W^{1,p}$ limit of warping function for $1\leq p<2$
    1. 3.1 $W^{1,p}$ limit function for $p<2$
    2. 3.2 Lower semi-continuous representative of the limit function
  4. 4 Positivity of the limit warping functions
    1. 4.1 $W^{1,2}$ regularity of limit of truncated warping functions
    2. 4.2 A $1$-sweepout of the warped product manifold ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$
    3. 4.3 Bound $\operatorname{MinA}$ from above by $L^{1}$-norm of warping function
    4. 4.4 Positivity of the limit of warping functions
    5. 4.5 Uniform systole positive lower bound
  5. 5 Nonnegative distributional scalar curvature of limit metric
    1. 5.1 $W^{1,p}$ limit Riemannian metric $g_{\infty}$
    2. 5.2 Nonnegative distributional scalar curvature of $g_{\infty}$
  6. A  $W^{1,2}$ convergence in ${\mathbb{S}}^{1}\times_{h}{\mathbb{S}}^{2}$ case
    1. A.1 Convergence of a subsequence
    2. A.2 Nonnegative distributional scalar curvature of the limit metric

##  1\.  Introduction

In [8] and [7], Gromov conjectured that a sequence of Riemannian manifolds
with nonnegative scalar curvature, $\operatorname{Scalar}\geq 0$, should have
a subsequence which converges in some weak sense to a limit space with some
generalized notion of “nonnegative scalar curvature”. In light of the examples
constructed by Basilio, Dodziuk, and Sormani in [2], the $\operatorname{MinA}$
condition in (2) below was added to prevent collapsing happening, and the
conjecture was made more precise at an IAS Emerging Topics Workshop co-
organized by Gromov and Sormani as follows [18]:

######  Conjecture 1.1.

Let $\\{M_{j}^{3}\\}_{j=1}^{\infty}$ be a sequence of closed oriented three
dimensional Riemannian manifolds without boundary satisfying

(1) $\operatorname{Scalar}_{j}\geq 0,\ \ {\rm Vol}(M_{j})\leq V,\ \
\operatorname{Diam}(M_{j})\leq D,$ (2)
$\operatorname{MinA}(M^{3}_{j})=\inf\\{\operatorname{Area}(\Sigma)\,:\,\Sigma\textrm{
closed min surf in }M_{j}^{3}\,\\}\geq A_{0}>0.$

Then there exists a subsequence which is still denoted as
$\\{M_{j}\\}_{j=1}^{\infty}$ that converges in the volume preserving intrinsic
flat sense to a three dimensional rectifiable limit space $M_{\infty}$.
Furthermore, $M_{\infty}$ is a connected geodesic metric space, that has
Euclidean tangent cones almost everywhere, and has nonnegative generalized
scalar curvature.

In a joint work with Jiewon Park [15], the authors confirmed Conjecture 1.1
for sequences of rotationally symmetric Riemannian manifolds
$(M^{3}_{j},g_{j})$. In our proof the $\operatorname{MinA}$ condition provides
a uniform lower bound for the warping functions in the closed region between
any two minimal surfaces. As a result, we can prevent counter examples like
the sequence of round spheres shrinking to a point, and we can also prevent
the formation of thin tunnels between two non-collapsed regions. The
regularity of the limit metric is high, and the convergence of the sequence of
warping functions is strong. In particular, in [15] we proved that the limit
warping function is Lipschitz and that the sequence of warping functions
converges to the limit function in the $W^{1,2}$ norm in closed regions away
from the two poles.

In this paper, we study the ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$
warped product case of the Conjecture 1.1. We consider the following:

######  Definition 1.2.

Let $\\{({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{j})\\}_{j=1}^{\infty}$ be a
sequence of Riemannian manifold such that

(3)
$g_{j}=g_{{\mathbb{S}}^{2}}+f_{j}^{2}g_{{\mathbb{S}}^{1}}=dr^{2}+\sin(r)^{2}d\theta^{2}+f_{j}^{2}d\varphi^{2},\text{
for }j=1,2,3,...$

where $g_{{\mathbb{S}}^{2}}$ and $g_{{\mathbb{S}}^{1}}$ are the standard
metrics on ${\mathbb{S}}^{2}$ and ${\mathbb{S}}^{1}$ respectively, and the
function $f_{j}:{\mathbb{S}}^{2}\to(0,\infty)$ is smooth for each $j$. Here
$r$ and $\theta$ are the geodesic polar coordinate for ${\mathbb{S}}^{2}$. We
also use the notation ${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ to
denote $({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{j})$.

We consider the convergence of the warping function and prove the sharp
regularity of the limit warping function in the following theorem:

######  Theorem 1.3.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product Riemannian manifolds such that each
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ has non-negative scalar
curvature. If we assume that

(4) ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\leq V\text{ and
}\operatorname{MinA}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\geq A>0,\
\ \forall j\in\mathbb{N},$

then we have the following:

  1. (i)

After passing to a subsequence if needed, the sequence of warping functions
$\\{f_{j}\\}_{j=1}^{\infty}$ converges to some limit function $f_{\infty}$ in
$L^{q}({\mathbb{S}}^{2})$ for all $q\in[1,\infty)$.

  2. (ii)

The limit function $f_{\infty}$ is in $W^{1,p}({\mathbb{S}}^{2})$, for all $p$
such that $1\leq p<2$.

  3. (iii)

The essential infimum of $f_{\infty}$ is strictly positive, i.e.
$\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0$.

  4. (iv)

If we allow $+\infty$ as a limit, then the limit

(5) $\overline{f_{\infty}}(x):=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}f_{\infty}$

exists for every $x\in{\mathbb{S}}^{2}$. Moreover, $\overline{f_{\infty}}$ is
lower semi-continuous and strictly positive everywhere on ${\mathbb{S}}^{2}$,
and $\overline{f_{\infty}}=f_{\infty}$ a.e. on ${\mathbb{S}}^{2}$.

The definition of essential infimum is given in Definition 4.6. In the proof
of convergence properties in items (i) and (ii) in Theorem 1.3, we only need
nonnegative scalar curvature condition and volume uniform upper bound
condition. In the proof of part (iii) of Theorem 1.3, we make essential use of
$\operatorname{MinA}$ condition combined with the spherical mean inequality
[Proposition 2.4], Min-Max minimal surface theory and a covering argument.
This is an interesting new way of applying the $\operatorname{MinA}$ condition
to prevent collapsing. Then the part (iv) follows from (iii) and an
interesting ball average monotonicity property [Proposition 2.6]. The ball
average monotonicity is obtained from spherical mean inequality by using the
trick as in the proof of Bishop-Gromov volume comparison theorem.

######  Remark 1.4.

The extreme example constructed by Sormani and authors in [19] shows that the
$W^{1,p}$ regularity for $1\leq p<2$ is sharp for the limit warping function
$f_{\infty}$.

By applying Theorem 1.3 and the spherical mean inequality [Proposition 2.4],
we obtain:.

######  Proposition 1.5.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product manifolds such that each
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ has non-negative scalar
curvature, and the sequence satisfies conditions in $(\ref{eqn-Intro-thm-main-
condition})$. Then there exists $j_{0}\in\mathbb{N}$ such that
$f_{j}(x)\geq\frac{e_{\infty}}{4}>0$, for all $j\geq j_{0}$ and
$x\in{\mathbb{S}}^{2}$, where $e_{\infty}=\inf_{{\mathbb{S}}^{2}}f_{\infty}>0$
obtained in Theorem 1.3.

As an application of Proposition 1.5, we have:

######  Corollary 1.6.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product manifolds such that each
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ has non-negative scalar
curvature, and the sequence satisfies conditions in $(\ref{eqn-Intro-thm-main-
condition})$. Then the systoles of
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$, for all $j\in\mathbb{N}$,
have a uniform positive lower bound given by
$\min\left\\{2\pi,\frac{e_{\infty}}{2}\pi\right\\}$, where
$e_{\infty}:=\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0$ obtained in Theorem
1.3.

The systole of a Riemannian manifold is defined to be the length of the
shortest closed geodesic in the manifold [Definition 4.16]. In order to
estimate systole of warped product manifolds:
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$, in Lemma 4.18 we establish
an interesting dichotomy property for closed geodesics in a general warped
product manifold $N\times_{f}{\mathbb{S}}^{1}$ with ${\mathbb{S}}^{1}$ as a
typical fiber, with metric tensor as $g=g_{N}+f^{2}g_{{\mathbb{S}}^{1}}$,
where $(N,g_{N})$ is a $n$-dimensional complete Riemannian manifold without
boundary and $f$ is a positive smooth function on $N$. The dichotomy property
in Lemma 4.18 has its own interests independently, and shall be useful in
other studies of closed geodesics in such warped product manifolds.

The convergence of the warping functions in Theorem 1.3 leads to the
convergence of the Riemannian metrics, we prove the following:

######  Theorem 1.7.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product Riemannian manifolds such that each
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ has non-negative scalar
curvature. If we assume that

(6) ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\leq V\text{ and
}\operatorname{MinA}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\geq A>0,\
\ \forall j\in\mathbb{N},$

Then there exists a subsequence $g_{j_{k}}$ and a (weak) warped product
Riemannian metric $g_{\infty}\in
W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for $p\in[1,2)$ such
that

(7) $g_{j_{k}}\rightarrow g_{\infty}\ \ \text{ in}\ \
L^{q}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0}),\ \ \forall
q\in[1,\infty).$

Theorem 1.7 is proved in §5.1. The definition of a (weak) warped product
Riemannian metric is given in Definition 5.1, and the spaces
$L^{q}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ and
$W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ are defined in
Definition 5.3. The $\operatorname{MinA}$ condition is used to prevent
$g_{j_{k}}$ converging to a non-metric tensor in
$W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$, with the help of the
non-collapsing property of $f_{\infty}$ in the item (iii) in Theorem 1.3.

In the limit space we calculate the scalar curvature as a distribution using
the definition by Lee and LeFloch [10], and we prove the following:

######  Theorem 1.8.

The limit metric $g_{\infty}$ obtained in Theorem 1.7 has nonnegative
distributional scalar curvature on ${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ in
the sense of Lee-LeFloch. [10]. Moreover, the total scalar curvatures of
$g_{j}$ converge to the distributional total scalar curvature of $g_{\infty}$.

Theorem 1.8 is proved in §5.2. In general, it is still an interesting and
difficult problem to formulate suitable notions of generalized (or weak)
nonnegative scalar curvature in Conjecture 1.1. A natural candidate is the
volume-limit notion of nonnegative scalar curvature. But recently Kazara and
Xu constructed a sequence of warped product metrics on
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ whose limit space does not have
nonnegative scalar curvature in the sense of volume-limit in Theorem 1.3 in
[9]. There are other candidates, like Gromov’s polyhedron comparison notion
[7, 12] and Burkhardt-Guim’s Ricci flow notion [4] of nonnegative scalar
curvature for $C^{0}$-metrics. However, as mentioned in Remark 1.4, the
$W^{1,p}$ regularity, for $1\leq p<2$, is the best regularity for our limit
metrics, and in general our limit metrics are not continuous. Lee and Lefloch
[10] defined the scalar curvature distribution for $W^{1,2}_{loc}$-metrics.
Our limit metric $g_{\infty}$ obtained in Theorem 1.7 does not satisfy the
regularity requirement in [10], but when we add up different terms in the
integrand, the divergent terms cancel with each other and the scalar curvature
is still well defined as a distribution. This is discussed in detail in Remark
5.18. Interestingly, we obtain the continuity of distributional total scalar
curvature in Theorem 1.8. More importantly, the scalar curvature distribution
of Lee-LeFloch enables us to see the concentration of scalar curvature on the
singular set, see §4.4 in [19].

In Appendix A, we study pre-compactness of the sequence of warped product
spheres over circle $(M^{3}_{j},g_{j})$, that is, $M^{3}_{j}$ are
diffeomorphic to ${\mathbb{S}}^{1}\times{\mathbb{S}}^{2}$ with warped product
metric tensors

(8) $g_{j}=g_{{\mathbb{S}}^{1}}+h_{j}^{2}g_{{\mathbb{S}}^{2}},\ \
\textrm{where}\ \ h_{j}:{\mathbb{S}}^{1}\rightarrow(0,\infty).$

The study of this case is similar to the rotationally symmetric case studied
in [15]. The key is to obtain a uniform bound for the norm of gradient of
$h_{j}$ from nonnegative scalar curvature condition [Lemma A.4]. By combining
this with uniform diameter upper bound and the $\operatorname{MinA}$
condition, we prove that a subsequence of $\\{h_{j}\\}^{\infty}_{j=1}$
converges in $C^{0}$ and $W^{1,2}$ sense to a bounded positive Lipschitz
function $h_{\infty}:{\mathbb{S}}^{1}\rightarrow(0,\infty)$ [Theorem A.1].
Moreover, we prove that the limit $W^{1,2}$ Riemannian metric
$g_{\infty}=g_{{\mathbb{S}}^{1}}+h^{2}_{\infty}g_{{\mathbb{S}}^{2}}$ has
nonnegative distributional scalar curvature in the sense of Lee-LeFloch
[Theorem A.2].

The proof of Theorem A.1 is similar to that of Theorems 4.1 and 4.8 in [15].
We include it here to show the difference with the rotationally symmetric case
and the difference with Theorem 1.3 and Theorem 1.7.

The proof of Theorem A.2 shows that in this case the regularity requirement in
Lee-LeFloch [10] is essential for the definition of the scalar curvature as a
distribution. This provides an interesting contrast with the proof of Theorem
1.8.

The article is organized as follows: in Section 2, we derive several analysis
properties of warping functions $f_{j}$ from the uniform geometric bounds of
metric $g_{j}$ as in (3). In particular, we show that metrics $g_{j}$ in (3)
have nonnegative scalar curvature if and only if the warping functions $f_{j}$
satisfy the differential inequality [Lemma 2.1]:

(9) $\Delta f_{j}\leq f_{j},\ \ \textrm{on}\ \ {\mathbb{S}}^{2},$

where $\Delta$ is the Lapacian on the standard round sphere
${\mathbb{S}}^{2}$, taken to be the trace of the Hessian. Moreover, a positive
number $V$ is a uniform upper bound of volumes of metrics $g_{j}$ in (3) if
and only if $f_{j}$ satisfy [Lemma 2.2]

(10) $\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}\leq\frac{V}{2\pi}.$

It is well-known that the spherical mean property of (sub, sup)-harmonic
functions plays important roles in the study of these functions. Inspired by
this, we prove a spherical mean inequality for functions $f_{j}$ satisfying
the differential inequality (9) [Proposition 2.4]. It turns out that the
spherical mean inequality is very important in the proof of non-collapsing
property in Section 4, in particular, in the proof of Proposition 4.10.
Furthermore, by employing the trick in the proof of Bishop-Gromov volume
comparison theorem, we prove a ball average monotonicity property for $f_{j}$
[Proposition 2.6], which helps us to obtain lower semi-continuity of the limit
warping function $f_{\infty}$ in Proposition 3.7.

In Section 3, we study the convergence of a sequence
$\\{f_{j}\\}^{\infty}_{j=1}$ of positive functions on ${\mathbb{S}}^{2}$
satisfying (9) and (10). We prove that there exists a subsequence of such
sequence $\\{f_{j}\\}$ and a function $f_{\infty}\in
W^{1,p}({\mathbb{S}}^{2})\,(1\leq p<2)$ such that the subsequence converges to
$f_{\infty}$ in $L^{q}({\mathbb{S}}^{2})$ for any $q\geq 1$ [Proposition 3.5].
The proof of this convergence result is very different from that in cases of
warped product metrics as in [15] and in (8). Because warping functions
$h_{j}$ in [15] and in (8) have one variable, whereas $f_{j}$ in (3) have two
variables, it is more difficult to obtain sub-convergence of $\\{f_{j}\\}$,
and we make use of the Moser-Trudinger inequality in (25) in [14]. The
regularity of the limit function $f_{\infty}$ is weaker than $h_{\infty}$. The
extreme example constructed by Sormani and authors in [19] shows that the
$W^{1,p}$ regularity for $1\leq p<2$ is sharp for $f_{\infty}$.

In Section 4, we use the $\operatorname{MinA}$ condition to show that the
limit function $f_{\infty}$ has positive essential infimum [Theorem 4.13] and
that the warping functions $f_{j}$ have a positive uniform lower bound
[Proposition 4.15]. This enables us to define weak warped product Riemnnian
metric $g_{\infty}$ on ${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ in Definition
5.1, and is crucial in the study of geometric convergence of warped product
circles over sphere with metric tensor as in (3). Moreover, as a consequence
of Proposition 4.15, we obtain a positive uniform lower bound for the systole
of the warped product manifolds
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ [Proposition 4.20].

The $\operatorname{MinA}$ condition can be viewed as a noncollapsing
condition. As shown in [15] and in Lemma A.6 below, it is not difficult to see
this in cases of metric tensors as in [15] and (8). In the case of metric
tensors as in (3), however, the implication of the $\operatorname{MinA}$
condition is much more complicated. We need to use the Min-Max minimal surface
theory of Marques and Neves (see e.g. [13]), the maximum principle for weak
solutions (Theorem 8.19 in [6]), and the spherical mean inequality obtained in
Proposition 2.4, in order to obtain noncollapsing from the
$\operatorname{MinA}$ condition.

In Section 5, we prove that a subsequence of $\\{g_{j}\\}^{\infty}_{j=1}$,
with $g_{j}$ as in (3) having nonnegative scalar curvatures and uniform upper
bounded volumes and satisfying $\operatorname{MinA}$ condition, converges to a
weak metric tensor $g_{\infty}\in
W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})\,(1\leq p<2)$ in the
sense of $L^{q}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for all $q\geq
1$ [Theorem 5.5]. Moreover, we prove that the limit metric $g_{\infty}$ has
nonnegative distributional scalar curvature in the sense of Lee-LeFloch
[Theorem 5.11].

Note that in the case of metric tensors as in [15] and (8), we need the
diameter uniform upper bound condition in addition to nonnegative scalar
curvature condition and the $\operatorname{MinA}$ condition for getting
convergence [Theorem 1.3 in [15] and Theorem A.1], whereas in the case of
metric tensors as in (3), we need the volume uniform upper bound condition
instead of the diameter uniform upper bound condition [Theorem 5.5].

Acknowledgements:

The authors would like to thank the Fields Institute for hosting the Summer
School on Geometric Analysis in July 2017 where we met Professor Christina
Sormani and she started to guide us working on the project concerning
compactness of manifolds with nonnegative scalar curvatures. We are grateful
to Professor Sormani for her constant encouragement and inspiring discussions.
In particular, Professor Sormani suggested us the method of spherical means,
and it turns out to be very useful in the study of warping functions in
Theorem 1.3. We thank Brian Allen for discussions and interest in this work.
Wenchuan Tian was partially supported by the AMS Simons Travel Grant.
Changliang Wang was partially supported by the Fundamental Research Funds for
the Central Universities and Shanghai Pilot Program for Basic Research.

##  2\.  Consequences of the geometric hypotheses on
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$

In this section we prove several consequences of the uniform geometric bounds.
In Subsection 2.1, we derive the differential inequality satisfied by the
warping function $f_{j}$ and prove that the uniform volume bounds on sequence
of Riemannian manifolds implies the uniform $L^{1}$ norm of the warping
function.

In Subsection 2.2, we prove the spherical mean inequality for the warping
function $f$ [Proposition 2.6], which is our main analytic tool. In Subsection
2.3, we prove a ball average monotonicity property for the warping function
$f$ [Proposition 2.4].

The implication of the $\operatorname{MinA}$ condition is more complicated we
discuss that in Section 4.

###  2.1. Basic consequences of the hypotheses

######  Lemma 2.1 (Non-negative scalar curvature condition).

The scalar curvature of warped product manifolds
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ are given by

(11) $\operatorname{Scalar}_{j}=2-2\frac{\Delta f_{j}}{f_{j}},$

where $\Delta$ is the Laplacian on ${\mathbb{S}}^{2}$ with respect to the
standard metric $g_{{\mathbb{S}}^{2}}$, taken to be the trace of the Hessian
(without the negative sign).

