[1]\fnmShota \surHamanaka

[1]\orgdivDepartment of Mathematics, \orgnameGraduate School of Science, Osaka University, \orgaddress\street1-1 Machikaneyama-cho, \cityToyonaka, \postcode560-0043, \stateOsaka, \countryJapan

Limit theorems for the total scalar curvature

0000-0003-1536-9620 [email protected] *

Abstract

We prove that the lower bound of the total scalar curvatures on a closed $n$ -manifold is preserved under the $W^{1,p}~{}(p>n)$ convergence of the Riemannian metrics provided that each scalar curvature is nonnegative. We also discuss certain weighted version of this type of theorem.

keywords:

Scalar curvature, Ricci flow, Heat flow, Weak notions of the scalar curvature lower bound

pacs:

[

MSC Classification]53C21, 53E20

1 Introduction

Gromov [8] proved the following “ $C^{0}$ -limit theorem”.

Theorem 1.1 ([8, p.1118] and [2]).

Let $M$ be a (possibly open) smooth manifold and $\kappa:M\rightarrow\mathbb{R}$ a continuous function. Consider a sequence of $C^{2}$ -Riemannian metrics $g_{i}$ on $M$ that converges to a $C^{2}$ -Riemannian metric $g$ in the local $C^{0}$ -sense. Assume that for all $i=1,2,\cdots$ the scalar curvature $R(g_{i})$ of $g_{i}$ satisfies $R(g_{i})\geq\kappa$ everywhere on $M.$ Then $R(g)\geq\kappa$ everywhere on $M.$

In contrast, let $M$ be the same as in the above theorem, for given a continuous function $\sigma:M\rightarrow\mathbb{R},$ the set $\{g\in\mathcal{M}|~{}R(g)\leq\sigma~{}\mathrm{on}~{}M\}$ is $C^{0}$ -dense in the set $\mathcal{M}$ of all smooth Riemannian metrics on $M$ ([15, Theorem B]). On the other hand, in our forthcoming paper (see the preprint by the author, arXiv:2301.05444v5), we will show that in a fixed conformal class, the upper bound of the total scalar curvature is preserved under $C^{0}$ -convergence of metric tensors. Gromov [8] proved the above theorem by using a gluing technique and the resolution of Geroch’s conjecture. Later, Bamler [2] gave an alternative proof of this theorem using the Ricci–DeTurck flow. On the other hand, Lee–Topping recently proved (in their preprint, arXiv:2203.01223v2) that non-negativity of scalar curvature is not preserved in dimension at least four under the topology of uniform convergence of Riemannian distance. For other studies about the behavior of the scalar curvature lower bound under various weak topologies, see, for example, [5, 9, 11].

As we will see below, we can observe that the point-wise version of the Gromov’s theorem (Theorem 1.1) is false in general. That is, we can easily construct an example of $C^{2}$ -Riemannian metrics $(g_{i})$ on a smooth manifold $M$ which satisfies the following: $g_{i}$ converges to a $C^{2}$ -Riemannian metric $g$ on $M$ in the $C^{0}$ -sense as $i\rightarrow\infty.$ And there is a point $p\in M$ such that for some $\kappa\in\mathbb{R},$

R(g_{i})(p)\geq\kappa~{}~{}\mathrm{for~{}all}~{}i\in\mathbb{N},

but

R(g)(p)<\kappa.

Indeed, Lohkamp gave an example in [4, Lecture Series 2, Counterexample 2.3.2].

Example 1.1 ([4, Lecture Series 2, Counterexample 2.3.2]. cf. Example 4.2 and 4.3 in this paper).

For each $i\in\mathbb{N},$ we define a smooth metric on $\mathbb{R}^{n}$ as

g_{i}:=\begin{cases}\exp{\left(2f_{i}\right)}\cdot g_{Eucl}&\mathrm{on}~{}D_{\alpha,i}:=\{x\in\mathbb{R}^{n}~{}|~{}|x|_{g_{Eucl}}^{2}\leq\alpha/i\},\\ g_{Eucl}&\mathrm{in}~{}\mathbb{R}^{n}\setminus D_{2\alpha,i}.\end{cases}

Here, $g_{Eucl}$ denotes the Euclidean metric on $\mathbb{R}^{n},$ $|x|_{g_{Eucl}}$ is the Euclidean distance from $x$ to the origin $o,$ the smooth function

f_{i}:=\frac{\alpha}{i}-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}

whose support is contained in $D_{2\alpha,i},$ and $\alpha\in\mathbb{R}$ is a positive constant. Then, using the following fact, we have $R(g_{i})(o)=C(n)>0.$ Moreover, $g_{i}$ converges to $g_{Eucl}$ in the uniformly $C^{0}$ -sense on $\mathbb{R}^{n}.$

Fact 1.1 (Conformal change of the scalar curvature).