Thus ${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ have nonnegative scalar
curvature if and only if

(12) $\Delta f_{j}\leq f_{j}.$

###### Proof.

By using the Ricci curvature formula for warped product metrics as in
Proposition 9.106 of [3], we can easily obtain the scalar curvature of
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ as
$\operatorname{Scalar}_{j}=2-2\frac{\Delta f_{j}}{f_{j}}.$ Then the second
claim directly follows, since $f_{j}>0$. ∎

######  Lemma 2.2 (Volume upper bound condition).

The warped product manifolds ${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$
have volume ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\leq V$
if and only if

(13) $\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi}.$

###### Proof.

The Riemannian volume measure of $g_{j}$ is given by

(14) $d{\rm vol}_{g_{j}}=f_{j}d{\rm vol}_{g_{{\mathbb{S}}^{2}}}d{\rm
vol}_{g_{{\mathbb{S}}^{1}}}.$

Thus the volume of ${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ is given
by

(15) ${\rm
Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})=\int_{{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}}f_{j}d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}d{\rm
vol}_{{\mathbb{S}}^{1}}=2\pi\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}.$

Then the claim directly follows. ∎

###  2.2. Spherical mean inequality

In this subsection, we prove a spherical mean inequality [Proposition 2.4] for
the smooth functions $f$ on ${\mathbb{S}}^{2}$ satisfying the differential
inequality $\Delta f\leq f$. By Lemma 2.1, this is equivalent to studying the
warping function of warped product manifolds
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ with nonnegative scalar
curvature. The spherical mean inequality plays an important role in the proof
of Proposition 4.10.

The derivation of the spherical mean value inequality is similar to that of
the mean value property of harmonic functions. We start with the following
lemma.

######  Lemma 2.3.

Let $f$ be a smooth function on ${\mathbb{S}}^{2}$. Consider the spherical
mean given by

(16) $\phi(r):=\fint_{\partial B_{r}(p)}fds,$

where $B_{r}(p)$ is the geodesic ball in the standard $\mathbb{S}^{2}$ with
center $p$ and radius $r$. The derivative of $\phi(r)$ satisfies

(17) $\frac{d}{dr}\phi(r)=\frac{1}{2\pi\sin r}\int_{B_{r}(p)}\Delta fd{\rm
vol}_{{\mathbb{S}}^{2}}.$

###### Proof.

Using the geodesic polar coordinate $(r,\theta)$ on $\mathbb{S}^{2}$ centered
at $p$, one can write $\phi(r)$ as

(18) $\phi(r)=\frac{\int^{2\pi}_{0}f(r,\theta)\sin rd\theta}{2\pi\sin
r}=\frac{\int^{2\pi}_{0}f(r,\theta)d\theta}{2\pi}.$

Then taking derivative with respective to $r$ gives

(19) $\displaystyle\phi^{\prime}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int^{2\pi}_{0}\frac{\partial f}{\partial
r}d\theta$ (20) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int^{2\pi}_{0}\langle\nabla
f,\partial_{r}\rangle$ (21) $\displaystyle=$ $\displaystyle\frac{1}{2\pi\sin
r}\int^{2\pi}_{0}\langle\nabla f,\partial_{r}\rangle\sin rd\theta$ (22)
$\displaystyle=$ $\displaystyle\frac{1}{2\pi\sin r}\int_{\partial
B_{r}(p)}\langle\nabla f,\partial_{r}\rangle ds$ (23)
$\displaystyle\overset{Stokes}{=}$ $\displaystyle\frac{1}{2\pi\sin
r}\int_{B_{r}(p)}\Delta fd{\rm vol}_{\mathbb{S}^{2}}.$

∎

Now we use Lemma 2.3 to prove the spherical mean inequality.

######  Proposition 2.4.

Let $f$ be a smooth function on ${\mathbb{S}}^{2}$ satisfying $\Delta f\leq
f$. Then for any fixed $p\in\mathbb{S}^{2}$ and
$0<r_{0}<r_{1}\leq\frac{\pi}{2}$, one has

(24) $\fint_{\partial B_{r_{1}}(p)}fds-\fint_{\partial
B_{r_{0}}(p)}fds\leq\frac{\|f\|_{L^{2}(\mathbb{S}^{2})}}{\sqrt{2\pi}}(r_{1}-r_{0}),$

where $B_{r}(p)$ is the geodesic ball in the $\mathbb{S}^{2}$ with center $p$
and radius $r$.

Moreover, by taking limit as $r_{0}{\rightarrow}0$, one has

(25) $\fint_{\partial
B_{r}(p)}fds-f(p)\leq\frac{\|f\|_{L^{2}(\mathbb{S}^{2})}}{\sqrt{2\pi}}r,$

for any $0<r\leq\frac{\pi}{2}$.

###### Proof.

By Lemma 2.3 and the assumption $\Delta f\leq f$, one has

(26) $\phi^{\prime}(r)\leq\frac{1}{2\pi\sin r}\int_{B_{r}(p)}fd{\rm
vol}_{\mathbb{S}^{2}}.$

Integrating this differential inequality for $r$ from $r_{0}$ to $r_{1}$ gives

(27) $\displaystyle\phi(r_{1})-\phi(r_{0})$ $\displaystyle\leq$
$\displaystyle\int^{r_{1}}_{r_{0}}\left(\frac{1}{2\pi\sin
r}\int_{B_{r}(p)}fd{\rm vol}_{\mathbb{S}^{2}}\right)dr$ (28)
$\displaystyle\leq$ $\displaystyle\int^{r_{1}}_{r_{0}}\left(\frac{1}{2\pi\sin
r}\|f\|_{L^{2}(\mathbb{S}^{2})}\sqrt{{\rm Area}(B_{r}(p))}\right)dr$ (29)
$\displaystyle=$
$\displaystyle\frac{\|f\|_{L^{2}(\mathbb{S}^{2})}}{\sqrt{2\pi}}\int^{r_{1}}_{r_{0}}\frac{\sqrt{1-\cos
r}}{\sin r}dr$ (30) $\displaystyle=$
$\displaystyle\frac{\|f\|_{L^{2}(\mathbb{S}^{2})}}{\sqrt{2\pi}}\int^{r_{1}}_{r_{0}}\frac{1}{\sqrt{1+\cos
r}}dr$ (31) $\displaystyle\leq$
$\displaystyle\frac{\|f\|_{L^{2}(\mathbb{S}^{2})}}{\sqrt{2\pi}}\int^{r_{1}}_{r_{0}}1dr\qquad\left(0<r_{0}<r_{1}\leq\frac{\pi}{2}\right)$
(32) $\displaystyle=$
$\displaystyle\frac{\|f\|_{L^{2}(\mathbb{S}^{2})}}{\sqrt{2\pi}}(r_{1}-r_{0}).$

∎

###  2.3. Ball average monotonicity

In this subsection, we further derive a ball average monotonicity [Proposition
2.6] for a smooth function on ${\mathbb{S}}^{2}$ satisfying $\Delta f\leq f$.
The proof uses the spherical mean inequality [Proposition 2.4] and the trick
as in the proof of Bishop-Gromov volume comparison theorem. This ball average
monotonicity is used in Proposition 3.7 to prove that the ball average limit
as $r\to 0$ exists everywhere for the limit function.

######  Lemma 2.5.

Let $f$ be a smooth function on ${\mathbb{S}}^{2}$ satisfying $\Delta f\leq f$
and $\|f\|_{L^{2}({\mathbb{S}}^{2})}\leq C\sqrt{2\pi}$, where $C$ is a
positive constant. For any fixed $x\in{\mathbb{S}}^{2}$, the spherical mean

(33) $\fint_{\partial B_{r}(x)}\left(f-Cr\right)=\frac{\int_{\partial
B_{r}(x)}(f-Cr)}{2\pi\sin r}$

is a non-increasing function in $r$ for $r\in(0,\frac{\pi}{2}]$

###### Proof.

The spherical mean inequality in Proposition 2.4 says that for any
$x\in{\mathbb{S}}^{2}$ and $0<r_{0}<r_{1}\leq\frac{\pi}{2}$,

(34) $\fint_{\partial B_{1}(x)}f-\fint_{\partial
B_{r_{0}}(x)}f\leq\frac{\|f\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}(r_{1}-r_{0})\leq
C(r_{1}-r_{0}).$

By rearranging this inequality, we obtain that for any fixed
$x\in{\mathbb{S}}^{2}$,

(35) $\fint_{\partial B_{r_{1}}(x)}(f-Cr_{1})\leq\fint_{\partial
B_{r_{0}}(x)}(f-Cr_{0}),\quad\forall 0<r_{0}\leq r_{1}\leq\frac{\pi}{2}.$

This completes the proof. ∎

Combine this spherical mean monotonicity with the trick as in the proof of
Bishop-Gromov volume comparison theorem, we obtain the following ball average
monotonicity.

######  Proposition 2.6.

Let $f$ be a smooth function on ${\mathbb{S}}^{2}$ satisfying $\Delta f\leq f$
and $\|f\|_{L^{2}({\mathbb{S}}^{2})}\leq C\sqrt{2\pi}$, then $\forall
0<r<R\leq\frac{\pi}{2}$,

(36) $\fint_{B_{R}(x)}\left(f(y)-Cd(y,x)\right)d{\rm
vol}(y)\leq\fint_{B_{r}(x)}\left(f(y)-Cd(y,x)\right)d{\rm vol}(y),$

where $d(y,x)$ is the distance between $y$ and $x$ in the standard
${\mathbb{S}}^{2}$.

###### Proof.

Step 1.

(37) $\displaystyle\int_{B_{r}(x)}\left(f(y)-Cd(y,x)\right)d{\rm vol}(y)$ (38)
$\displaystyle=$ $\displaystyle\int^{r}_{0}\left(\int_{\partial
B_{s}(x)}(f-Cs)\right)ds$ (39) $\displaystyle=$
$\displaystyle\int^{r}_{0}(2\pi\sin s)\left(\fint_{\partial
B_{s}(x)}(f-Cs)\right)ds$ (40) $\displaystyle\geq$
$\displaystyle\fint_{\partial B_{r}(x)}(f-Cr)\cdot\int^{r}_{0}2\pi\sin
sds\quad(\text{by}\ \ (\ref{eqn: spherical mean non-increasing})\ \
\text{and}\ \ s\leq r)$ (41) $\displaystyle=$ $\displaystyle{\rm
Vol}(B_{r}(x))\fint_{\partial B_{r}(x)}(f-Cr).$

So

(42) $\fint_{B_{r}(x)}(f(y)-Cd(y,x))d{\rm vol}(y)\geq\fint_{\partial
B_{r}(x)}(f(y)-Cr)$

Step 2. Let $A_{r,R}(x)=B_{R}(x)\setminus B_{r}(x)$. Similar as in step 1, we
have

(43) $\displaystyle\int_{A_{r,R}(x)}(f(y)-Cd(y,x))d{\rm vol}(y)$ (44)
$\displaystyle=$ $\displaystyle\int^{R}_{r}\left(\int_{\partial
B_{s}(x)}(f-Cs)d\sigma\right)ds$ (45) $\displaystyle=$
$\displaystyle\int^{R}_{r}(2\pi\sin s)\left(\fint_{\partial
B_{s}(x)}(f-Cs)d\sigma\right)ds$ (46) $\displaystyle\leq$
$\displaystyle\fint_{\partial B_{r}(x)}(f-Cr)d\sigma\cdot\int^{R}_{r}(2\pi\sin
s)ds\quad(\text{by}\ \ (\ref{eqn: spherical mean non-increasing})\ \
\text{and}\ \ s\geq r)$ (47) $\displaystyle=$ $\displaystyle{\rm
vol}(A_{r,R}(x))\fint_{\partial B_{r}(x)}(f-Cr)d\sigma$

So

(48) $\fint_{A_{r,R}(x)}(f(y)-Cd(y,x))d{\rm vol}(y)\leq\fint_{\partial
B_{r}(x)}(f-Cr)d\sigma.$

Step 3. By combining (42) and (48), we obtain that for
$0<r<R\leq\frac{\pi}{2}$

(49) $\fint_{A_{r,R}(x)}(f(y)-Cd(y,x))d{\rm
vol}(y)\leq\fint_{B_{r}(x)}(f(y)-Cd(y,x))d{\rm vol}(y).$

Step 4.

(50) $\displaystyle\int_{B_{R}(x)}(f-Cd(y,x))d{\rm vol}(y)$ (51)
$\displaystyle=$ $\displaystyle\int_{B_{r}(x)}(f-Cd(y,x))d{\rm
vol}(y)+\int_{A_{r,R}(x)}(f-Cd(y,x))d{\rm vol}(y)$ (53) $\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}(f-Cd(y,x))d{\rm vol}(y)$ $\displaystyle+{\rm
Vol}(A_{r,R}(x))\cdot\fint_{B_{r}(x)}(f(y)-Cd(y,x))d{\rm vol}(y)$ (54)
$\displaystyle=$ $\displaystyle\left({\rm Vol}(B_{r}(x))+{\rm
vol}(A_{r,R}(x))\right)\fint_{B_{r}(x)}(f(y)-Cd(y,x))d{\rm vol}(y)$ (55)
$\displaystyle=$ $\displaystyle{\rm
Vol}(B_{R}(x))\fint_{B_{r}(x)}(f(y)-Cd(y,x))d{\rm vol}(y).$

This completes the proof. ∎

##  3\. $W^{1,p}$ limit of warping function for $1\leq p<2$

In this section, we study the $L^{q}$ pre-compactness of a sequence of
positive smooth functions $f_{j}$ satisfying the inequalities

(56) $\Delta f_{j}\leq f_{j},\quad\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi},\quad\forall j\in\mathbb{N}.$

Here $V$ is a positive constant. By Lemmas 2.1 and 2.2, the inequlities in
(56) are equivalent to the requirements that the Riemannian manifolds
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ have nonnegative scalar
curvature and uniform volume upper bound.

In Subsection 3.1, we prove that a sequence of positive smooth functions
$f_{j}$ on ${\mathbb{S}}^{2}$ satisfying requirements in (56) has a convergent
subsequence in $L^{q}({\mathbb{S}}^{2})$ for any $1\leq q<+\infty$, and that
the limit function is in $W^{1,p}({\mathbb{S}}^{2})$ for any $1\leq p<2$
[Proposition 3.5].

In Subsection 3.2, we apply the ball average monotonicity property obtained in
Proposition 2.6 to prove that the limit function has a lower semi-continuous
representative [Proposition 3.7, Remark 3.8].

###  3.1. $W^{1,p}$ limit function for $p<2$

We first derive the gradient estimate for the sequence of function $\ln f_{j}$
in Lemma 3.1, which is used to obtain $L^{p}$ estimate for $f_{i}$ by using
Moser-Trudinger inequality in Lemma 3.2.

######  Lemma 3.1.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive functions on
${\mathbb{S}}^{2}$ satisfying

(57) $\Delta f_{j}\leq f_{j},\quad\forall j\in\mathbb{N}.$

We have

(58) $\|\nabla\ln f_{j}\|_{L^{2}({\mathbb{S}}^{2})}^{2}\leq{\rm
Vol}({\mathbb{S}}^{2}),\quad\forall j\in\mathbb{N}.$

###### Proof.

Note that

(59) $\Delta\ln f_{j}=\frac{\Delta f_{j}}{f_{j}}-\frac{|\nabla
f_{j}|^{2}}{f_{j}^{2}}.$

By equation (59) and the assumption, we have

(60) $|\nabla\ln f_{j}|^{2}=\frac{|\nabla f_{j}|^{2}}{f_{j}^{2}}=\frac{\Delta
f_{j}}{f_{j}}-\Delta\ln f_{j}\leq 1-\Delta\ln f_{j}.$

Integrating it over ${\mathbb{S}}^{2}$, and using Stokes’ theorem, we get

(61) $\|\nabla\ln
f_{j}\|_{L^{2}({\mathbb{S}}^{2})}^{2}=\int_{{\mathbb{S}}^{2}}|\nabla\ln
f_{j}|^{2}\leq{\rm Vol}({\mathbb{S}}^{2}).$

∎

######  Lemma 3.2.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive functions on
${\mathbb{S}}^{2}$ satisfying

(62) $\Delta f_{j}\leq f_{j},\quad\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi},\quad\forall j\in\mathbb{N}.$

Then we have

(63) $\|f_{j}\|_{L^{p}({\mathbb{S}}^{2})}^{p}\leq
4\pi\exp\left(\frac{Vp}{8\pi^{2}}+\frac{p^{2}}{4}\right),$

for all $j\in\mathbb{N}$ and $p\in[1,+\infty)$.

###### Proof.

By the Moser-Trudinger inequality (inequality (25) in [14]), for any smooth
function $\psi:{\mathbb{S}}^{2}\to\mathbb{R}$ we have

(64) $\int_{{\mathbb{S}}^{2}}e^{\psi}d{\rm vol}_{{\mathbb{S}}^{2}}\leq
4\pi\exp\left(\frac{1}{4\pi}\int_{{\mathbb{S}}^{2}}\left(\psi+\frac{1}{4}|\nabla\psi|^{2}\right)d{\rm
vol}_{{\mathbb{S}}^{2}}\right).$

Here $\nabla$ is the Levi-Civita connection of the standard metric
$g_{{\mathbb{S}}^{2}}$ and $d{\rm vol}_{{\mathbb{S}}^{2}}$ is the volume form
on ${\mathbb{S}}^{2}$ with respect to the standard metric
$g_{{\mathbb{S}}^{2}}$. Take $\psi=p\ln f_{j}$, then we have

(65) $\displaystyle\|f_{j}\|_{L^{p}({\mathbb{S}}^{2})}^{p}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}}^{2}}f_{j}^{p}d{\rm vol}_{{\mathbb{S}}^{2}}$
(66) $\displaystyle=$ $\displaystyle\int_{{\mathbb{S}}^{2}}e^{p\ln f_{j}}d{\rm
vol}_{{\mathbb{S}}^{2}}$ (67) $\displaystyle\leq$ $\displaystyle
4\pi\exp\left(\frac{1}{4\pi}\int_{{\mathbb{S}}^{2}}\left(p\ln
f_{j}+\frac{p^{2}}{4}|\nabla\ln f_{j}|^{2}\right)d{\rm
vol}_{{\mathbb{S}}^{2}}\right).$

By the fact that $\ln x\leq x,\forall x>0$, we have

(68) $\int_{{\mathbb{S}}^{2}}\ln
f_{j}\leq\int_{{\mathbb{S}}^{2}}f_{j}\leq\frac{V}{2\pi}.$

On the hand, by Lemma 3.1 we have

(69) $\int_{{\mathbb{S}}^{2}}|\nabla\ln f_{j}|^{2}\leq{\rm
vol}({\mathbb{S}}^{2})=4\pi.$

This completes the proof. ∎

Next, we show that such sequence of function is uniformly bounded in
$W^{1,p}({\mathbb{S}}^{2})$ for $p\in[1,2)$.

######  Lemma 3.3.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive functions on
${\mathbb{S}}^{2}$ satisfying

(70) $\Delta f_{j}\leq f_{j},\quad\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi},\quad\forall j\in\mathbb{N}.$

Then the sequence is uniformly bounded in $W^{1,p}({\mathbb{S}}^{2})$ for
$p\in[1,2)$, i.e. for each $p\in[1,2)$, there exists a constant $C(p)$ such
that

(71) $\|f_{j}\|_{W^{1,p}({\mathbb{S}}^{2})}\leq C(p),\quad\forall
j\in\mathbb{N}.$

###### Proof.

For any $1\leq p<2$,

(72) $|\nabla f_{j}|^{p}=|\nabla\ln f_{j}|^{p}\cdot|f_{j}|^{p}.$

The Cauchy-Schwarz inequality implies that

(73) $\displaystyle\|\nabla f_{j}\|_{L^{p}(\mathbb{S}^{2})}$ (74)
$\displaystyle=$ $\displaystyle\left(\int_{\mathbb{S}^{2}}|\nabla\ln
f_{j}|^{p}\cdot|f_{j}|^{p}\right)^{\frac{1}{p}}$ (75) $\displaystyle\leq$
$\displaystyle\|\nabla\ln
f_{j}\|_{L^{2}(\mathbb{S}^{2})}\cdot\|f_{j}\|_{L^{\frac{2p}{2-p}}(\mathbb{S}^{2})}$
(76) $\displaystyle\leq$ $\displaystyle\|\nabla\ln
f_{j}\|_{L^{2}(\mathbb{S}^{2})}\cdot\left(\|f_{j}\|_{L^{\frac{2p}{2-p}}(\mathbb{S}^{2})}+{\rm
Vol}({\mathbb{S}}^{2})\right)$ (77) $\displaystyle\leq$
$\displaystyle\left({\rm
vol}(\mathbb{S}^{2})\right)^{\frac{1}{2}}\left((4\pi)^{\frac{2-p}{2p}}\exp\left(\frac{V}{8\pi^{2}}+\frac{p}{2(2-p)}\right)+{\rm
Vol}({\mathbb{S}}^{2})\right).$

Here in the last step, we used Lemma 3.1 and Lemma 3.2. Moreover, by Lemma 3.2
again, for each $p\in[1,2)$, $\|f_{j}\|_{L^{p}({\mathbb{S}}^{2})}$ is
uniformly bounded for all $j\in\mathbb{N}$. Hence for each $p\in[1,2)$,
$\|f_{j}\|_{W^{1,p}({\mathbb{S}}^{2})}$ is uniformly bounded for all
$j\in\mathbb{N}$. ∎

We use the uniform $W^{1,p}({\mathbb{S}}^{2})$ bound to prove convergence in
the following lemma.

######  Lemma 3.4.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive functions on
${\mathbb{S}}^{2}$ satisfying

(78) $\Delta f_{j}\leq f_{j},\quad\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi},\quad\forall j\in\mathbb{N}.$

Then for each fixed $p\in[1,2)$, there exists a subsequence
$\\{f_{j^{(p)}_{k}}\\}_{k=1}^{\infty}$ and $f_{\infty,p}\in
W^{1,p}(\mathbb{S}^{2})$ such that

(79) $f_{j^{(p)}_{k}}\rightarrow f_{\infty,p},\quad\text{in}\ \
L^{q}({\mathbb{S}}^{2}),$

for each $1\leq q<\frac{2p}{2-p}$.

Moreover, for any $\varphi\in C^{\infty}({\mathbb{S}}^{2})$,

$\int_{{\mathbb{S}}^{2}}\left(f_{j^{(p)}_{k}}\varphi+\langle\nabla
f_{j^{(p)}_{k}},\nabla\varphi\rangle\right)d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}\rightarrow\int_{{\mathbb{S}}^{2}}\left(f_{\infty,p}\varphi+\langle\nabla
f_{\infty,p},\nabla\varphi\right)d{\rm vol}_{g_{{\mathbb{S}}^{2}}},$

as $j^{(p)}_{k}\rightarrow\infty$, where $\nabla f_{\infty,p}$ is the weak
gradient of $f_{\infty,p}$.

###### Proof.

For each fixed $p\in[1,2)$, by using Rellich-Kondrachov compactness theorem,
the uniform estimate of Sobolev norms in Lemma 3.3 implies that there exists a
subsequence of $\\{f_{j}\\}$, which is still denoted by $\\{f_{j}\\}$,
converging to $f_{\infty,p}$ in $L^{q}({\mathbb{S}}^{2})$ for $1\leq
q<\frac{2p}{2-p}$. Then by the weak compactness in $L^{p}$ space (see, e.g.
Theorem 1.42 in [5]), we can obtain that $f_{\infty,p}\in
W^{1,p}({\mathbb{S}}^{2})$. Indeed, $\|f_{j}\|_{W^{1,p}({\mathbb{S}}^{2})}\leq
C$ for all $j\in\mathbb{N}$ implies that $\|f_{j}\|_{L^{p}({\mathbb{S}}^{2})}$
and $\|\nabla f_{j}\|_{L^{p}({\mathbb{S}}^{2})}$ are both uniformly bounded.
Then the weak compactness in $L^{p}$ space implies that there exist a further
subsequence, denoted by $f_{j^{(p)}_{k}}$, and $X\in
L^{p}({\mathbb{S}}^{2},{\rm T}{\mathbb{S}}^{2})$ such that

(80) $\nabla f_{j^{(p)}_{k}}\rightharpoonup X\quad\text{in}\ \
L^{p}({\mathbb{S}}^{2},{\rm T}{\mathbb{S}}^{2}),$

i.e.

(81) $\int_{{\mathbb{S}}^{2}}\langle\nabla f_{j^{(p)}_{k}},Y\rangle d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}\rightarrow\int_{{\mathbb{S}}^{2}}\langle
X,Y\rangle d{\rm vol}_{g_{{\mathbb{S}}^{2}}},\quad\forall Y\in
C^{\infty}({\mathbb{S}}^{2},{\rm T}{\mathbb{S}}^{2}).$

On the other hand,

(82) $\int_{{\mathbb{S}}^{2}}\langle\nabla f_{j^{(p)}_{k}},Y\rangle d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}=\int_{{\mathbb{S}}^{2}}f_{j}{\rm div}Yd{\rm
vol}_{g_{{\mathbb{S}}^{2}}}\rightarrow\int_{{\mathbb{S}}^{2}}f_{\infty,p}{\rm
div}Yd{\rm vol}_{g_{{\mathbb{S}}^{2}}},$

since $f_{j^{(p)}_{k}}\rightarrow f_{\infty,p}$ in $L^{p}$. Thus,

(83) $\int_{{\mathbb{S}}^{2}}f_{\infty,p}{\rm div}Yd{\rm
vol}_{g_{{\mathbb{S}}^{2}}}=\int_{{\mathbb{S}}^{2}}\langle X,Y\rangle d{\rm
vol}_{g_{{\mathbb{S}}^{2}}},\quad\forall Y\in C^{\infty}({\mathbb{S}}^{2},{\rm
T}{\mathbb{S}}^{2}).$

Therefore, $X=\nabla f_{\infty,p}$ is the gradient of $f_{\infty,p}$ in the
sense of distribution, and so $f_{\infty,p}\in
W^{1,p}({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$. For any $\varphi\in
C^{\infty}({\mathbb{S}}^{2})$, by taking $Y=\nabla\varphi$ in (81), we obtain

(84) $\int_{{\mathbb{S}}^{2}}\left(f_{j^{(p)}_{k}}\varphi+\langle\nabla
f_{j^{(p)}_{k}},\nabla\varphi\rangle\right)d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}\rightarrow\int_{{\mathbb{S}}^{2}}\left(f_{\infty,p}\varphi+\langle\nabla
f_{\infty,p},\nabla\varphi\right)d{\rm vol}_{g_{{\mathbb{S}}^{2}}}.$

∎

Now we use Lemma 3.4 and diagonal argument to find a subsequence converging in
$L^{q}$ for all $q\geq 1$ and prove the following proposition:

######  Proposition 3.5.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive functions on
${\mathbb{S}}^{2}$ satisfying

(85) $\Delta f_{j}\leq f_{j},\quad\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi},\quad\forall j\in\mathbb{N}.$

Then there exists a subsequence $\\{f_{j_{k}}\\}_{k=1}^{\infty}$ and
$f_{\infty}\in W^{1,p}(\mathbb{S}^{2})$ for all $p\in[1,2)$, such that

(86) $f_{j_{k}}\rightarrow f_{\infty},\quad\text{in}\ \
L^{q}({\mathbb{S}}^{2}),\ \ \forall q\in[1,\infty).$

Moreover, for any $\varphi\in C^{\infty}({\mathbb{S}}^{2})$,

(87) $\int_{{\mathbb{S}}^{2}}\left(f_{j_{k}}\varphi+\langle\nabla
f_{j_{k}},\nabla\varphi\rangle\right)d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}\rightarrow\int_{{\mathbb{S}}^{2}}\left(f_{\infty}\varphi+\langle\nabla
f_{\infty},\nabla\varphi\rangle\right)d{\rm vol}_{g_{{\mathbb{S}}^{2}}},$

as $j_{k}\rightarrow\infty$, where $\nabla f_{\infty}$ is the weak gradient of
$f_{\infty}$.

###### Proof.

The proof is a diagonal argument. We apply Lemma 3.4 for
$p=2-\frac{1}{n+1},n=1,2,3,\dots$.

For $n=1$, by applying Lemma 3.4 to $\\{f_{j}\\}_{j=1}^{\infty}$ and
$p=2-\frac{1}{2}$, we obtain a subsequence, denoted by $f_{j^{(1)}_{k},1}$,
and $f_{\infty,1}\in W^{1,2-\frac{1}{2}}$ such that

(88) $f_{j^{(1)}_{k},1}\rightarrow f_{\infty,1}\ \ \text{in}\ \
L^{q}({\mathbb{S}}^{2}),\ \ \forall 1\leq q<6,\ \ \text{as}\ \
k\rightarrow\infty.$

For $n=2$, by applying Lemma 3.4 to the subsequence
$\left\\{f_{j^{(1)}_{k},1}\right\\}_{k=1}^{\infty}$ and $p=2-\frac{1}{3}$, we
obtain a subsequence,
$\left\\{f_{j^{(2)}_{k},2}\right\\}_{k=1}^{\infty}\subset\left\\{f_{j^{(1)}_{k},1}\right\\}_{k=1}^{\infty}$,
and $f_{\infty,2}\in W^{1,2-\frac{1}{3}}$ such that

(89) $f_{j^{(2)}_{k},2}\rightarrow f_{\infty,2}\ \ \text{in}\ \
L^{q}({\mathbb{S}}^{2}),\ \ \forall 1\leq q<10,\ \ \text{as}\ \
k\rightarrow\infty.$

Then by repeating this process for $n=3,4,5,\dots$, we can obtain a family of
decreasing subsequence
$\left\\{f_{j^{(n)}_{k},n}\right\\}_{k=1}^{\infty}\subset\left\\{f_{j^{(n-1)}_{k},n-1}\right\\}_{k=1}^{\infty}$
and $f_{\infty,n}\in W^{1,2-\frac{1}{n+1}}$ for all $n\in\mathbb{N}$, such
that for each fixed $n\in\mathbb{N}$

(90) $f_{j^{(n)}_{k},n}\rightarrow f_{\infty,n}\ \ \text{in}\ \
L^{q}({\mathbb{S}}^{2}),\ \ \forall 1\leq q<4n+2,\ \ \text{as}\ \
k\rightarrow\infty.$

Now we take the diagonal subsequence
$\left\\{f_{j_{k}}:=f_{f_{j^{(k)}_{k},k}}\mid k\in\mathbb{N}\right\\}$. By the
construction of $f_{j_{k}}$ and $4k+2\rightarrow+\infty$ as
$k\rightarrow+\infty$, we have that $\\{f_{j_{k}}\\}$ is a Cauchy sequence in
$L^{q}({\mathbb{S}}^{2})$ for all $q\in[1,\infty)$. Thus there exists
$f_{\infty}\in L^{q}({\mathbb{S}}^{2})$ such that

(91) $f_{j_{k}}\rightarrow f_{\infty}\ \ \text{in}\ \ \in
L^{q}({\mathbb{S}}^{2}),\ \ \text{as}\ \ k\rightarrow\infty,\ \ \forall
q\in[1,\infty).$

Then by the uniqueness of $L^{2}$ limit, $f_{\infty}=f_{\infty,n}$ in
$L^{2}({\mathbb{S}}^{2})$ for all $n\in\mathbb{N}$. Furthermore, because
$f_{\infty,n}\in W^{1,2-\frac{1}{n+1}}({\mathbb{S}}^{2})$ and
$2-\frac{1}{n+1}\rightarrow 2^{-}$ as $n\rightarrow\infty$, we see that the
$L^{p}$ norm of the weak derivative of $f_{\infty}$ is bounded for any
$p\in[1,2)$. Thus $f_{\infty}\in W^{1,p}({\mathbb{S}}^{2})$ for all
$p\in[1,2)$.

Finally, the last claim in (87) follows from that
$\left\\{f_{j_{k}}\right\\}_{k=1}^{\infty}\subset\left\\{f_{j^{(1)}_{k},1}\right\\}_{k=1}^{\infty}$
and the corresponding convergence in Lemma 3.4 for $p=2-\frac{1}{2}$, in
particular for the subsequence
$\left\\{f_{j^{(1)}_{k},1}\right\\}_{k=1}^{\infty}$. ∎

######  Remark 3.6.

The extreme example constructed by Christina Sormani and authors in [19] shows
that $W^{1,p}$ regularity for $p<2$ is the best regularity we can expect for
$f_{\infty}$ in general (see Lemma 3.4 in [19]).

###  3.2. Lower semi-continuous representative of the limit function

For the limit function $f_{\infty}$ obtained in Proposition 3.5, Lebesgue-
Besicovitch differential theorem implies that

(92) $\lim_{r\rightarrow 0}\fint_{B_{r}(x)}f_{\infty}d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}=f_{\infty}(x)$

holds for a.e. $x\in{\mathbb{S}}^{2}$ with respect to the volume measure
$d{\rm vol}_{g_{{\mathbb{S}}^{2}}}$. In Proposition 3.7, by applying the ball
average monotonicity property in Proposition 2.6, we will show that the limit
of ball average in (92) actually exists for all $x\in{\mathbb{S}}^{2}$, and
that the limit produces a lower semi-continuous function.

######  Proposition 3.7.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of smooth positive functions on
${\mathbb{S}}^{2}$ satisfying

(93) $\Delta f_{j}\leq f_{j},\quad\int_{{\mathbb{S}}^{2}}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}\leq\frac{V}{2\pi},\quad\forall j\in\mathbb{N}.$

Then the limit function, $f_{\infty}$, obtained in Proposition 3.5, has the
following properties.

  1. (i)

For each fixed $x\in{\mathbb{S}}^{2}$, the ball average

(94) $\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y)$

is non-increasing in $r\in\left(0,\frac{\pi}{2}\right)$, where $C$ is a
positive real number such that
$\sup_{j\in\mathbb{N}}\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}\leq C\sqrt{2\pi}$.
Note that the existence of such $C$ is guaranteed by Lemma 3.2.

  2. (ii)

Consequently, the limit

(95) $\overline{f_{\infty}}(x):=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}f_{\infty}=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y)$

exists, allowing $+\infty$ as a limit, for every $x\in{\mathbb{S}}^{2}$.
Moreover, $\overline{f_{\infty}}$ is a lower semi-continuous function on
${\mathbb{S}}^{2}$.

###### Proof.

By Lemma 3.2, there exists $C\in\mathbb{R}$ such that

(96) $\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}\leq C\sqrt{2\pi},\ \ \forall
j\in\mathbb{N}.$

Then by applying Proposition 2.6 to functions $f_{j}$, we obtain that for any
fixed $x\in{\mathbb{S}}^{2}$

(97) $\fint_{B_{R}(x)}(f_{j}(y)-Kd(y,x))d{\rm
vol}(y)\leq\fint_{B_{r}(x)}(f_{j}(y)-Cd(y,x))d{\rm vol}(y)$

holds for any $0<r<R<\frac{\pi}{2}$ and all $j\in\mathbb{N}$.

By Proposition 3.5 $f_{j}\rightarrow f_{\infty}$ in $L^{1}({\mathbb{S}}^{2})$.
Then for any fixed $x\in{\mathbb{S}}^{2}$, and any fixed
$0<r<R<\frac{\pi}{2}$, by taking the limit as $j\rightarrow+\infty$, we obtain

(98) $\fint_{B_{R}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm
vol}(y)\leq\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y),$

So for each fixed $x\in{\mathbb{S}}^{2}$, the ball average

(99) $\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y)$

is non-increasing for $r\in\left(0,\frac{\pi}{2}\right)$. Therefore, for any
$x\in{\mathbb{S}}^{2}$ the limit

(100) $\lim_{r\rightarrow
0}\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y)$

exists as a finite number or $+\infty$.

On the other hand, by direct calculation

(101) $\fint_{B_{r}(x)}d(y,x)d{\rm vol}(y)=\frac{\int^{r}_{0}2\pi s\sin
sds}{\int^{r}_{0}2\pi\sin(s)ds}=\frac{\sin r-r\cos r}{1-\cos r}\rightarrow 0,$

as $r\rightarrow 0$. Thus the limit

(102) $\overline{f_{\infty}}(x):=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}f_{\infty}=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y)$

exists for all $x\in{\mathbb{S}}^{2}$.

For each fixed $0<r<\frac{\pi}{2}$, we have that
$\fint_{B_{r}(x)}(f_{\infty}(y)-Cd(y,x))d{\rm vol}(y)$ is a continuous
function of $x\in{\mathbb{S}}^{2}$, since $f_{\infty}\in
L^{2}({\mathbb{S}}^{2})$, $Cd(y,x)\leq C\pi$, and ${\rm
Area}(B_{r}(x))=2\pi\sin r$ for all $x\in{\mathbb{S}}^{2}$. Then by the
monotonicity in (98), we have

(103)
$\overline{f_{\infty}}(x)=\sup_{r>0}\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm
vol}(y).$

In other words, $\overline{f_{\infty}}$ is the supremum of a sequence of
continuous function. Thus $\overline{f_{\infty}}$ is lower semi-continuous. ∎

######  Remark 3.8.

Recall that by (92), $\lim\limits_{r\rightarrow
0}\fint_{B_{r}(x)}f_{\infty}d{\rm vol}_{g_{{\mathbb{S}}^{2}}}=f_{\infty}(x)$
hold for a.e. $x\in{\mathbb{S}}^{2}$, thus
$\overline{f_{\infty}}(x)=f_{\infty}(x)$ holds for a.e.
$x\in{\mathbb{S}}^{2}$. So as a $W^{1,p}$ function, $f_{\infty}$ has a lower
semi-continuous representative $\overline{f_{\infty}}$.

##  4\. Positivity of the limit warping functions

In this section, we prove that the limit warping function $f_{\infty}$ has a
positive essential infimum, provided that the Riemannian manifold
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ satisfies both requirements
in (56) and the $\operatorname{MinA}$ condition [Theorem 4.13]. The main tools
we use in the proof of Theorem 4.13 include the maximum principle, the Min-Max
minimal surface theory of Marques and Neves, and the spherical mean inequality
we obtained in Proposition 2.4.

The maximum principle for weak solutions (Theorem 8.19 in [6]) requires
$W^{1,2}$ regularity, but in general we only have $f_{\infty}\in
W^{1,p}({\mathbb{S}}^{2})$ for $p<2$ [Remark 3.6]. To overcome this
difficulty, in Subsection 4.1, we consider the truncation of warping functions
$\bar{f}^{K}_{j}$ as defined in Definition 4.1, and obtain a
$W^{1,2}({\mathbb{S}}^{2})$ limit function $\bar{f}^{K}_{\infty}$ for the
sequence of truncated function $\bar{f}^{K}_{j}$ [Lemma 4.4]. This enables us
to apply maximum principle for weak solutions (Theorem 8.19 in [6]) to
$\bar{f}^{K}_{\infty}$, and prove that either $\inf\bar{f}^{K}_{\infty}>0$ or
$\bar{f}^{K}_{\infty}\equiv 0$ on ${\mathbb{S}}^{2}$ [Proposition 4.7].

In Subsection 4.3, we use Min-Max minimal surface theory of Marques and Neves
and the spherical mean inequality in Proposition 2.4 to obtain an upper bound
for $\operatorname{MinA}({\mathbb{S}}\times_{f}{\mathbb{S}}^{1})$ in terms of
$L^{1}$ norm of the warping function $f$, provided that the $L^{2}$ norm of
$f$ is sufficiently small [Proposition 4.10].