For a Riemannian metric $g$ and a $C^{2}$ -function $\phi,$ set $\bar{g}:=e^{2\phi}g,$ then

R(\bar{g})=e^{-2\phi}R(g)-2(n-1)~{}e^{-2\phi}\Delta_{g}\phi-(n-2)(n-1)~{}e^{-2\phi}~{}|d\phi|_{g}^{2}.

The proof of this fact is a straightforward calculation. Note that the scalar curvature lower bounds are guaranteed only at the origin $o$ in this example. But we can never take a metric whose scalar curvature is bounded from below by some positive constant on a small neighborhood of $o$ and nonnegative on the whole manifold $\mathbb{R}^{n},$ and which is equal to the Euclidean metric outside a compact subset of $\mathbb{R}^{n}.$ Indeed, if such a metric exists, then we can construct a metric on the $n$ -dimensional torus whose scalar curvature is nonnegative everywhere and positive somewhere. But, this is impossible by the resolution of Geroch’s conjecture (see [7, 16, 17]). Of course, we can apply Theorem 1.1 to this sequence $(g_{i})$ and $M=\mathbb{R}^{n}.$ But, we can only take the lower bound $\kappa$ such that $\sup_{\mathbb{R}^{n}}\kappa\leq 0$ in this case because the support of $f_{i}$ shrinks as $i\rightarrow\infty.$ Hence we can only obtain the trivial fact $R(g_{Eucl})=0\geq\kappa$ even though we use Theorem 1.1.

In this paper we investigate some total scalar curvature versions of Theorem 1.1. More precisely, we will consider the following problem: Let $M$ be a (possibly non-compact) smooth manifold, $\{g_{i}\}_{i\in\mathbb{N}}$ a sequence of $C^{2}$ -Riemannian metrics and $g$ a $C^{2}$ -Riemannian metric on $M.$ If $g_{i}$ converges to $g$ in some sense (with respect to $g$ ) and

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq\kappa~{}~{}~{}~{}\mathrm{for~{}all}~{}~{}i

for some constant $\kappa\in\mathbb{R}.$ Here $R_{g_{i}},\,d\mathrm{vol}_{g_{i}}$ denote respectively the scalar curvature of $g_{i}$ and the Riemannian volume measure of $g_{i}.$ Then, does

\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa

hold? Or, more generally, let we consider that a sequence of $C^{2}$ metrics $g_{i}$ on $M$ converging to a $C^{2}$ metric in some sense and a sequence of functions $f_{i}$ on $M$ converging to a function $f$ in some sense that satisfy

\int_{M}R(g_{i})e^{-f_{i}}\,d\mathrm{vol}_{g_{i}}\geq\kappa~{}~{}~{}~{}\mathrm{for~{}all}~{}~{}i

for some constant $\kappa\in\mathbb{R}.$ Then, we ask whether

\int_{M}R(g)e^{-f}\,d\mathrm{vol}_{g}\geq\kappa

holds or not.

We emphasize that we will only consider the situation that metrics converge to some metric with respect to certain topology which is weaker than $C^{2},$ each metric is at least $C^{2},$ and the underlying manifolds we consider are assumed to be smooth.

If $M^{2}$ is closed (i.e., compact without boundary) surface, from the Gauss-Bonnet theorem,

\int_{M^{2}}R(g)\,d\mathrm{vol}_{g}=4\pi\chi(M)

for each Riemannian metric $g$ on $M.$ Here $\chi(M)$ denotes the Euler characteristic of $M.$ Hence it is sufficient to consider the above problem (unweighted case) in dimension $n\geq 3$ and, unless otherwise mentioned, we will assume below that the dimension of manifolds are greater than or equal to three.