In Subsection 4.4, we use Proposition 4.7 and Proposition 4.10 to prove
Theorem 4.13. Moreover, as an application of Theorem 4.13, we obtain a
positive uniform lower bound for warping functions $f_{j}$, if the warped
product manifolds ${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ satisfy
requirements in (56) and the $\operatorname{MinA}$ condition [Proposition
4.15].

###  4.1. $W^{1,2}$ regularity of limit of truncated warping functions

We define the truncation of a function firstly:

######  Definition 4.1.

Let $f:{\mathbb{S}}^{2}\to\mathbb{R}$ be a positive smooth function. Let $K>0$
be a real number, for each $x\in{\mathbb{S}}^{2}$, we define

(104) $\bar{f}^{K}(x)=\begin{cases}f(x),&\text{ if }\ \ f(x)<K,\\\ K,&\text{
if }\ \ f(x)\geq K.\end{cases}$

Then $\bar{f}^{K}$ is a positive continuous function on ${\mathbb{S}}^{2}$
with the maximal value not greater than $K$.

From the definition we can prove the following lemma:

######  Lemma 4.2.

Let $f:{\mathbb{S}}^{2}\to\mathbb{R}$ be a positive smooth function, and let
$K>0$ be a regular value of the function $f$. If

(105) $\Delta f\leq f$

then for all $u\in W^{1,2}({\mathbb{S}}^{2})$ such that $u\geq 0$ we have

(106) $-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}^{K}\rangle\leq\int_{{\mathbb{S}}^{2}}u\bar{f}^{K}.$

###### Proof.

By Theorem 4.4 from [5], we have for all $K>0$

(107) $\nabla\bar{f}^{K}=\begin{cases}\nabla f,&\text{a.e. on
}\\{f(x)<K\\},\\\ 0,&\text{a.e. on }\\{f(x)\geq K\\}.\end{cases}$

As a result we have

(108) $\begin{split}-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}^{K}\rangle&=-\int_{\\{f<K\\}}\langle\nabla u,\nabla
f\rangle\\\ &=\int_{\\{f<K\\}}u\Delta
f-\int_{\partial\\{f<K\\}}u\partial_{\nu}f.\end{split}$

Here, since $K$ is a regular value of $f$, from the Regular Level Set Theorem
we know that the level set $\\{f=K\\}=\partial\\{f<K\\}$ is am embedded
submanifold of dimension $1$ in ${\mathbb{S}}^{2}$. Hence we can apply Stokes’
theorem to get the last step. Moreover, since $\nu$ is the outer unit normal
vector on the boundary of the set $\\{f<K\\}$, we have

(109) $\partial_{\nu}f\geq 0.$

Hence we can drop the boundary term to get the inequality

(110) $-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}^{K}\rangle\leq\int_{\\{f<K\\}}u\Delta f.$

Since

(111) $\Delta f\leq f,$

we have

(112) $-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}^{K}\rangle\leq\int_{\\{f<K\\}}u\Delta
f\leq\int_{\\{f<K\\}}uf\leq\int_{{\mathbb{S}}^{2}}u\bar{f}^{K}.$

This finishes the proof. ∎

We can prove similar results for a sequence of functions:

######  Lemma 4.3.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of smooth positive function
defined on ${\mathbb{S}}^{2}$. If

(113) $\Delta f_{j}\leq f_{j},\ \ \forall j\in\mathbb{N},$

then there exists $K>0$ such that for all $u\in W^{1,2}({\mathbb{S}}^{2})$
with $u\geq 0$ we have

(114) $-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}_{j}^{K}\rangle\leq\int_{{\mathbb{S}}^{2}}u\bar{f}_{j}^{K}\quad\forall
j\in\mathbb{N}.$

Moreover, we can choose $K$ as large as we want.

###### Proof.

Note that if $0<K\leq\inf\limits_{x\in{\mathbb{S}}^{2}}f_{j}(x)$ for some $i$
then we have $\bar{f}_{j}^{K}(x)=K$. On the other hand, if
$\sup\limits_{x\in{\mathbb{S}}^{2}}f(x)\leq K$ for some $i$ then
$\bar{f}_{j}^{K}(x)=f_{j}(x)$. Either way the inequality (114) holds.

In general, by Sard’s theorem, for each function $f_{j}$, the critical values
of $f_{j}$ has measure zero, and the union of all the critical sets for each
of the function also has measure zero. As a result, there exists $K>0$ such
that for each $f_{j}$ either $K$ is a regular value or
$f_{j}^{-1}(\\{K\\})=\emptyset$. By Lemma 4.2 we get inequality (114).
Moreover, we can choose $K$ as large as we want. This finishes the proof. ∎

Next we prove similar results for the limit function, but before that we need
to consider the regularity of the limit function:

######  Lemma 4.4.

Let $K>0$ be a real number. Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of
positive smooth functions on ${\mathbb{S}}^{2}$ satisfying

(115) $\Delta f_{j}\leq f_{j},\quad\forall j\in\mathbb{N}.$

Then the sequence $\\{\bar{f}^{K}_{j}\\}_{j=1}^{\infty}$ is uniformly bounded
in $W^{1,2}({\mathbb{S}}^{2})$:

(116) $\|\bar{f}^{K}_{j}\|_{W^{1,2}({\mathbb{S}}^{2})}\leq 2K{\rm
vol}({\mathbb{S}}^{2}).$

As a result, there exists $\bar{f}_{\infty}^{K}\in W^{1,2}({\mathbb{S}}^{2})$
such that $\bar{f}^{K}_{j}$ converges to $\bar{f}^{K}_{\infty}$ in
$L^{2}({\mathbb{S}}^{2})$, and that $\bar{f}^{K}_{j}$ converges to
$\bar{f}^{K}_{\infty}$ weakly in $W^{1,2}({\mathbb{S}}^{2})$.

###### Proof.

By definition of the cutoff in Definition 4.1, we get

(117) $\|\bar{f}_{j}^{K}\|_{L^{2}({\mathbb{S}}^{2})}\leq K\sqrt{{\rm
vol}({\mathbb{S}}^{2})}.$

By Theorem 4.4 from [5], we have for all $K>0$ and for each $i$

(118) $\nabla\bar{f}^{K}_{j}=\begin{cases}\nabla f_{j},&\text{a.e. on
}\\{f_{j}(x)<K\\},\\\ 0,&\text{a.e. on }\\{f_{j}(x)\geq K\\}.\end{cases}$

Hence

(119)
$\begin{split}\|\nabla\bar{f}_{j}\|^{2}_{L^{2}({\mathbb{S}}^{2})}&=\int_{\\{f_{j}<K\\}}|\nabla
f_{j}|^{2}\\\ &=\int_{\\{f_{i}<K\\}}|f_{j}|^{2}|\nabla\ln f_{j}|^{2}\\\ &\leq
K^{2}\int_{\\{f_{j}<K\\}}|\nabla\ln f_{j}|^{2}\\\ &\leq K^{2}\|\nabla\ln
f_{j}\|^{2}\\\ &\leq K^{2}{\rm vol}({\mathbb{S}}^{2}),\end{split}$

where the last step follows from Lemma 3.1. Combine inequalities (117) and
(119) then we get the desired results. ∎

Now we prove the following proposition concerning the limit function:

######  Lemma 4.5.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive smooth functions on
${\mathbb{S}}^{2}$ satisfying

(120) $\Delta f_{j}\leq f_{j},\quad\forall j\in\mathbb{N}.$

Let $K>0$ be a real number that satisfies the requirement in Lemma 4.3. Let
$\bar{f}^{K}_{\infty}\in W^{1,2}({\mathbb{S}}^{2})$ be the limit function as
in Lemma 4.4. Then $\bar{f}^{K}_{\infty}$ satisfies the inequality

(121) $-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}_{\infty}^{K}\rangle\leq\int_{{\mathbb{S}}^{2}}u\bar{f}_{\infty}^{K},$

for all $u\in W^{1,2}({\mathbb{S}}^{2})$ such that $u\geq 0$.

###### Proof.

By Lemma 4.4 we know that $\bar{f}^{K}_{j}$ converges to
$\bar{f}^{K}_{\infty}$ in $L^{2}({\mathbb{S}}^{2})$, and that
$\bar{f}^{K}_{j}$ converges to $\bar{f}^{K}_{\infty}$ weakly in
$W^{1,2}({\mathbb{S}}^{2})$. As a result, for any $u\in
W^{1,2}({\mathbb{S}}^{2})$ we have that

(122)
$\int_{{\mathbb{S}}^{2}}u\bar{f}_{j}^{K}\to\int_{{\mathbb{S}}^{2}}u\bar{f}_{\infty}^{K},\
\ \text{ as }\ \ j\to\infty,$

and that

(123) $\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}_{j}^{K}\rangle\to\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}_{\infty}^{K}\rangle,\ \ \text{ as }j\to\infty.$

As a result, by (114) we have for all $u\in W^{1,2}({\mathbb{S}}^{2})$ such
that $u\geq 0$

(124) $-\int_{{\mathbb{S}}^{2}}\langle\nabla
u,\nabla\bar{f}_{\infty}^{K}\rangle\leq\int_{{\mathbb{S}}^{2}}u\bar{f}_{\infty}^{K}.$

Hence by Theorem 8.19 in [6], we have that either the essential infimum of
$\bar{f}^{K}_{\infty}$ is bounded away from zero or $\bar{f}^{K}_{\infty}$ is
the zero function. This finishes the proof. ∎

We need the definition of essential infimum of a function:

######  Definition 4.6.

Consider the standard ${\mathbb{S}}^{2}$ and use $m$ to denote the standard
volume measure in ${\mathbb{S}}^{2}$. Let $U$ be an open subset of
${\mathbb{S}}^{2}$ . Let $f:U\to\mathbb{R}$ be measurable. Define the set

(125) $U_{f}^{ess}=\\{a\in\mathbb{R}:m(f^{-1}(-\infty,a))=0\\}.$

We use $\inf_{U}f$ to denote the essential infimum of $f$ in $U$ and define

(126) $\inf_{U}f=\sup U^{ess}_{f}$

Finally, we apply the maximum principle for weak solution to prove the
following property for the essential infimum of $f_{\infty}$.

######  Proposition 4.7.

Let $\\{f_{j}\\}_{j=1}^{\infty}$ be a sequence of positive smooth functions on
${\mathbb{S}}^{2}$ satisfying

(127) $\Delta f_{j}\leq f_{j},\quad\forall j\in\mathbb{N}.$

If we further assume that $f_{j}\to f_{\infty}$ in $L^{2}({\mathbb{S}}^{2})$
for some $f_{\infty}$, then either the essential infimum of $f_{\infty}$ is
bounded away from zero or $f_{\infty}=0$ a.e. on ${\mathbb{S}}^{2}$.

###### Proof.

Since $\|f_{j}-f_{\infty}\|_{L^{2}({\mathbb{S}}^{2})}\to 0$ as $j\to\infty$,
choose a subsequence if needed, then we have $f_{j}\to f_{\infty}$ poiintwise
almost everywhere in ${\mathbb{S}}^{2}$. Let $K>0$ be a real number that
satisfies the requirement in Lemma 4.3. Construct a truncated sequence
$\\{\bar{f}^{K}_{j}\\}_{j=1}^{\infty}$ as in Definition 4.1. By Lemma 4.4,
choose a subsequence if needed, there exists $\bar{f}^{K}_{\infty}\in
W^{1,2}({\mathbb{S}}^{2})$ such that $\bar{f}^{K}_{j}$ converges to
$\bar{f}^{K}_{\infty}$ in $L^{2}({\mathbb{S}}^{2})$ norm. As a result, choose
a subsequence if needed we have $\bar{f}_{j}^{K}\to\bar{f}_{\infty}^{K}$
pointwise almost everywhere in ${\mathbb{S}}^{2}$.

It suffices to show that if the essential infimum
$\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}=0$ then
$\bar{f}^{K}_{\infty}=f_{\infty}=0$ in ${\mathbb{S}}^{2}$. We assume that
$\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}=0$. Since for each $j$ we have
$0<\bar{f}_{j}^{K}\leq f_{j}$, we have
$0\leq\inf\limits_{{\mathbb{S}}^{2}}\bar{f}_{j}^{K}\leq\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}=0$.
This implies that for any $\delta,\delta^{\prime}>0$, we have

(128)
$m\left(\left(\bar{f}^{K}_{\infty}\right)^{-1}(-\infty,\delta)\right)>0,$

and

(129)
$m\left(\left(\bar{f}^{K}_{\infty}\right)^{-1}(-\infty,-\delta^{\prime})\right)=0.$

Let $N$ be the north pole of $\mathbb{S}^{2}$, and $S$ be the south pole.
$B_{\frac{\pi}{2}}(N)$ and $B_{\frac{\pi}{2}}(S)$ are upper and lower
hemispheres respectively. Then either

(130) $\inf\limits_{B_{\frac{\pi}{2}}(N)}\bar{f}^{K}_{\infty}=0,$

or

(131) $\inf\limits_{B_{\frac{\pi}{2}}(S)}\bar{f}^{K}_{\infty}=0.$

Without loss of generality we assume that
$\inf\limits_{B_{\frac{\pi}{2}}(N)}\bar{f}^{K}_{\infty}=0$. Since
$\bar{f}^{K}_{\infty}\geq 0$ in ${\mathbb{S}}^{2}$, for any $r>\frac{\pi}{2}$,
and $\epsilon>0$ such that $r+\epsilon<\pi$ we have

(132)
$\inf_{B_{r}(N)}\bar{f}^{K}_{\infty}=\inf_{B_{r+\epsilon}(N)}\bar{f}^{K}_{\infty}=0.$

Now by Lemma 4.5, $\bar{f}^{K}_{\infty}$ satisfies

(133) $(\Delta-1)\bar{f}^{K}_{\infty}\leq 0,$

on $B_{r+\epsilon}(N)$ in the weak sense. Hence by the strong maximum
principle for weak solutions (see Theorem 8.19 in [6]), the equality in (132)
implies that $\bar{f}^{K}_{\infty}$ is constant on $B_{r}(N)$. This is true
for any $r>\frac{\pi}{2}$, thus $\bar{f}^{K}_{\infty}\equiv 0$ on
$\mathbb{S}^{2}$. Moreover, since $K>0$, for almost every
$x\in{\mathbb{S}}^{2}$ we have,

(134) $\lim_{j\to\infty}\bar{f}_{j}^{K}=\lim_{j\to\infty}f_{j}=0,$

and hence $f_{\infty}=0$ a.e. on ${\mathbb{S}}^{2}$. This finishes the proof.
∎

###  4.2. A $1$-sweepout of the warped product manifold
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$

Because we will apply the Min-Max minimal surface theory to get an upper bound
for $\operatorname{MinA}$ in §4.3, in this subsection we briefly recall some
basic notions in geometric measure theory following Marques and Neves [13],
and construct a $1$-sweepout for ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$,
which will be used in the proof in Lemma 4.11. For an excellent survey and
more details about these materials we refer to [13] and references therein.

A $k$-current $T$ on $\mathbb{R}^{J}$ is a continuous linear functional on the
space of compactly supported smooth $k$-forms:
$\mathcal{D}^{k}(\mathbb{R}^{J})$. Its boundary $\partial T$ is a
$(k-1)$-current that is defined as $\partial T(\phi):=T(d\phi)$ for
$\phi\in\mathcal{D}^{k-1}(\mathbb{R}^{J})$. A $k$-current $T$ is said to be an
integer multiplicity $k$-current if it can be written as

(135)
$T(\phi)=\int_{S}\langle\phi(x),\tau(x)\rangle\theta(x)d\mathcal{H}^{k},\quad\phi\in\mathcal{D}^{k}(\mathbb{R}^{J}),$

where $S$ is a $\mathcal{H}^{k}$-measurable countable $k$-rectifiable set,
that is $S\subset S_{0}\cup_{j\in\mathbb{N}}S_{j}$ with
$\mathcal{H}^{k}(S_{0})=0$ and $S_{j}$ is an embedded $k$-dimensional
$C^{1}$-submanifold for all $j\in\mathbb{N}$, $\theta$ is a
$\mathcal{H}^{k}$-integrable $\mathbb{N}$-valued function, and $\tau$ is a
$k$-form such that $\tau(x)$ is a volume form for $T_{x}S$ at $x$ where a
$k$-dimensional tangent space $T_{x}S$ is well-defined. Note that this tangent
space $T_{x}S$ is well-defined for $\mathcal{H}^{k}$-a.e. $x\in S$, provided
$\mathcal{H}^{k}(S\cap K)<+\infty$ for every compact set
$K\subset\mathbb{R}^{J}$. Also note that the form $\tau$ give an orientation
for $T_{x}S$. The mass of an integer multiplicity $k$-current $T$ is defined
as

(136) ${\bf M}(T):=\sup\\{T(\phi)\mid\phi\in\mathcal{D}^{k}(\mathbb{R}^{J}),\
\ |\phi|\leq 1\\},$

where $|\phi|$ is the pointwise maximal norm of a form $\phi$.

In particular, a $k$-dimensional embedded smooth submanifold of
$\mathbb{R}^{J}$ can be viewed as an integer multiplicity $k$-current by
integrating a $k$-form over it. Its current boundary is given by its usual
boundary, and its mass is the $k$-dimensional volume of the submanifold.

Let $M$ be a manifold embedded in $\mathbb{R}^{J}$. The space of integral
$k$-currents on $M$, denoted by ${\bf I}_{k}(M)$, is defined to be the space
of $k$-current such that both $T$ and $\partial T$ are integer multiplicity
currents with finite mass and support contained in $M$. The space of
$k$-cycles, denoted by $\mathcal{Z}_{k}(M)$, is defined to be the space of
those $T\in{\bf I}_{k}(M)$ so that $T=\partial Q$ for some $Q\in{\bf
I}_{k+1}(M)$.

A rectifiable $k$-varifold V is defined to be a certain Radon measure on
$\mathbb{R}^{J}\times G_{k}(\mathbb{R}^{J})$, where $G_{k}(\mathbb{R}^{J})$ is
the Grassmannian of $k$-planes in $\mathbb{R}^{J}$. An integral $k$-current
$T\in{\bf I}_{k}(M)$ given as in $(\ref{eqn-integer-current})$ naturally
associates a rectifiable $k$-varifold, denoted by $|T|$, as

(137) $|T|(A)=\int_{S\cap\pi(TS\cap A)}\theta(x)d\mathcal{H}^{k}.$

Here $\pi$ is the natural projection map from $\mathbb{R}^{J}\times
G_{k}(\mathbb{R}^{J})$ to $\mathbb{R}^{J}$, and $TS$ is rank-$k$ tangent
bundle of $S$ consisting of $T_{x}S$ at $x\in S$ where its $k$-dimensional
tangent plane can be well defined. Note that: in the varifold expression
$(\ref{eqn-varifold})$ of $|T|$, we forget the orientation of $S$ determined
by the $k$-form $\tau$ in the current expression $(\ref{eqn-integer-current})$
of $T$.