On the other hand, if $M^{2n}$ is a closed complex $n~{}(\mathrm{real}~{}2n)$ -manifold ( $n\geq 1$ ) and $g,g_{i}~{}(i=1,2,\cdots)$ are Kähler metrics on $M.$ Let $\omega$ and $\omega_{i}$ be the Kähler forms of $g$ and $g_{i}$ respectively. Assume that $g_{i}$ converges to $g$ (hence $\omega_{i}$ converges to $\omega$ ) in the $C^{0}$ -sense as $i\rightarrow\infty.$ Then

\begin{split}\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}&=\frac{4\pi}{(n-1)!}(c_{1}(M)\smallsmile[\omega_{i}^{n-1}](M))\\ &\longrightarrow\frac{4\pi}{(n-1)!}(c_{1}(M)\smallsmile[\omega^{n-1}](M))=\int_{M}R(g)\,d\mathrm{vol}_{g}~{}~{}~{}(i\rightarrow\infty),\end{split}

where $c_{1}(M)$ denotes the first Chern class of $M.$ Note that we assumed here that the limiting metric $g$ is Kähler metric on $M$ as well, but, in our main theorem 3, we will not assume that the limiting metric $g$ is a Ricci soliton. Although it deviates a bit from our subject, the following interesting result about lower bounds of scalar curvature integrals is also known.

Theorem 1.2 ([4, Lecture Series 1, Theorem 4.1]).

Let $M$ be a compact $n$ -dimensional manifold $(n\geq 3)$ carrying a hyperbolic metric $g_{0}.$ There is a neighborhood $\mathcal{U}$ of $g_{0}$ in the space of all Riemannian metrics with the $C^{2}$ -topology such that for any $g\in\mathcal{U},$

\int_{M}\left(R(g)_{-}\right)^{n/2}d\mathrm{vol}_{g}\geq\int_{M}\left(R(g_{0})_{-}\right)^{n/2}\,d\mathrm{vol}_{g_{0}}

and equality if and only if $g$ is isometric to $g_{0}.$ Here, $R(g)_{-}:=\max\left\{-R(g),~{}0\right\}.$

Our first main result in this paper is the following.

Main Theorem 1.

Let $p>n.$ Let $M^{n}$ be a closed manifold of dimension $n\geq 3$ and $g$ a $C^{2}$ Riemannian metric on $M.$ Assume that $(g_{i})$ is a sequence of $C^{2}$ Riemannian metrics on $M$ such that $g_{i}$ converges to $g$ on $M$ in the $W^{1,p}$ -sense as $i\rightarrow\infty,$

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq\kappa~{}~{}~{}\mathrm{for~{}some~{}constant}~{}\kappa\in\mathbb{R}

and $R(g_{i})\geq 0$ on $M$ for each $i.$ Then

\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa.

Remark 1.1.

From the assumption $R(g_{i})\geq 0,$ of course the only meaningful case is $\kappa\geq 0.$ For the case of $\kappa=0,$ this claim follows from Theorem 1.1. However, Main Theorem 1 states that this statement still holds for all $\kappa\geq 0.$

Here, we said that a sequence of metrics $(g_{i})_{i}$ converges to a metric $g$ on $M$ in the $W^{1,p}$ -sense if $g_{i}$ and all first weak derivatives of it respectively converge to those of $g$ with respect to the $L^{p}$ -norm of $g.$ (Since $M$ is compact, if $(g_{i})$ converges in the $W^{1,p}$ -sense with respect to $g,$ then it also converges $W^{1,p}$ -sense with respect to any fixed reference metric on $M.$ ) Note that from Morrey’s embedding, there is a continuous embedding: $W^{1,p}\hookrightarrow C^{0,1-\frac{n}{p}}$ if $p>n.$ Therefore the same statement of Main Theorem 1 still holds even though one replace $W^{1,p}~{}(p>n)$ with $C^{0,\alpha}$ for all $\alpha\in(0,1].$ On the other hand, in Main Theorem 1, if we weaken the assumption from $W^{1,p}$ -convergence to $C^{0}$ -convergence, then the same statement (without the assumption that each $g_{i}$ has nonnegative scalar curvature) no longer holds true in general. Indeed, we will give an example (Example 4.3) on every closed Riemannian $n$ -manifold for $n\geq 3.$

As a corollary of Main Theorem 1 and Theorem 1.1, we obtain the following.

Corollary 1.1.

Let $p>n,$ and let $\mathcal{M}$ be the space of all $C^{2}$ -Riemannian metrics on a closed manifold $M.$ For any nonnegative continuous function $\sigma\in C^{0}(M,\mathbb{R}_{\geq 0})$ and constant $\kappa\in\mathbb{R},$ the subspace

\left\{g\in\mathcal{M}~{}\middle|~{}\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa,~{}R(g)\geq\sigma~{}\mathrm{on}~{}M\right\}~{}(\subset\mathcal{M})

is closed in $\mathcal{M}$ with respect to the $W^{1,p}$ -topology.