The space ${\bf I}_{k}(M)$ can be endowed with various metrics and have
different induced topologies. Given $T,S\in{\bf I}_{k}(M)$, the flat metric is
defined by

$\mathcal{F}(T,S):=\inf\left\\{{\bf M}(Q)+{\bf M}(R)\mid T-S=Q+\partial R,\ \
Q\in{\bf I}_{k}(M),\ \ R\in{\bf I}_{k+1}(M)\right\\}$

and induces the flat topology on ${\bf I}_{k}(M)$. We also denote
$\mathcal{F}(T):=\mathcal{F}(T,0)$ and have

(138) $\mathcal{F}(T)\leq{\bf M}(T),\quad\forall T\in{\bf I}_{k}(M).$

For $T,S\in{\bf I}_{k}(M)$, the F-metric is defined by Pitts in [16] as:

(139) ${\bf F}(S,T):=\mathcal{F}(S-T)+{\bf F}(|S|,|T|),$

where ${\bf F}(|S|,|T|)$ is the F-metric on the associated varifolds defined
on page 66 in [16] as:

${\bf F}(|S|,|T|):=\sup\left\\{|S|(f)-|T|(f)\mid f\in
C_{c}(G_{k}(\mathbb{R}^{J})),\ \ |f|\leq 1,\ \ {\rm Lip}(f)\leq 1\right\\}.$

Recall that (see page 66 in [16])

(140) ${\bf F}(|S|,|T|)\leq{\bf M}(S-T),$

and hence

(141) ${\bf F}(S,T)\leq 2{\bf M}(S-T),\quad\forall S,T\in{\bf I}_{k}(M).$

For the Min-Max theory for minimal surfaces, the space of mod $2$ integral
$k$-currents and mod $2$ $k$-cycles are also needed. They are denoted by ${\bf
I}_{k}(M;\mathbb{Z}_{2})$ and $\mathcal{Z}_{k}(M;\mathbb{Z}_{2})$,
respectively, and defined by an equivalence relation: $T\equiv S$ if $T-S=2Q$
for $T,S,Q\in{\bf I}_{k}(M)$. The notions of boundary, mass and metrics
defined above for ${\bf I}_{k}(M)$ can be extended to ${\bf
I}_{k}(M;\mathbb{Z}_{2})$. For a $n$-dimensional manifold $M$, the Constancy
Theorem (Theorem 26.27 in [17]) says that if $T\in{\bf
I}_{n}(M;\mathbb{Z}_{2})$ has $\partial T=0$, then either $T=M$ or $T=0$.

Then we recall some basic facts about the topology of
$\mathcal{Z}_{k}(M;\mathcal{F};\mathbb{Z}_{2})$, that is
$\mathcal{Z}_{k}(M;\mathbb{Z}_{2})$ endowed with flat metric. Their proofs can
be found in [13], also see [1]. Let $n$ be the dimension of the manifold $M$.
Then ${\bf I}_{n}(M;\mathcal{F};\mathbb{Z}_{2})$ is contractible and the
continuous map

(142) $\partial:{\bf
I}_{n}(M;\mathcal{F};\mathbb{Z}_{2})\rightarrow\mathcal{Z}_{n-1}(M;\mathcal{F};\mathbb{Z}_{2})$

is a $2$-fold covering map. The homotopy groups are:

(143)
$\pi_{k}\left(\mathcal{Z}_{n-1}(M;\mathcal{F};\mathbb{Z}_{2}),0\right)=\begin{cases}0,&\text{when}\
\ k\geq 2,\\\ \mathbb{Z}_{2},&\text{when}\ \ k=1.\end{cases}$

For the calculation of the fundamental group, one notes that the map

(144) $\displaystyle
P:\pi_{1}\left(\mathcal{Z}_{n-1}(M;\mathcal{F};\mathbb{Z}_{2}),0\right)$
$\displaystyle\rightarrow$ $\displaystyle\\{0,M\\}$ (145)
$\displaystyle\left[\gamma\right]$ $\displaystyle\mapsto$
$\displaystyle\tilde{\gamma}(1)$

is an isomorphism. Here $\gamma$ is a loop in
$\mathcal{Z}_{n-1}(M;\mathcal{F};\mathbb{Z}_{2})$ with
$\gamma(0)=\gamma(1)=0$, and $\tilde{\gamma}$ is the unique lift to ${\bf
I}_{n}(M;\mathcal{F};\mathbb{Z}_{2})$ with $\tilde{\gamma}(0)=0$. Then by
applying Hurewicz Theorem, one can obtain:

(146)
$H^{1}\left(\mathcal{Z}_{n-1}(M;\mathcal{F};\mathbb{Z}_{2});\mathbb{Z}_{2}\right)=\mathbb{Z}_{2}=\\{0,\bar{\lambda}\\}.$

The the action of the fundamental cohomology class $\bar{\lambda}$ on a
homology class induced by a loop is nonzero if and only if the loop is
homotopically non-trivial.

We take the following definition of $1$-sweepout from [13].

######  Definition 4.8.

A continuous map $\Phi:{\mathbb{S}}^{1}\rightarrow\mathcal{Z}_{n-1}(M;{\bf
F};\mathbb{Z}_{2})$ is called a $1$-sweepout if $\Phi^{*}(\bar{\lambda})\neq
0\in H^{1}({\mathbb{S}}^{1},\mathbb{Z}_{2})$.

Here $\mathcal{Z}_{n-1}(M;{\bf F};\mathbb{Z}_{2})$ is the space
$\mathcal{Z}_{n-1}(M;\mathbb{Z}_{2})$ endowed with the F-metric given in
$(\ref{eqn-F-metric})$.

Now we return back our warped product manifold
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$, that is
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ with Riemannian metric

(147) $g=g_{{\mathbb{S}}^{2}}+f^{2}g_{{\mathbb{S}}^{1}}.$

For each fixed $x\in{\mathbb{S}}^{2}$, we construct a $1$-sweepout of
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ consisting of tori
$\\{\Sigma_{x,r}:=\partial B_{r}(x)\times{\mathbb{S}}^{1}\mid 0\leq
r\leq\pi\\}$ , where $B_{r}(x)$ denotes the geodesic ball on
${\mathbb{S}}^{2}$ centered at $x$ with radius $r$. In other words, we
consider the map

(148) $\displaystyle\Phi:[0,\pi]$
$\displaystyle\rightarrow{\mathcal{Z}_{2}(\mathbb{S}^{2}\times_{f}\mathbb{S}^{1};{\bf
F};\mathbb{Z}_{2}}),$ $\displaystyle r$
$\displaystyle\mapsto\partial\left(B_{r}(x)\times{\mathbb{S}}^{1}\right)=\partial
B_{r}(x)\times{\mathbb{S}}^{1}.$

######  Lemma 4.9.

The map $\Phi$ given in $(\ref{eqn-tori-sweepout})$ provides a $1$-sweepout of
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ as in Definition 4.8.

###### Proof.

Clearly, $\Phi(0)=\Phi(\pi)=0$, and hence $\Phi$ can be viewed as a map from
${\mathbb{S}}^{1}$ to
$\mathcal{Z}_{2}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1};{\bf
F};\mathbb{Z}_{2})$ by identifying the end points of the interval $[0,\pi]$.

Now we show the continuity of the map $\Phi$ on $[0,\pi]$. This is clear for
$r\in(0,\pi)$, since $\partial B_{r}(x)$ varies smoothly for $r\in(0,\pi)$.
Then the continuity at $t=0$ follows from the inequality in $(\ref{eqn-F-
metric-less-than-mass})$ and the estimate:

(149)
${\mathbf{M}}(\Phi(r)-\Phi(0))={\mathbf{M}}(\Phi(r))={\mathbf{M}}\left(\partial
B_{r}(x)\times{\mathbb{S}}^{1}\right)=f\cdot 4\pi^{2}\sin r\rightarrow 0,$

as $r\rightarrow 0$, since the warping function $f$ is smooth on
${\mathbb{S}}^{2}$. The continuity at $t=\pi$ follows similarly, since $\sin
r\rightarrow 0$ as $r\rightarrow\pi$.

Because by the definition flat metric is less than or equal to F-metric,
$\Phi$ is also continuous if we endow the flat metric on
$\mathcal{Z}_{2}(M;\mathbb{Z}_{2})$. So $\Phi$ is a loop in
$\mathcal{Z}_{2}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1};\mathcal{F};\mathbb{Z}_{2})$,
and represents a non-trivial element:

(150) $\left[\Phi\right]\neq
0\in\pi_{1}\left(\mathcal{Z}_{2}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1};\mathcal{F};\mathbb{Z}_{2})\right).$

This is because by the definition of the map $\Phi$ we have that the unique
lift $\tilde{\Phi}$ of $\Phi$ with $\tilde{\Phi}(0)=0$ is given by

(151) $\displaystyle\tilde{\Phi}:[0,\pi]$
$\displaystyle\rightarrow{\mathcal{Z}_{3}(\mathbb{S}^{2}\times_{f}\mathbb{S}^{1};\mathcal{F};\mathbb{Z}_{2}}),$
$\displaystyle r$ $\displaystyle\mapsto B_{r}(x)\times{\mathbb{S}}^{1},$

and has $\tilde{\Phi}(\pi)={\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$.
Consequently, $\Phi^{*}(\bar{\lambda})\neq 0$, and so $\Phi$ is a
$1$-sweepout. ∎

###  4.3. Bound $\operatorname{MinA}$ from above by $L^{1}$-norm of warping
function

In this subsection, we derive an upper bound for
$\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})$ in terms of
$\|f\|_{L^{1}({\mathbb{S}}^{2})}$, provided that
$\|f\|_{L^{2}({\mathbb{S}}^{2})}$ is small relative to
$\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})$.

######  Proposition 4.10.

Let ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ be a warped product
Riemannian manifolds with metric tensor as in (3) that has nonnegative scalar
curvature and
$\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})\geq A>0$. If
$\|f\|_{L^{2}({\mathbb{S}}^{2})}<\frac{A}{2^{\frac{3}{2}}\pi^{\frac{5}{2}}}$,
then we have $\|f\|_{L^{1}({\mathbb{S}}^{1})}\geq\frac{A}{100\pi}$.

Recall that $\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})$
is the infimum of areas of closed embedded minimal surfaces in
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$. Proposition 4.10 is crucial in
the proof of Theorem 4.13 below. In order to prove Proposition 4.10, we first
prove the following two lemmas.

First of all, we use the Min-Max minimal surface theory of Marques and Neves
to bound $\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})$
from above by areas of some tori in
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$.

######  Lemma 4.11.

Let ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ be a warped product
Riemannian manifold with metric tensor as in (3). For each
$x\in{\mathbb{S}}^{2}$, there exists a torus $\Sigma_{x,r_{x}}=\partial
B_{r_{x}}(x)\times\mathbb{S}^{1}\subset\mathbb{S}^{2}\times_{f}\mathbb{S}^{1}$,
$0<r_{x}<\pi$, whose area is not less than
$\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})$, i.e.

(152) ${\rm
Area}(\Sigma_{x,r_{x}})\geq\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}),$

where $B_{r_{x}}(x)$ is the geodesic ball in the standard ${\mathbb{S}}^{2}$
centered at $x$ with radius $r_{x}$.

###### Proof.

We will use Min-Max minimal surface theory of Marques and Neves to prove the
lemma.

For each fixed point $x\in{\mathbb{S}}^{2}$, by Lemma 4.9, the map $\Phi$ in
$(\ref{eqn-tori-sweepout})$ gives a $1$-sweepout of
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ as in Definition 4.8. For
$r\in[0,\pi]$, the image $\Phi(r)=\partial
B_{r}(x)\times{\mathbb{S}}^{1}=:\Sigma_{x,r}$ are tori in
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ with mass:

(153) ${\mathbf{M}}(\Phi(r))={\rm Area}(\Sigma_{x,r})=2\pi\int_{\partial
B_{r}(x)}fds.$

Clearly, ${\mathbf{M}}(\Phi(r))$ is a continuous function of $r$ on $[0,\pi]$
with ${\mathbf{M}}(\Phi(0))={\mathbf{M}}(\Phi(\pi))=0$. Thus there exist
$r_{x}\in(0,\pi)$ such that

(154) ${\mathbf{M}}(\Phi(r_{x}))=\max\\{{\mathbf{M}}(\Phi(r))\mid 0\leq
r\leq\pi\\}.$

Let $\Pi$ be the homotopy class of the $1$-sweepout $\Phi$, which consists of
all continuous maps
$\Phi^{\prime}:[0,\pi]\rightarrow\mathcal{Z}_{2}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1};{\bf
F};\mathbb{Z}_{2})$ with $\Phi^{\prime}(0)=\Phi^{\prime}(\pi)$ such that
$\Phi$ and $\Phi^{\prime}$ are homotopic to each other in the flat topology.
By Lemma 2.2.6 in [13], the width

(155) ${\bf L}(\Pi)=\inf_{\Phi^{\prime}\in\Pi}\sup_{r\in[0,\pi]}\\{{\bf
M}(\Phi^{\prime}(r))\\}>0,$

since $\Phi$ is a $1$-sweepout and so $\Pi$ is a non-trivial homotopy class.
Then Min-Max Theorem of Marques-Neves (see Theorem 2.2.7 in [13]) implies that
there exists a smooth embedded minimal surface $\Sigma$ in
$\mathbb{S}^{2}\times_{f}\mathbb{S}^{1}$ achieving the width, i.e. ${\rm
Area}(\Sigma)={\bf L}(\Pi)>0$.

Finally, by the definitions of the width in (155) and $\operatorname{MinA}$,
and by the choice of $\Sigma_{x,r_{x}}$, we have

(156) ${\rm Area}(\Sigma_{x,r_{x}})\geq{\bf L}(\Pi)={\rm
Area}(\Sigma)\geq\operatorname{MinA}(\mathbb{S}^{2}\times\mathbb{S}^{1}).$

Because $x$ is an arbitrary point on ${\mathbb{S}}^{2}$, this completes the
proof. ∎

Next, we apply Lemma 4.11 and the spherical mean inequality from Proposition
2.4 to prove the following lemma.

######  Lemma 4.12.

Let ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ be a warped product
Riemannian manifold with metric tensors as in (3) that have non-negative
scalar curvatures and
$\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})\geq A>0$. If
$\|f\|_{L^{2}({\mathbb{S}}^{2})}<\frac{A}{2^{\frac{3}{2}}\pi^{\frac{5}{2}}}$,
then there exists a set $\mathcal{H}\subset{\mathbb{S}}^{2}$ satisfying that
for each $x\in\mathcal{H}$ there exists $0<r_{x}\leq\frac{\pi}{2}$ such that

  1. (i)

${\rm
Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)\geq\frac{1}{2}{\rm
Area}({\mathbb{S}}^{2})$,

  2. (ii)

and

(157) $\fint_{\partial B_{r}(x)}fds\geq\frac{A}{2(2\pi)^{2}}$

holds for all $r\in[0,r_{x}]$.

###### Proof.

For any point $x\in{\mathbb{S}}^{2}$, we denote its antipodal point by
$\bar{x}$. By Lemma 4.11, for any $x\in{\mathbb{S}}^{2}$, there exists
$0<r_{x}<\pi$ such that the torus $\Sigma_{x,r_{x}}=\partial
B_{r_{x}}(x)\times{\mathbb{S}}^{1}$ in
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ has area

(158) ${\rm
Area}(\Sigma_{x,r_{x}})\geq\operatorname{MinA}({\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1})\geq
A.$

Since ${\rm Area}(\Sigma_{x,r_{x}})=2\pi\int_{\partial B_{r_{x}}(x)}fds$, we
have

(159) $2\pi\int_{\partial B_{r_{x}}(x)}fds\geq A.$

Thus, we have

(160) $\int_{\partial B_{r_{x}}(x)}fds\geq\frac{A}{2\pi}.$

Now if $0<r_{x}\leq\frac{\pi}{2}$, then we include the point $x$ in the set
$\mathcal{H}$, and if $r_{x}>\frac{\pi}{2}$, then we include its antipodal
point $\bar{x}$ in the set $\mathcal{H}$, and we set $r_{\bar{x}}=\pi-
r_{x}<\frac{\pi}{2}$. Then we still have

(161) $\int_{\partial B_{r_{\bar{x}}}(\bar{x})}fds=\int_{\partial
B_{r_{x}}(x)}fds\geq\frac{A}{2\pi},$

since $\partial B_{r_{\bar{x}}}(\bar{x})=\partial B_{r_{x}}(x)$.

By the construction of the set $\mathcal{H}\subset{\mathbb{S}}^{2}$,
$\mathcal{H}$ contains at least one of any pair of antipodal points on
${\mathbb{S}}^{2}$, and for any $x\in\mathcal{H}$, there exists
$0<r_{x}\leq\frac{\pi}{2}$ such that

(162) $\int_{\partial B_{r_{x}}(x)}fds\geq\frac{A}{2\pi}.$

Then we have that the area of the open set
$\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)$ is at least half of
the area of the whole sphere ${\mathbb{S}}^{2}$, i.e.

(163)
$\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)\geq\frac{1}{2}\operatorname{Area}({\mathbb{S}}^{2}).$

Indeed, otherwise, we have

(164)
$\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(\bar{x})\right)=\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)<\frac{1}{2}\operatorname{Area}({\mathbb{S}}^{2}).$

On the other hand, because for each $x\in{\mathbb{S}}^{2}$ either $x$ or
$\bar{x}$ is contained in $\mathcal{H}$, we have

(165)
${\mathbb{S}}^{2}=\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)\cup\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(\bar{x})\right).$

So

(166) $\displaystyle\operatorname{Area}({\mathbb{S}}^{2})$ $\displaystyle=$
$\displaystyle\operatorname{Area}\left(\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)\cup\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(\bar{x})\right)\right)$
(167) $\displaystyle\leq$
$\displaystyle\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)+\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(\bar{x})\right)$
(168) $\displaystyle<$
$\displaystyle\frac{1}{2}\operatorname{Area}({\mathbb{S}}^{2})+\frac{1}{2}\operatorname{Area}({\mathbb{S}}^{2})=\operatorname{Area}({\mathbb{S}}^{2}).$

This gives a contradiction. So we have
$\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)\geq\frac{1}{2}\operatorname{Area}({\mathbb{S}}^{2})$.

Because ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$ has non-negative scalar
curvature, by Lemma 2.1, we have $\Delta f\leq f$. Then by the spherical mean
inequality in Proposition 2.4, for any
$x\in\mathcal{H}\subset{\mathbb{S}}^{2}$ and any $0\leq r\leq
r_{x}(\leq\frac{\pi}{2})$ we have that

(169) $\fint_{\partial B_{r_{x}}(x)}fds-\fint_{\partial
B_{r}(x)}fds\leq\frac{\|f\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}(r_{x}-r)\leq\frac{A}{2(2\pi)^{2}},$

since
$\|f\|_{L^{2}({\mathbb{S}}^{2})}\leq\frac{A}{2^{\frac{3}{2}}\pi^{\frac{5}{2}}}$
and $r_{x}-r\leq\frac{\pi}{2}$. By rearrange the inequality, we obtain that
for any $x\in\mathcal{H}$ and any $0\leq r\leq r_{x}$,

(170) $\displaystyle\fint_{\partial B_{r}(x)}fds$ $\displaystyle\geq$
$\displaystyle\fint_{\partial B_{r_{x}}(x)}fds-\frac{A}{2(2\pi)^{2}}$ (171)
$\displaystyle=$ $\displaystyle\frac{1}{2\pi\sin r_{x}}\int_{\partial
B_{r_{x}}(x)}fds-\frac{A}{2(2\pi)^{2}}$ (172) $\displaystyle\geq$
$\displaystyle\frac{1}{2\pi}\int_{\partial
B_{r_{x}}(x)}fds-\frac{A}{2(2\pi)^{2}}$ (173) $\displaystyle\geq$
$\displaystyle\frac{A}{(2\pi)^{2}}-\frac{A}{2(2\pi)^{2}}=\frac{A}{2(2\pi)^{2}}.$

∎

We now apply Lemma 4.12 and Vitali covering theorem to prove Proposition 4.10:

###### Proof of Proposition 4.10.

By Lemma 4.12, there exists a set $\mathcal{H}\subset{\mathbb{S}}^{2}$ such
that

(174) ${\rm
Area}(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x))\geq\frac{1}{2}{\rm
Area}({\mathbb{S}}^{2}),$

and for any $x\in\mathcal{H}$, there exists $r_{x}\leq\frac{\pi}{2}$ such that

(175) $\fint_{\partial B_{r}(x)}f\geq\frac{A}{2(2\pi)^{2}}$

holds for all $r\in[0,r_{x}]$.

By the Vitali covering theorem, there exists a countable sequence of points
$\\{x_{i}\mid i\in\mathbb{N}\\}\subset\mathcal{H}$ such that the collection of
balls $\\{B_{\frac{r_{x_{i}}}{10}}(x_{i})\\}$ are disjoint with each other,
and that

(176)
$\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\subset\underset{i\in\mathbb{N}}{\cup}B_{\frac{r_{x_{i}}}{2}}(x_{i}).$

By Lemma 4.12 we have

(177) $\frac{A}{8\pi^{2}}\leq\fint_{\partial B_{r}(x_{i})}f=\frac{1}{2\pi\sin
r}\int_{\partial B_{r}(x_{i})}fds,\quad\forall r\in[0,r_{x_{i}}].$

As a result, we have

(178) $\frac{A}{4\pi}\sin r\leq\int_{\partial B_{r}(x_{i})}fds,\quad\forall
r\in[0,r_{x_{i}}].$

Integrating this inequality from $0$ to $\frac{r_{x_{i}}}{10}$ gives

(179)
$\displaystyle\frac{A}{8\pi^{2}}\operatorname{Area}(B_{\frac{r_{x_{i}}}{10}})$
$\displaystyle=$
$\displaystyle\frac{A}{8\pi^{2}}\int^{\frac{r_{x_{i}}}{10}}_{0}2\pi\sin rdr$
(180) $\displaystyle\leq$
$\displaystyle\int^{\frac{r_{x_{i}}}{10}}_{0}\left(\int_{\partial
B_{r}(x_{i})}fds\right)dr$ (181) $\displaystyle=$
$\displaystyle\int_{B_{\frac{r_{x_{i}}}{10}}(x_{i})}f{\rm
vol}_{{\mathbb{S}}^{2}}.$

Then by summing the above inequalities for $i\in\mathbb{N}$ together, we
obtain

(182)
$\frac{A}{8\pi^{2}}\sum^{+\infty}_{i=1}\operatorname{Area}(B_{\frac{r_{x_{i}}}{10}})\leq\sum^{+\infty}_{i=1}\int_{B_{\frac{r_{x_{i}}}{10}}(x_{i})}f{\rm
vol}_{{\mathbb{S}}^{2}}\leq\|f\|_{L^{1}({\mathbb{S}}^{2})},$

since $\\{B_{\frac{r_{x_{i}}}{10}}(x_{i})\mid i\in\mathbb{N}\\}$ are disjoint
balls. In the standard ${\mathbb{S}}^{2}$ we have

(183)
$\operatorname{Area}\left(B_{\frac{r_{x_{i}}}{10}}(x_{i})\right)\geq\frac{1}{25}\operatorname{Area}\left(B_{\frac{r_{x_{i}}}{2}}(x_{i})\right).$

As a result, we have

(184) $\displaystyle\|f\|_{L^{1}({\mathbb{S}}^{2})}$ $\displaystyle\geq$
$\displaystyle\frac{A}{8\pi^{2}}\sum^{+\infty}_{i=1}\operatorname{Area}\left(B_{\frac{r_{x_{i}}}{10}}\right)$
(185) $\displaystyle\geq$
$\displaystyle\frac{A}{200\pi^{2}}\sum^{+\infty}_{i=1}\operatorname{Area}\left(B_{\frac{r_{x_{i}}}{2}}(x_{i})\right)$
(186) $\displaystyle\geq$
$\displaystyle\frac{A}{200\pi^{2}}\operatorname{Area}\left(\underset{i\in\mathbb{N}}{\cup}B_{\frac{r_{x_{i}}}{2}}(x_{i})\right)$
(187) $\displaystyle\geq$
$\displaystyle\frac{A}{200\pi^{2}}\operatorname{Area}\left(\underset{x\in\mathcal{H}}{\cup}B_{\frac{r_{x}}{10}}(x)\right)$
(188) $\displaystyle\geq$
$\displaystyle\frac{A}{200\pi^{2}}\frac{1}{2}\operatorname{Area}({\mathbb{S}}^{2})=\frac{A}{100\pi}.$

This completes the proof. ∎

###  4.4. Positivity of the limit of warping functions

In this subsection, we use Proposition 4.7 and Proposition 4.10 to prove
Theorem 1.3, we restate it here for the convenience of the reader

######  Theorem 4.13.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product manifolds such that each
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ has non-negative scalar
curvature. If we assume that

(189) ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\leq V\text{
and }\operatorname{MinA}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\geq
A>0,\forall j\in\mathbb{N},$

then we have the following:

  1. $(i)$

After passing to a subsequence if needed, the sequence of warping functions
$\\{f_{j}\\}_{j=1}^{\infty}$ converges to some limit function $f_{\infty}$ in
$L^{q}({\mathbb{S}}^{2})$ for all $q\in[1,\infty)$.

  2. $(ii)$

The limit function $f_{\infty}$ is in $W^{1,p}({\mathbb{S}}^{2})$, for all $p$
such that $1\leq p<2$.

  3. $(iii)$

The essential infimum of $f_{\infty}$ is strictly positive, i.e.
$\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0$.

  4. $(iv)$

If we allow $+\infty$ as a limit, then the limit

(190) $\overline{f_{\infty}}(x):=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}f_{\infty}$

exists for every $x\in{\mathbb{S}}^{2}$. Moreover, $\overline{f_{\infty}}$ is
lower semi-continuous and strictly positive everywhere on ${\mathbb{S}}^{2}$,
and $\overline{f_{\infty}}=f_{\infty}$ a.e. on ${\mathbb{S}}^{2}$.

###### Proof.

$(i)$ By Lemma 2.1 and Lemma 2.2, the nonnegative scalar curvature condition
and ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{2})\leq V$ imply
that the sequence of warping functions $\\{f_{j}\\}_{j=1}^{\infty}$ satisfies
the hypothesis in Proposition 3.5. By applying Proposition 3.5, we get the
desired convergence.

$(ii)$ By applying Proposition 3.5 we get that $f_{\infty}\in
W^{1,p}({\mathbb{S}}^{2})$, for all $p\in[1,2)$.

$(iii)$ We prove $\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0$ by
contradiction. Recall that $\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}$ is the
essential infimum of $f_{\infty}$ as defined in Definition 4.6. First note
that $f_{\infty}\geq 0$, since $f_{j}>0,\forall j\in\mathbb{N}$. Assume that
$\inf\limits_{\mathbb{S}^{2}}f_{\infty}=0$, then by Proposition 4.7 we have
$f_{\infty}=0$ almost everywhere in ${\mathbb{S}}^{2}$ and hence

(191) $f_{j}\rightarrow 0\ \ \text{in}\ \ L^{2}({\mathbb{S}}^{2}),\ \
\text{as}\ \ j\rightarrow+\infty.$

Therefore, for all sufficiently large $j$, we have
$\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}<\frac{A}{2^{\frac{3}{2}}\pi^{\frac{5}{2}}}$.
Then by Proposition 4.10, we have
$\|f_{j}\|_{L^{1}({\mathbb{S}}^{2})}\geq\frac{A}{100\pi}>0$ for all
sufficiently large $j\in\mathbb{N}$. This contradicts with that
$f_{j}\rightarrow 0$ in $L^{2}({\mathbb{S}}^{2})$ as $j\rightarrow+\infty$ in
(191). This finishes the proof of part $(ii)$.

$(iv)$ Because warping functions $f_{i}$ satisfy the requirements in
Proposition 3.7, the existence of the limit

(192) $\overline{f_{\infty}}(x):=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}f_{\infty},$

the lower semi-continuity of $\overline{f_{\infty}}$ and
$\overline{f_{\infty}}=f_{\infty}$ a.e. on ${\mathbb{S}}^{2}$ directly follow
from Proposition 3.7.

Thus we only need to prove that $\overline{f_{\infty}}(x)>0$ for all
$x\in{\mathbb{S}}^{2}$. Let

(193) $e_{\infty}:=\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0.$

By the continuity of the distance funciton $d(y,x)$, there exists
$0<r_{0}<\frac{\pi}{2}$ such that for all $x\in{\mathbb{S}}^{2}$ we have

(194) $f_{\infty}(y)-Cd(y,x)>\frac{e_{\infty}}{2},\ \ \text{ for a.e.}\ \ y\in
B_{r_{0}}(x).$

As a result, we have

(195) $\fint_{B_{r_{0}}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm
vol}(y)>\frac{e_{\infty}}{2},\ \ \forall x\in{\mathbb{S}}^{2}.$

Then because in Proposition 3.7 we proved that for each fixed
$x\in{\mathbb{S}}^{2}$ the ball average
$\fint_{B_{r_{0}}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y)$ is non-
increasing in $r\in\left(0,\frac{\pi}{2}\right)$, and

(196) $\lim_{r\rightarrow 0}\fint_{B_{r}(x)}f_{\infty}=\lim_{r\rightarrow
0}\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm vol}(y),$

we have that for each fixed $x\in{\mathbb{S}}^{2}$,

(197) $\displaystyle\overline{f_{\infty}}(x)$ $\displaystyle:=$
$\displaystyle\lim_{r\rightarrow 0}\fint_{B_{r}(x)}f_{\infty}$ (198)
$\displaystyle=$
$\displaystyle\sup_{0<r<\frac{\pi}{2}}\fint_{B_{r}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm
vol}(y)$ (199) $\displaystyle\geq$
$\displaystyle\fint_{B_{r_{0}}(x)}\left(f_{\infty}(y)-Cd(y,x)\right)d{\rm
vol}(y)$ (200) $\displaystyle>$ $\displaystyle\frac{e_{\infty}}{2}>0.$

This completes the proof of theorem. ∎

######  Remark 4.14.

Theorem 4.13 implies that the limit function $f_{\infty}$ has a everywhere
positive lower semi-continuous representative $\overline{f_{\infty}}$ as a
function in $W^{1,p}({\mathbb{S}}^{2})$ for $1\leq p<2$. For the rest of
paper, $f_{\infty}\in W^{1,p}({\mathbb{S}}^{2})$ will always denote this
everywhere positive lower semi-continuous representative.

We end this section with Proposition 4.15 below. The proof of Proposition 4.15
uses Theorem 4.13 and the spherical mean inequality from Proposition 2.4. The
positive uniform lower bound for warping functions $f_{j}$ obtained in
Proposition 4.15 is important in proving geometric convergences of the
sequence of warped product manifolds
$\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ in our
next paper.

######  Proposition 4.15.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product manifolds with metric tensors as in (3) that have
non-negative scalar curvature and satisfy

(201) ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\leq V\text{
and }\operatorname{MinA}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\geq
A>0,\forall j\in\mathbb{N}.$

Let $e_{\infty}:=\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0$. Then there
exists $j_{0}\in\mathbb{N}$ such that $f_{j}(x)\geq\frac{e_{\infty}}{4}>0$,
for all $j\geq j_{0}$ and all $x\in{\mathbb{S}}^{2}$.

###### Proof.

By Lemma 2.1, the non-negativity of scalar curvature of
${\mathbb{S}}^{2}\times_{f_{i}}{\mathbb{S}}^{1}$ implies that

(202) $\Delta f_{j}\leq f_{j},\quad\forall j\in\mathbb{N}.$

Therefore, by the spherical mean inequality in Proposition 2.4, we have

(203) $f_{j}(x)\geq\fint_{\partial
B_{s}(x)}f_{j}ds-\frac{\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}s,\quad\forall
s\in\left(0,\frac{\pi}{2}\right),x\in{\mathbb{S}}^{2},j\in\mathbb{N}.$

Then multiplying the inequality by $\operatorname{Area}(\partial
B_{s}(x))=2\pi\sin(s)$ gives us

(204) $2\pi\sin(s)f_{j}(x)\geq\int_{\partial
B_{s}(x)}f_{j}ds-\frac{\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}2\pi\sin(s)s,$

for all $s\in\left(0,\frac{\pi}{2}\right),x\in{\mathbb{S}}^{2}$ and
$j\in\mathbb{N}$. Let

(205) $V(r):={\rm vol}(B_{r}(x))=\int^{r}_{0}2\pi\sin sds=2\pi(1-\cos r),$

and let $e_{\infty}:=\inf_{{\mathbb{S}}^{2}}f_{\infty}$ denote the essential
infimum of the limit function $f_{\infty}$ which is strictly positive by
Theorem 4.13.

Now integrating the inequality (204) with respect to $s$ from $0$ to
$r<\frac{\pi}{2}$ gives us

(206) $\displaystyle V(r)f_{j}(x)$ $\displaystyle\geq$
$\displaystyle\int_{B_{r}(x)}f_{j}d{\rm
vol}_{{\mathbb{S}}^{2}}-\frac{\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}\int^{r}_{0}2\pi
s\sin sds$ (208) $\displaystyle\geq$
$\displaystyle\int_{B_{r}(x)}f_{\infty}d{\rm
vol}_{{\mathbb{S}}^{2}}-\|f_{\infty}-f_{j}\|_{L^{1}({\mathbb{S}}^{2})}$
$\displaystyle-\sqrt{2\pi}\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}(\sin r-r\cos r)$
(210) $\displaystyle\geq$ $\displaystyle
e_{\infty}V(r)-\|f_{\infty}-f_{j}\|_{L^{1}({\mathbb{S}}^{2})}$
$\displaystyle-\sqrt{2\pi}\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}(\sin r-r\cos
r).$

Then by dividing the inequality by $V(r)$ we obtain

(211) $f_{j}(x)\geq
e_{\infty}-\frac{\|f_{\infty}-f_{j}\|_{L^{1}({\mathbb{S}}^{2})}}{V(r)}-\frac{\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}\frac{\sin
r-r\cos r}{1-\cos r},$

for all $0<r<\frac{\pi}{2},x\in{\mathbb{S}}^{2}$ and $j\in\mathbb{N}$. By
Lemma 3.2 we have $\sup\limits_{j}\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}<\infty$,
and by direct calculation we have that

(212) $\lim_{r\rightarrow 0}\frac{\sin r-r\cos r}{1-\cos r}=0,$

we can choose $0<r_{1}<\frac{\pi}{2}$ such that

(213) $\left|\frac{\|f_{j}\|_{L^{2}({\mathbb{S}}^{2})}}{\sqrt{2\pi}}\frac{\sin
r_{1}-r_{1}\cos r_{1}}{1-\cos r_{1}}\right|<\frac{e_{\infty}}{2},\quad\forall
j\in\mathbb{N}.$

Moreover, because $f_{j}\rightarrow f_{\infty}$ in $L^{1}({\mathbb{S}}^{2})$,
we can choose $j_{0}\in\mathbb{N}$ such that

(214)
$\frac{\|f_{\infty}-f_{j}\|_{L^{1}({\mathbb{S}}^{2})}}{V(r_{1})}\leq\frac{e_{\infty}}{4},\quad\forall
j\geq j_{0}.$

Finally by combining (211), (213) and (214) together, we conclude that
$f_{j}(x)\geq\frac{e_{\infty}}{4}>0$ for all $j\geq j_{0}$ and
$x\in{\mathbb{S}}^{2}$. ∎

###  4.5. Uniform systole positive lower bound

In this subsection, as an application of non-collapsing of warping functions
$f_{j}$ obtained in Proposition 4.15, we derive a uniform positive lower bound
for the systole of the sequence of warped product manifolds
${\mathbb{S}}^{2}\times_{f_{i}}{\mathbb{S}}^{1}$ satisfying assumptions in
Proposition 4.15.

######  Definition 4.16 (Systole).

The systole of a Riemannian manifold $(M,g)$, which is denoted by $sys(M,g)$
is defined to be the length of the shortest closed geodesic in $M$.

######  Remark 4.17.

People may usually consider so-called $\pi_{1}$-systole that is the length of
a shortest non-contractible closed geodesic. But in the study of compactness
problem of manifolds with nonnegative scalar curvature, we also need to take
into account contractible closed geodesic, for example, in a dumbell, which is
diffeomorphic to ${\mathbb{S}}^{3}$, we may have a short contractible closed
geodesic.

First of all we derive an interesting dichotomy property for closed geodesics
in warped product manifolds: $N\times_{f}{\mathbb{S}}^{1}$, that is, the
product manifold $N\times{\mathbb{S}}^{1}$ endowed with the metric
$g=g_{N}+f^{2}g_{{\mathbb{S}}^{1}}$, where $(N,g_{N})$ is a $n$-dimensional
(either compact or completep non-compact) Riemannian manifold without
boundary, and $f$ is a positive smooth function on $N$.

######  Lemma 4.18.

There is a dichotomy for closed geodesics in $N\times_{f}{\mathbb{S}}^{1}$,
that is, a closed geodesic in $N\times_{f}{\mathbb{S}}^{1}$ either wraps
around the fiber ${\mathbb{S}}^{1}$, or is a geodesic in the base $N$.

###### Proof.

Let $\varphi\in[0,2\pi]$ is a coordinate on the fiber ${\mathbb{S}}^{1}$. The
warped product metric $g$ then can be written as

(215) $g=g_{N}+f^{2}d\varphi^{2}.$

Let

(216) $\gamma(t)=(\gamma_{N}(t),\varphi(t))\ \ t\in[0,1]$

be a closed geodesic in ${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$, and
without loss of generality, we assume $\varphi(0)=0$. We have two possible
cases as following:

Case 1: $\varphi([0,1])=[0,2\pi]$. In this case, clearly, the geodesic wraps
around the fiber ${\mathbb{S}}^{1}$.

Case 2: $\varphi([0,1])\neq[0,2\pi]$. In this case, we show that
$\varphi([0,1])=\\{0\\}$ by a proof by contradiction, and then clearly,
$\gamma$ is a closed geodesic on base $N\cong N\times\\{\varphi=0\\}$.
Otherwise, we have

(217) $0<\varphi_{0}:=\max\\{\varphi(t)\mid t\in[0,1]\\}<2\pi.$

Moreover, there exists $0<t_{0}<1$ such that $\varphi(t_{0})=\varphi_{0}$,
since $\varphi(1)=\varphi(0)=0$ due to the closeness of the geodesic $\gamma$.
Consequently, $t_{0}$ is a critical point of the function $\varphi(t)$, i.e.
$\varphi^{\prime}(t_{0})=0$. As a result, the tangent vector of the geodesic
at $t_{0}$, $\gamma^{\prime}(t_{0})=(\gamma^{\prime}_{N}(t_{0}),0)$, is
tangent to $N\times\\{\varphi=\varphi_{0}\\}$. On the other hand, there is a
geodesic contained in $N\times\\{\varphi=\varphi_{0}\\}$ that passes through
the point $(\gamma_{N}(t_{0}),\varphi_{0})$ and is tangent to
$(\gamma^{\prime}_{N}(t_{0}),0)$ at this point. Then by the uniqueness of the
geodesic with given tangent vector at a point, and the fact that base $N$ is
totally geodesic in the warped product manifold $N\times_{f}{\mathbb{S}}^{1}$,
which can be seen easily by Koszul’s formula, or see Proposition 9.104 in [3],
we can obtain $\varphi([0,1])=\\{\varphi_{0}\\}$, and this contradicts with
$\varphi(0)=0$. ∎

By the dichotomy of closed geodesics in Lemma 4.18, we can obtain a lower
bound estimate for the systole of $N\times_{f}{\mathbb{S}}^{1}$.

######  Lemma 4.19.

The systole of the warped product Riemannian manifold
$N\times_{f}{\mathbb{S}}^{1}$ is greater than or equal to
$\min\left\\{sys(N,g_{N}),2\pi\min\limits_{{\mathbb{S}}^{2}}f\right\\}$.

###### Proof.

Let $\gamma(t)=(r(t),\theta(t),\varphi(t)),t\in[0,1],$ is a closed geodesic in
${\mathbb{S}}^{2}\times_{f}{\mathbb{S}}^{1}$. By Lemma 4.18, $\gamma$ either
wraps around the fiber ${\mathbb{S}}^{1}$, or $\gamma$ is a closed geodesic in
the base manifold $(N,g_{N})$.

If $\gamma$ wraps around the fiber ${\mathbb{S}}^{1}$, then
$\varphi([0,1])=[0,2\pi]$, and so the length of $\gamma$:

(218) $\displaystyle L(\gamma)=\int^{1}_{0}|\gamma^{\prime}(t)|_{g}dt$
$\displaystyle\geq$
$\displaystyle\int^{1}_{0}f(\gamma(t))|\varphi^{\prime}(t)|dt$ (219)
$\displaystyle\geq$
$\displaystyle\min\limits_{{\mathbb{S}}^{2}}f\int^{1}_{0}|\varphi^{\prime}(t)|dt$
(220) $\displaystyle\geq$ $\displaystyle 2\pi\min\limits_{{\mathbb{S}}^{2}}f.$

If $\gamma$ is a closed geodesic in the base $(N,g_{N})$, then by the
definition of systole, the length of $\gamma$ is greater than or equal to
$sys(N,g_{N})$.

These estimates of length of closed geodesics imply the lower bound of systole
in the conclusion. ∎

By combining the lower bound estimate of systole in Lemma 4.19 and Proposition
4.15, we immediately have the following uniform lower bound for systoles.

######  Proposition 4.20.

Let $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}_{j=1}^{\infty}$ be a
sequence of warped product manifolds with metric tensors as in (3) that have
non-negative scalar curvature and satisfy

(221) ${\rm Vol}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\leq V\text{
and }\operatorname{MinA}({\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1})\geq
A>0,\forall j\in\mathbb{N}.$

Let $e_{\infty}:=\inf\limits_{{\mathbb{S}}^{2}}f_{\infty}>0$. Then the
systoles of ${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$, for all
$j\in\mathbb{N}$, have a uniform positive lower bound given by
$\min\left\\{2\pi,\frac{e_{\infty}}{2}\pi\right\\}$.

###### Proof.

First note that the base manifold of the sequence of the warped product
manifolds is the standard $2$-sphere, and its systole is equal to $2\pi$,
since the image of a closed geodesic in
$({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$ is always a great circle.

Then note that $e_{\infty}>0$ follows from the item $(iii)$ in Theorem 4.13.
For each $j\in\mathbb{N}$, by Lemma 4.19, the systole of
${\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}$ has a lower bound given by
$\min\left\\{2\pi,2\pi\min\limits_{{\mathbb{S}}^{2}}f_{j}\right\\}$. Then by
Proposition 4.15,
$\min\limits_{{\mathbb{S}}^{2}}f_{j}\geq\frac{e_{\infty}}{4}$ holds for all
$j\in\mathbb{N}$. Hence the conclusion follows and we complete the proof. ∎

##  5\. Nonnegative distributional scalar curvature of limit metric

Now we use the positive limit function $f_{\infty}$ obtained in Theorem 4.13
to define a weak warped product metrics:

######  Definition 5.1.

Let $f_{\infty}$ be a function defined on ${\mathbb{S}}^{2}$ such that it is
almost everywhere positive and finite on ${\mathbb{S}}^{2}$. We further assume
that $f_{\infty}\in W^{1,p}({\mathbb{S}}^{2})$ for $1\leq p<2$. Define

(222) $g_{\infty}:=g_{{\mathbb{S}}^{2}}+f_{\infty}^{2}g_{{\mathbb{S}}^{1}},$

to be a (weak) warped product Riemannian metric on
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ in the sense of defining an inner
product on the tangent space at (almost) every point of
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$.

######  Remark 5.2.

In general, $g_{\infty}$ is only defined almost everywhere in
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ with respect to the standard product
volume measure $d{\rm vol}_{g_{{\mathbb{S}}^{2}}}d{\rm
vol}_{g_{{\mathbb{S}}^{1}}}$, since $f_{\infty}$ may have value as $+\infty$
on a measure zero set in ${\mathbb{S}}^{2}$. Note that we allow $+\infty$ as
ball average limit in Proposition 3.7. For example, in the extreme example
constructed by Christina Sormani and authors in [19], the limit warping
function equal to $+\infty$ at two poles of ${\mathbb{S}}^{2}$.

In Subsection 5.1, we show $W^{1,p}$ regularity of the weak metric tensor
$g_{\infty}$ defined in Definition 5.1 for $1\leq p<2$ [Proposition 5.4], and
prove that the warped product metrics
$g_{j}=g_{{\mathbb{S}}^{2}}+f^{2}_{j}g_{{\mathbb{S}}^{1}}$ converge to
$g_{\infty}$ in the $L^{q}$ sense for any $1\leq q<+\infty$ [Theorem 5.5].

In Subsection 5.2, we show that the limit weak metric $g_{\infty}$ has
nonnegative distributional scalar curvature in the sense of Lee-LeFloch
[Theorem 5.11].

###  5.1. $W^{1,p}$ limit Riemannian metric $g_{\infty}$

we prove the regularity of the metric tensor. Before that we need the
following definition:

######  Definition 5.3.

We define $L^{p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ as the set of
all tensors defined almost everywhere on
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ such that its $L^{p}$ norm measured
in terms of $g_{0}$ is finite where $g_{0}$ is the isometric product metric

(223) $g_{0}=g_{{\mathbb{S}}^{2}}+g_{{\mathbb{S}}^{1}}\textrm{ on
}{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}.$

We define $W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ as the set
of all tensors, $h$, defined almost everywhere on
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ such that both the $L^{p}$ norm of
$h$ and the $L^{p}$ norm of $\overline{\nabla}h$ measured in terms of $g_{0}$
are finite where $\overline{\nabla}$ is the connection corresponding to the
metric $g_{0}$.

Now we prove the regularity of the metric tensor $g_{\infty}$ defined in
Definition 5.1:

######  Proposition 5.4 (Regularity of the metric tensor).

The Riemannian metric tensor $g_{\infty}$ as in Definition 5.1 satisfies

(224) $g_{\infty}\in W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$

for all $p\in[1,2)$ in the sense of Definition 5.3.

###### Proof.

Using the background metric, $g_{0}$, we have

(225)
$\displaystyle\|g_{\infty}\|_{L^{p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})}$
$\displaystyle=$
$\displaystyle(2\pi)^{\frac{1}{p}}\|(2+f^{4}_{\infty})^{\frac{1}{2}}\|_{L^{p}({\mathbb{S}}^{2})}$
(226) $\displaystyle\leq$
$\displaystyle(2\pi)^{\frac{1}{p}}\|\sqrt{2}+f^{2}_{\infty}\|_{L^{p}({\mathbb{S}}^{2})}$
(227) $\displaystyle\leq$
$\displaystyle(2\pi)^{\frac{1}{p}}\left(\sqrt{2}(4\pi)^{\frac{1}{p}}+\|f_{\infty}\|^{2}_{L^{2p}({\mathbb{S}}^{2})}\right)$

is finite, since by the assumption, $f_{\infty}\in W^{1,p}({\mathbb{S}}^{2})$
for any $p\in[1,2)$, and Sobolev embedding theorem, we have $f_{\infty}\in
L^{2p}({\mathbb{S}}^{2})$ for any $p\in[1,\infty)$.

Now for the gradient estimate, we fix an arbitrary $p\in[1,2)$. We use
$\overline{\nabla}$ to denote the connection of the background metric $g_{0}$.
Clearly, we have

(228)
$\overline{\nabla}g_{\infty}=\overline{\nabla}g_{{\mathbb{S}}^{2}}+\overline{\nabla}f^{2}_{\infty}\otimes
g_{{\mathbb{S}}^{1}}+f^{2}_{\infty}\overline{\nabla}g_{{\mathbb{S}}^{1}}.$

and

(229) $\overline{\nabla}g_{{\mathbb{S}}^{2}}=0,\text{ and
}\overline{\nabla}g_{{\mathbb{S}}^{1}}=0.$

Moreover, since $\overline{\nabla}f^{2}_{\infty}=2f_{\infty}\nabla f_{\infty}$
we have

(230) $\overline{\nabla}g_{\infty}=2f_{\infty}\nabla f_{\infty}\otimes
g_{{\mathbb{S}}^{1}},$

where $\nabla f_{\infty}$ is the gradient of $f_{\infty}$ on
$({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$. As a result, we have

(231)
$\displaystyle\|\overline{\nabla}g_{\infty}\|^{p}_{L^{p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})}$
$\displaystyle=$ $\displaystyle
2\pi\int_{{\mathbb{S}}^{2}}2^{p}f^{p}_{\infty}|\nabla f_{\infty}|^{p}d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}$ (232) $\displaystyle=$ $\displaystyle
2^{p+1}\pi\|f_{\infty}\|_{L^{pq^{*}}({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})}\cdot\|\nabla
f_{\infty}\|_{L^{pq}({\mathbb{S}}^{2})},$

where $q>1$ is chosen so that $pq<2$, and $q^{*}=\frac{q}{q-1}$. Then again by
Sobolev embedding theorem we have $f_{\infty}\in L^{q}$ for any
$p\in[1,\infty)$, thus we obtain that
$\|\overline{\nabla}g_{\infty}\|_{L^{p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})}$
is finite for any $p\in[1,2)$. This completes the proof. ∎

Then we apply Proposition 3.5 to prove Theorem 1.7 which concerns the $L^{q}$
pre-compactness of warped product circles over sphere with non-negative scalar
curvature. We restate Theorem 1.7 as follows:

######  Theorem 5.5.

Let $\\{g_{j}=g_{{\mathbb{S}}^{2}}+f^{2}_{j}g_{{\mathbb{S}}^{1}}\mid
j\in\mathbb{N}\\}$ be a sequence of warped Riemannian metrics on
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ satisfying requirements in (4). Then
there exists a subsequence $g_{j_{k}}$ and a (weak) warped Riemannian metric
$g_{\infty}\in W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for
$p\in[1,2)$ as in Definition 5.1 such that

(233) $g_{j_{k}}\rightarrow g_{\infty}\ \ \text{ in}\ \
L^{q}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0}),\ \ \forall
q\in[1,\infty).$

###### Proof.

By Lemma 2.1 and Lemma 2.2, the assumptions in (4) for $g_{j}$ implies that
the warping functions $f_{j}$ satisfy the assumptions in Proposition 3.5.
Thus, by applying Proposition 3.5, we have that there exists a subsequence
$f_{j_{k}}$ of warping functions and $f_{\infty}\in W^{1,p}({\mathbb{S}}^{2})$
for all $1\leq p<2$, such that

(234) $f_{j_{k}}\rightarrow f_{\infty},\ \ \text{in}\ \
L^{q}({\mathbb{S}}^{2}),\ \ \forall q\in[1,\infty).$

Let $g_{\infty}:=g_{{\mathbb{S}}^{2}}+f^{2}_{\infty}g_{{\mathbb{S}}^{1}}$.
Then by Proposition 5.4, we have

(235) $g_{\infty}\in W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})\ \
\forall 1\leq p<2.$

Moreover, because

(236) $g_{j}-g_{\infty}=(f_{j}^{2}-f_{\infty}^{2})g_{{\mathbb{S}}^{1}},$

we have that for any $q\in[1,\infty)$,

(237)
$\displaystyle\|g_{j_{k}}-g_{\infty}\|_{L^{q}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})}$
(238) $\displaystyle=$
$\displaystyle(2\pi)^{\frac{1}{q}}\|f^{2}_{j_{k}}-f^{2}_{\infty}\|_{L^{q}({\mathbb{S}}^{2})}$
(239) $\displaystyle=$
$\displaystyle(2\pi)^{\frac{1}{q}}\|(f_{j_{k}}-f_{\infty})\cdot(f_{j_{k}}+f_{\infty})\|_{L^{q}({\mathbb{S}}^{2})}$
(240) $\displaystyle\leq$
$\displaystyle(2\pi)^{\frac{1}{q}}\|f_{j_{k}}-f_{\infty}\|_{L^{2q}({\mathbb{S}}^{2})}\cdot\|f_{j_{k}}+f_{\infty}\|_{L^{2q}({\mathbb{S}}^{2})}$
(241) $\displaystyle\rightarrow$ $\displaystyle 0,\ \ \text{as}\ \
j_{k}\rightarrow\infty,$

since $f_{j_{k}}\rightarrow f_{\infty}$ in $L^{2q}({\mathbb{S}}^{2})$ for any
$q\in[1,\infty)$. ∎

######  Remark 5.6.

As showed by the example constructed by Christina Sormani and authors in [19],
$g_{\infty}\in W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for
$1\leq p<2$ is the best regularity we can expect in general for the limit weak
Riemannian metric $g_{\infty}$, see Proposition 3.6 and Remark 3.8 in [19].

###  5.2. Nonnegative distributional scalar curvature of $g_{\infty}$

Building upon work of Mardare-LeFloch [11], Dan Lee and Philippe LeFloch
defined a notion of distributional scalar curvature for smooth manifolds that
have a metric tensor which is only $L^{\infty}_{loc}\cap W^{1,2}_{loc}$. See
Definition 2.1 of [10] which we review below in Definition 5.7.

In Theorem 5.5 we proved that if a sequence of smooth warped product circles
over the sphere $\\{{\mathbb{S}}^{2}\times_{f_{j}}{\mathbb{S}}^{1}\\}$ with
non-negative scalar curvature have uniform bounded volumes, then a subsequence
of the smooth warped product metric
$g_{j}=g_{{\mathbb{S}}^{2}}+f^{2}_{j}g_{{\mathbb{S}}^{1}}$ converges to a weak
warped product metric
$g_{\infty}=g_{{\mathbb{S}}^{2}}+f^{2}_{\infty}g_{{\mathbb{S}}^{1}}\in
W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})(1\leq p<2)$ in the sense
of $L^{q}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for any $q\geq 1$.
For the rest of this section, we use $g_{\infty}$ to denote such limit metric.
We use $g_{0}=g_{{\mathbb{S}}^{2}}+g_{{\mathbb{S}}^{1}}$ as a background
metric .

In Theorem 5.11, we prove that this limit (weak) metric $g_{\infty}$ has
nonnegative distributional scalar curvature in the sense of Lee-LeFloch . In
Remarks 5.9-5.10, we discuss how the metric tensors studied by Lee and LeFloch
have stronger regularity than the regularity of $g_{\infty}$ but their
definition of distributional scalar curvature is still valid in our case.

First we recall Definition 2.1 in the work of Lee-LeFloch [10]. In their
paper, they assume that

######  Definition 5.7 (Lee-LeFloch).

Let $M$ be a smooth manifold endowed with a smooth background metric, $g_{0}$.
Let $g$ be a metric tensor defined on $M$ with $L^{\infty}_{loc}\cap
W^{1,2}_{loc}$ regularity and locally bounded inverse $g^{-1}\in
L^{\infty}_{loc}$.

The scalar curvature distribution $\operatorname{Scalar}_{g}$ is defined as a
distributions in $M$ such that for every test function $u\in
C^{\infty}_{0}(M)$

(242)
$\langle\operatorname{Scalar}_{g},u\rangle:=\int_{M}\left(-V\cdot\overline{\nabla}\left(u\frac{d\mu_{g}}{d\mu_{g_{0}}}\right)+Fu\frac{d\mu_{g}}{\,d\mu_{0}}\right)\,d\mu_{0},$

where the dot product is taken using the metric $g_{0}$, $\overline{\nabla}$
is the Levi-Civita connection of $g_{0}$, $d\mu_{g}$ and $d\mu_{g_{0}}$ are
volume measure with respect to $g$ and $g_{0}$ respectively, $V$ is a vector
field given by

(243) $V^{k}:=g^{ij}\Gamma^{k}_{ij}-g^{ik}\Gamma^{j}_{ji},$

where

(244)
$\Gamma^{k}_{ij}:=\frac{1}{2}g^{kl}\left(\overline{\nabla}_{i}g_{jl}+\overline{\nabla}_{j}g_{il}-\overline{\nabla}_{l}g_{ij}\right),$
(245)
$F:=\overline{R}-\overline{\nabla}_{k}g^{ij}\Gamma^{k}_{ij}+\overline{\nabla}_{k}g^{ik}\Gamma^{j}_{ji}+g^{ij}\left(\Gamma^{k}_{kl}\Gamma^{l}_{ij}-\Gamma^{k}_{jl}\Gamma^{l}_{ik}\right),$

and

(246)
$\overline{R}:=g^{ij}\left(\partial_{k}\overline{\Gamma}^{k}_{ij}-\partial_{i}\overline{\Gamma}^{k}_{kj}+\overline{\Gamma}^{l}_{ij}\overline{\Gamma}^{k}_{kl}-\overline{\Gamma}^{l}_{kj}\overline{\Gamma}^{k}_{il}\right).$

The Riemannian metric $g$ has nonnegative distributional scalar curvature, if
$\langle\operatorname{Scalar}_{g},u\rangle\geq 0$ for every nonnegative test
function $u$ in the integral in (242).

######  Definition 5.8 (Distributional total scalar curvature).

For a weak metric $g$ having the regularity as in Definition 5.7, we define
the distributional total scalar curvature of $g$ to be
$\langle\operatorname{Scalar}_{g},1\rangle$, which is obtained by setting the
test function $u\equiv 1$ in the integration in (242).

Note that for a $C^{2}$-metric, the distributional total scalar curvature is
exactly the usual total scalar curvature.

######  Remark 5.9.

By the regularity assumption for the Riemannian metric $g$ in the work of Lee-
LeFloch [10], one has the regularity $\Gamma^{k}_{ij}\in L^{2}_{loc}$, $V\in
L^{2}_{loc},F\in L^{1}_{loc}$, and the density of volume measure $d\mu_{g}$
with respect to $\,d\mu_{0}$ is

(247) $\tfrac{d\mu_{g}}{\,d\mu_{0}}\in L^{\infty}_{loc}\cap W^{1,2}_{loc}.$

Thus

(248)
$FirstInt_{g}=\int_{M}\left(-V\cdot\overline{\nabla}\left(u\frac{d\mu_{g}}{d\mu_{g_{0}}}\right)\right)\,d\mu_{0}$

and

(249)
$SecondInt_{g}=\int_{M}\left(Fu\frac{d\mu_{g}}{\,d\mu_{0}}\right)\,d\mu_{0}.$

are both finite.

######  Remark 5.10.

Our limit metric is less regular than the metrics studied by Lee-LeFloch in
[10]. Recall that in Proposition 5.4 we showed $g_{\infty}\in
W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for $1\leq p<2$, and as
shown by the extreme example constructed in [19], in general $g_{\infty}\notin
W^{1,2}_{loc}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$, see Proposition
3.6 in [19].

In Remark 5.18 below we show that in genenral both integrals in (248) and
(249) may be divergent. However, in Theorem 5.11 below, we show that in our
case the sum of (248) and (249) is still well-defined since the singularity
cancels out when we add them up.

We are ready to prove Theorem 1.8. We restate it as follows:

######  Theorem 5.11.

The limit metric $g_{\infty}$ obtained in Theorem 5.5 has nonnegative
distributional scalar curvature on ${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ in
the sense of Lee-LeFloch as in Definition 5.7. In particular, (242) is finite
and nonnegative for any nonnegative test function, $u\in
C^{\infty}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$. Moreover, the total
scalar curvatures of $g_{j}$ converge to the distributional total scalar
curvature of $g_{\infty}$.

The proof of Theorem 5.11 consists of straightforward but technical
calculations. For the convenience of readers, we provide some details of the
calculations in the following lemmas.

We use $g_{0}=g_{{\mathbb{S}}^{2}}+g_{{\mathbb{S}}^{1}}$ as background metric,
and use coordinate $\\{r,\theta,\varphi\\}$ on
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$, where $(r,\theta)$ is a polar
coordinate on ${\mathbb{S}}^{2}$ and $\varphi$ is a coordinate on
${\mathbb{S}}^{1}$. The corresponding local frame of the tangent bundle is
$\\{\partial_{r},\partial_{\theta},\partial_{\varphi}\\}$. In this coordinate
system, both $g_{0}$ and $g_{\infty}$ are diagonal and given as

(250) $g_{0}=\begin{pmatrix}1&0&0\\\ 0&\sin^{2}r&0\\\ 0&0&1\\\
\end{pmatrix}\textrm{ and }g_{\infty}=\begin{pmatrix}1&0&0\\\ 0&\sin^{2}r&0\\\
0&0&f_{\infty}^{2}(r,\theta)\\\ \end{pmatrix}.$

First of all, by the formula of Christoffel symbols:

(251)
$\overline{\Gamma}^{i}_{jk}=\frac{1}{2}(g_{0})^{il}\left(\frac{\partial(g_{0})_{il}}{\partial
x^{k}}+\frac{\partial(g_{0})_{lk}}{\partial
x^{j}}-\frac{\partial(g_{0})_{jk}}{\partial x^{l}}\right),$

one can easily obtain the following lemma:

######  Lemma 5.12.

The Christoffel symbols of the Levi-Civita connection $\overline{\nabla}$ of
the background metric $g_{0}=g_{{\mathbb{S}}^{2}}+g_{{\mathbb{S}}^{1}}$, in
the coordinate $\\{r,\theta,\varphi\\}$, all vanish except

(252) $\overline{\Gamma}^{r}_{\theta\theta}=-\sin r\cos r,$

and

(253) $\overline{\Gamma}^{\theta}_{r\theta}=\overline{\Gamma}^{\theta}_{\theta
r}=\frac{\cos r}{\sin r}.$

Then by Lemma 5.12, the formula

(254)
$\overline{\nabla}_{i}(g_{\infty})_{jl}=\partial_{i}\left((g_{\infty})_{jl}\right)-\overline{\Gamma}^{p}_{ij}(g_{\infty})_{pl}-\overline{\Gamma}^{q}_{il}(g_{\infty})_{jq},$

and the diagonal expression of $g_{\infty}$ in (250), one can obtain the
following lemma:

######  Lemma 5.13.

For the limit metric, $g_{\infty}$, with the background metric, $g_{0}$, the
Christoffel symbols defined by Lee-LeFloch as in (244), in the coordinate
$\\{r,\theta,\varphi\\}$, all vanish except

(255)
$\Gamma^{r}_{\varphi\varphi}=-f_{\infty}\partial_{r}f_{\infty},\quad\Gamma^{\theta}_{\varphi\varphi}=-\frac{1}{\sin^{2}r}f_{\infty}\partial_{\theta}f_{\infty},$

and

(256) $\Gamma^{\varphi}_{r\varphi}=\Gamma^{\varphi}_{\varphi
r}=\frac{\partial_{r}f_{\infty}}{f_{\infty}},\quad\Gamma^{\varphi}_{\theta\varphi}=\Gamma^{\varphi}_{\varphi\theta}=\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}.$

Note also that

######  Lemma 5.14.

Note that the volume forms are:

(257) $d\mu_{0}=\,dr\wedge\sin(r)\,d\theta\wedge\,d\varphi$

and

(258) $d\mu_{\infty}=dr\wedge\sin(r)\,d\theta\wedge
f_{\infty}(r,\theta)\,d\varphi$

which are both defined almost everywhere. In particular,

(259) $\frac{\,d\mu_{\infty}}{\,d\mu_{0}}=f_{\infty}(r,\theta)$

is in $W^{1,p}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$ for $p<2$.

###### Proof.

The first claim holds away from $r=0$ and $r=\pi$ by the definition of volume
form, and the second claim holds almost everywhere on
$({\mathbb{S}}^{2}\times{\mathbb{S}}^{1},g_{0})$. So
$d\mu_{\infty}=f_{\infty}d\mu_{0}$ almost everywhere which gives us the third
claim. The rest follows from Proposition 3.5. ∎

Now we are ready to compute the vector field $V$ and the function $F$ defined
by Lee-LeFloch as in (243) and $(\ref{defn-F})$.

######  Lemma 5.15.

For the limit metric $g_{\infty}$ with the background metric $g_{0}$, the
vector field $V$ defined in (243), in the local frame
$\\{\partial_{r},\partial_{\theta},\partial_{\varphi}\\}$, is given by

(260)
$V=\left(-2\frac{\partial_{r}f_{\infty}}{f_{\infty}},-\frac{2}{\sin^{2}r}\frac{\partial_{\theta}f_{\infty}}{f_{\infty}},0\right).$

Furthermore

(261)
$-V\cdot\overline{\nabla}\left(u\frac{\,d\mu_{\infty}}{\,d\mu_{0}}\right)=2\frac{\partial_{r}f_{\infty}}{f_{\infty}}\partial_{r}(uf_{\infty})+\frac{2}{\sin^{2}r}\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}\partial_{\theta}(uf_{\infty}).$

###### Proof.

By plugging the non-vanishing Christoffel symbols in Lemma 5.13 into

(262) $V^{k}:=g^{ij}_{\infty}\Gamma^{k}_{ij}-g^{ik}_{\infty}\Gamma^{j}_{ji},$

we get

(263) $\displaystyle V^{r}$ $\displaystyle=$ $\displaystyle
g^{\varphi\varphi}_{\infty}\Gamma^{r}_{\varphi\varphi}-g^{rr}_{\infty}\Gamma^{\varphi}_{\varphi
r}$ (264) $\displaystyle=$
$\displaystyle\frac{1}{(f_{\infty})^{2}}(-f_{\infty}\partial_{r}f_{\infty})-\frac{\partial_{r}f_{\infty}}{f_{\infty}}=-2\frac{\partial_{r}f_{\infty}}{f_{\infty}}.$

Also

(265) $\displaystyle V^{\theta}$ $\displaystyle=$ $\displaystyle
g^{\varphi\varphi}_{\infty}\Gamma^{\theta}_{\varphi\varphi}-g^{\theta\theta}_{\infty}\Gamma^{\varphi}_{\varphi\theta}$
(266) $\displaystyle=$
$\displaystyle\frac{1}{f^{2}_{\infty}}\left(-\frac{1}{\sin^{2}r}f_{\infty}\partial_{\theta}f_{\infty}\right)-\frac{1}{\sin^{2}r}\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}=-\frac{2}{\sin^{2}r}\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}.$
(267)
$V^{\varphi}=g^{ij}_{\infty}\Gamma^{\varphi}_{ij}-g^{\varphi\varphi}_{\infty}\Gamma^{j}_{j\varphi}=0.$

By Lemma A.8, we now see that,

(268)
$\displaystyle\overline{\nabla}\left(u\frac{\,d\mu_{\infty}}{\,d\mu_{0}}\right)$
$\displaystyle=$ $\displaystyle\overline{\nabla}\left(uf_{\infty}\right)$
(269) $\displaystyle=$
$\displaystyle\partial_{r}(uf_{\infty})\frac{\partial}{\partial
r}+\frac{1}{\sin^{2}r}\partial_{\theta}(uf_{\infty})\frac{\partial}{\partial\theta}+\partial_{\varphi}(uf_{\infty})\frac{\partial}{\partial\varphi}$

Thus

(270)
$-V\cdot\overline{\nabla}\left(u\frac{\,d\mu_{\infty}}{\,d\mu_{0}}\right)=2\frac{\partial_{r}f_{\infty}}{f_{\infty}}\partial_{r}(uf_{\infty})+\frac{2}{\sin^{2}r}\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}\partial_{\theta}(uf_{\infty})$

∎

######  Lemma 5.16.

For the limit metric $g_{\infty}$ with the background metric $g_{0}$, the
function $F$ defined in (245) is given by

(271)
$F=2-2\left(\frac{\partial_{r}f_{\infty}}{f_{\infty}}\right)^{2}-\frac{2}{\sin^{2}r}\left(\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}\right)^{2}=2-2\frac{1}{(f_{\infty})^{2}}|\nabla
f_{\infty}|^{2}.$

Furthermore,

(272)
$\left(Fu\frac{\,d\mu_{\infty}}{d\mu_{0}}\right)=2uf_{\infty}-2\frac{u}{f_{\infty}}|\nabla
f_{\infty}|^{2}.$

Here $|\nabla f_{\infty}|$ is the norm of weak gradient of $f_{\infty}$ with
respect to the standard metric $g_{{\mathbb{S}}^{2}}$.

###### Proof.

First note that from the expression of $\overline{R}$ in (246) and the
Christofell symbols calculated in Lemma 5.12, one can easily see that

(273) $\overline{R}=R_{g_{{\mathbb{S}}^{2}}}=2.$

Also recall that

(274)
$\overline{\nabla}_{i}g^{jl}_{\infty}=\partial_{i}(g^{jl}_{\infty})+\overline{\Gamma}^{j}_{ip}g^{pl}_{\infty}+\overline{\Gamma}^{l}_{iq}g^{jq}_{\infty}.$

Then by Lemmas 5.12 and 5.13, one has

(275) $\displaystyle F$ $\displaystyle:=$
$\displaystyle\overline{R}-(\overline{\nabla}_{k}g^{ij})\Gamma^{k}_{ij}+(\overline{\nabla}_{k}g^{ik})\Gamma^{j}_{ji}+g^{ij}(\Gamma^{k}_{kl}\Gamma^{l}_{ij}-\Gamma^{k}_{jl}\Gamma^{l}_{ik})$
(280) $\displaystyle=$ $\displaystyle
2-\overline{\nabla}_{r}g^{\varphi\varphi}\Gamma^{r}_{\varphi\varphi}-\overline{\nabla}_{\theta}g^{\varphi\varphi}\Gamma^{\theta}_{\varphi\varphi}-2\overline{\nabla}_{\varphi}g^{r\varphi}\Gamma^{\varphi}_{r\varphi}-2\overline{\nabla}_{\varphi}g^{\theta\varphi}\Gamma^{\varphi}_{\theta\varphi}$
$\displaystyle+\overline{\nabla}_{k}g^{rk}\Gamma^{\varphi}_{\varphi
r}+\overline{\nabla}_{k}g^{\theta k}\Gamma^{\varphi}_{\varphi\theta}$
$\displaystyle+\cancel{g^{\varphi\varphi}\Gamma^{\varphi}_{\varphi
r}\Gamma^{r}_{\varphi\varphi}}+\bcancel{g^{\varphi\varphi}\Gamma^{\varphi}_{\varphi\theta}\Gamma^{\theta}_{\varphi\varphi}}$
$\displaystyle-g^{\varphi\varphi}\Gamma^{r}_{\varphi\varphi}\Gamma^{\varphi}_{r\varphi}-\bcancel{g^{\varphi\varphi}\Gamma^{\theta}_{\varphi\varphi}\Gamma^{\varphi}_{\varphi\theta}}-g^{rr}\Gamma^{\varphi}_{r\varphi}\Gamma^{\varphi}_{r\varphi}-\cancel{g^{\varphi\varphi}\Gamma^{\varphi}_{\varphi
r}\Gamma^{r}_{\varphi\varphi}}$
$\displaystyle-g^{\theta\theta}\Gamma^{\varphi}_{\theta\varphi}\Gamma^{\varphi}_{\theta\varphi}-g^{\varphi\varphi}\Gamma^{\varphi}_{\varphi\theta}\Gamma^{\theta}_{\varphi\varphi}$
(290) $\displaystyle=$ $\displaystyle
2-\left(\partial_{r}(g^{\varphi\varphi})+2\overline{\Gamma}^{\varphi}_{r\varphi}g^{\varphi\varphi}\right)\Gamma^{r}_{\varphi\varphi}-\left(\partial_{\theta}(g^{\varphi\varphi})+2\overline{\Gamma}^{\varphi}_{\theta\varphi}g^{\varphi\varphi}\right)\Gamma^{\theta}_{\varphi\varphi}$
$\displaystyle-2\left(\partial_{\varphi}(g^{r\varphi})+\overline{\Gamma}^{r}_{\varphi\varphi}g^{\varphi\varphi}+\overline{\Gamma}^{\varphi}_{\varphi
r}g^{rr}\right)\Gamma^{\varphi}_{r\varphi}$
$\displaystyle-2\left(\partial_{\varphi}(g^{\theta\varphi})+\overline{\Gamma}^{\theta}_{\varphi\varphi}g^{\varphi\varphi}+\overline{\Gamma}^{\varphi}_{\varphi\theta}g^{\theta\theta}\right)\Gamma^{\varphi}_{\theta\varphi}$
$\displaystyle+\left(\partial_{r}(g^{rr})+\overline{\Gamma}^{r}_{rr}g^{rr}+\overline{\Gamma}^{r}_{rr}g^{rr}\right)\Gamma^{\varphi}_{\varphi
r}$
$\displaystyle+\left(\partial_{\theta}(g^{r\theta})+\overline{\Gamma}^{r}_{\theta\theta}g^{\theta\theta}+\overline{\Gamma}^{\theta}_{\theta
r}g^{rr}\right)\Gamma^{\varphi}_{\varphi r}$
$\displaystyle+\left(\partial_{\varphi}(g^{r\varphi})+\overline{\Gamma}^{r}_{\varphi\varphi}g^{\varphi\varphi}+\overline{\Gamma}^{\varphi}_{\varphi
r}g^{rr}\right)\Gamma^{\varphi}_{\varphi r}$
$\displaystyle+\left(\partial_{r}(g^{\theta
r})+\overline{\Gamma}^{\theta}_{rr}g^{rr}+\overline{\Gamma}^{r}_{r\theta}g^{\theta\theta}\right)\Gamma^{\varphi}_{\varphi\theta}$
$\displaystyle+\left(\partial_{\theta}(g^{\theta\theta})+\overline{\Gamma}^{\theta}_{\theta\theta}g^{\theta\theta}+\overline{\Gamma}^{\theta}_{\theta\theta}g^{\theta\theta}\right)\Gamma^{\varphi}_{\varphi\theta}$
$\displaystyle+\left(\partial_{\varphi}(g^{\theta\varphi})+\overline{\Gamma}^{\theta}_{\varphi\varphi}g^{\varphi\varphi}+\overline{\Gamma}^{\varphi}_{\varphi\theta}g^{\theta\theta}\right)\Gamma^{\varphi}_{\varphi\theta}$
$\displaystyle-g^{\varphi\varphi}\Gamma^{r}_{\varphi\varphi}\Gamma^{\varphi}_{r\varphi}-g^{\varphi\varphi}\Gamma^{\varphi}_{\varphi\theta}\Gamma^{\theta}_{\varphi\varphi}-g^{rr}\Gamma^{\varphi}_{r\varphi}\Gamma^{\varphi}_{r\varphi}-g^{\theta\theta}\Gamma^{\varphi}_{\varphi\theta}\Gamma^{\varphi}_{\varphi\theta}$
(294) $\displaystyle=$ $\displaystyle
2-(-2)\frac{\partial_{r}f_{\infty}}{(f_{\infty})^{3}}\left(-f_{\infty}\partial_{r}f_{\infty}\right)-(-2)\frac{\partial_{\theta}f_{\infty}}{(f_{\infty})^{3}}\left(-\frac{1}{\sin^{2}r}f_{\infty}\partial_{\theta}f_{\infty}\right)$
$\displaystyle+\left(-\frac{\cos r}{\sin r}+\frac{\cos r}{\sin
r}\right)\Gamma^{\varphi}_{\varphi
r}-\frac{1}{(f_{\infty})^{2}}(-f_{\infty}\partial_{r}f_{\infty})\left(\frac{\partial_{r}f_{\infty}}{f_{\infty}}\right)$
$\displaystyle-\frac{1}{(f_{\infty})^{2}}\left(-\frac{1}{\sin^{2}r}f_{\infty}\partial_{\theta}f_{\infty}\right)\left(\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}\right)$
$\displaystyle-\left(\frac{\partial_{r}f_{\infty}}{f_{\infty}}\right)^{2}-\frac{1}{\sin^{2}r}\left(\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}\right)^{2}$
(295) $\displaystyle=$ $\displaystyle
2-2\left(\frac{\partial_{r}f_{\infty}}{f_{\infty}}\right)^{2}-\frac{2}{\sin^{2}r}\left(\frac{\partial_{\theta}f_{\infty}}{f_{\infty}}\right)^{2}$
(296) $\displaystyle=$ $\displaystyle 2-2\frac{1}{(f_{\infty})^{2}}|\nabla
f_{\infty}|^{2}.$

We immediately obtain our second claim by applying Lemma A.8. ∎

######  Lemma 5.17.

For $g$ being our limit metric tensor $g_{\infty}$ and a smooth nonnegative
test function $u$, the integrals in (248) and (249) are given by

(297) $\displaystyle\quad FirstInt_{g_{\infty}}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}}\left(-V\cdot\overline{\nabla}\left(u\frac{\,d\mu_{\infty}}{\,d\mu_{0}}\right)\right)\,d\mu_{0}$
(298) $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}}^{2}}\left(2\langle\nabla
f_{\infty},\nabla\bar{u}\rangle+2\frac{\bar{u}}{f_{\infty}}|\nabla
f_{\infty}|^{2}\right)d{\rm vol}_{g_{{\mathbb{S}}^{2}}},$

and

(299) $\displaystyle\quad SecondInt_{g_{\infty}}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}}\left(Fu\frac{\,d\mu_{\infty}}{d\mu_{0}}\right)\,d\mu_{0}$
(300) $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}}^{2}}\left(2\bar{u}f_{\infty}-2\frac{\bar{u}}{f_{\infty}}|\nabla
f_{\infty}|^{2}\right)d{\rm vol}_{g_{{\mathbb{S}}^{2}}},$

where

(301) $\bar{u}(r,\theta)=\int^{2\pi}_{0}u(r,\theta,\varphi)d\varphi,$

$\nabla f_{\infty}$ and $\nabla\bar{u}$ are (weak) gradients of functions
$f_{\infty}$ and $\bar{u}$ on standard sphere
$({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$ respectively, and
$\langle\cdot,\cdot\rangle$ is the Riemannian metric on
$({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$.

###### Proof.

By integrating the formulas in Lemma 5.15 and Lemma 5.16, one can easily
obtain the integrals in (298) and (300). ∎

######  Remark 5.18.

As explained in Remark 3.6, $f_{\infty}\in W^{1,p}$ for any $1\leq p<2$, which
is obtained in in Proposition 3.5, is the best regularity for $f_{\infty}$ in
general, and we cannot expect $f_{\infty}$ is in
$W^{1,2}_{loc}({\mathbb{S}}^{2})$. So the integral
$\int_{{\mathbb{S}}^{2}}\frac{\bar{u}}{f_{\infty}}|\nabla f_{\infty}|^{2}d{\rm
vol}_{g_{{\mathbb{S}}^{2}}}$ appearing in both (298) and (300) may be
divergent (c.f. Lemma 4.16 in [19]). But if we sum the integrants in (298) and
(300) firstly and then integrate, then this possible divergent integrant terms
cancel out and we obtain a finite integral as in the following lemma.

######  Lemma 5.19.

For the limit metric
$g_{\infty}=g_{{\mathbb{S}}^{2}}+f^{2}_{\infty}g_{{\mathbb{S}}^{1}}$, the
scalar curvature distribution $\operatorname{Scalar}_{g_{\infty}}$ defined in
Definition 5.7 can be expressed, for every test function $u\in
C^{\infty}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$, as the integral

(302)
$\langle\operatorname{Scalar}_{g_{\infty}},u\rangle=\int_{{\mathbb{S}}^{2}}\left(2\langle\nabla
f_{\infty},\nabla\bar{u}\rangle+2f_{\infty}\bar{u}\right)d{\rm
vol}_{g_{{\mathbb{S}}^{2}}},$

and this is finite for any test function $u\in
C^{\infty}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$. Here $\bar{u}$ is defined
as in (350), $\nabla f_{\infty}$ and $\nabla\bar{u}$ are (weak) gradients of
functions $f_{\infty}$ and $\bar{u}$ on standard sphere
$({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$ respectively, and
$\langle\cdot,\cdot\rangle$ is the Riemannian metric on
$({\mathbb{S}}^{2},g_{{\mathbb{S}}^{2}})$.

###### Proof.

The expression in (302) immediately follows from the expressions in (298) and
(300) and Definition 5.7. The finiteness of the integral in (302) follows from
that $f_{\infty}\in W^{1,p}({\mathbb{S}}^{2})$ for $1\leq p<2$ as proved in
Proposition 3.5. ∎

We now apply these lemmas to prove Theorem 5.11:

###### Proof.

By the expression (11) of the scalar curvature of
${\mathbb{S}}^{2}\times_{f_{i}}{\mathbb{S}}^{1}$, we have that for any test
function $u\in C^{\infty}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$,

(303)
$\displaystyle\int_{{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}}\operatorname{Scalar}_{g_{j}}ud{\rm
vol}_{g_{j}}$ $\displaystyle=$