As our second main result in this paper, we will prove the following theorem in a more general setting.

Main Theorem 2.

Let $p>n^{2}/2.$ Suppose that $M^{n}$ is a closed $n$ -manifold ( $n\geq 2$ ), $g$ is a $C^{2}$ Riemannian metric on $M,$ and $(g_{i})$ is a sequence of $C^{2}$ Riemannian metrics on $M$ such that $g_{i}$ converges to $g$ on $M$ in the $W^{1,p}$ -sense as $i\rightarrow\infty.$ Let $(f_{i})$ be a family of functions on $M$ and $f$ a function on $M.$ Assume the following:

(1)

There ia a positive constant $\Lambda>0$ such that $f$ and $f_{i}~{}(i\geq 0)$ are $\Lambda$ -Lipschitz functions on $M,$
(2)

$f_{i}\overset{C^{0}}{\longrightarrow}f$ uniformly on $M,$
(3)

$R(g_{i})\geq 0$ on $M$ for all $i,$
(4)

$\int_{M}R(g_{i})\,e^{-f_{i}}d\mathrm{vol}_{g_{i}}\geq\kappa~{}~{}(\kappa\in\mathbb{R}).$

Then

\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}\geq\kappa.

This is non-trivial even in the two-dimensional case because $f_{i}$ is non-constant in general, hence we cannot use the Gauss-Bonnet theorem. For example, $(1)$ and $(2)$ are automatically satisfied in case that $e^{-f_{i}}d\mathrm{vol}_{g_{i}}=d\mathrm{vol}_{g_{0}}=e^{-f}d\mathrm{vol}_{g}$ for some $C^{0}$ metric $g_{0}$ , and $g_{i}\overset{C^{1}}{\longrightarrow}g.$ Indeed, since each $f_{i}$ can be locally written as $f_{i}=\log(\sqrt{detg_{i}})-\log(\sqrt{detg_{0}})$ ( $f$ is also represented in the same form) and $g_{i}\overset{C^{1}}{\longrightarrow}g,$ so the norm of the first derivatives $(|\nabla f_{i}|_{g})_{i}$ are uniformly bounded and $f_{i}\rightarrow f$ uniformly on $M.$ And, we speculate the assumptions $R(g_{i})\geq 0~{}(i=1,2,\cdots)$ in Main Theorem 1 and 2 can be not needed. As a corollary of Main Theorem 2, we can obtain the following and from it, we can also define a new weak notion of scalar curvature lower bounds.

Corollary 1.2 ( $=$ Corollary 3.4).

Let $p>n^{2}/2$ and $\kappa$ a constant. Suppose that $M$ is a closed manifold of dimension $n\geq 2,$ $g$ is a $C^{2}$ Riemannian metric on $M$ and $(g_{i})$ is a sequence of $C^{2}$ metric on $M.$ Assume the following:

(1)

a sequence $(\phi_{i})$ of nonnegative continuous functions on $M$ satisfying: for any positive constant $a>0$ there is a positive constant $\Lambda>0$ such that $\log(\phi_{i}+a)$ is $\Lambda$ -Lipschitz on $M$ for all $i,$
(2)

$(\phi_{i})$ converges to some nonnegative continuous function $\phi$ in the uniformly $C^{0}$ -sense on $M,$
(3)

$R(g_{i})\geq 0$ on $M$ for each $i,$
(4)

$\int_{M}R(g_{i})\phi_{i}\,d\mathrm{vol}_{g_{i}}\geq\kappa\int_{M}\phi_{i}\,d\mathrm{vol}_{g_{i}},$
(5)

$g_{i}$ converges to $g$ in the $W^{1,p}$ -sense.

Then

\int_{M}R(g)\phi\,d\mathrm{vol}_{g}\geq\kappa\int_{M}\phi\,d\mathrm{vol}_{g}.

Remark 1.2.

From $(3),(5)$ and Remark 3.3, it is known that $\int_{M}R(g)\psi\,d\mathrm{vol}_{g}\geq 0$ for all smooth nonnegative function $\psi.$ Hence it is reasonable to consider case that $\kappa\geq 0$ in this setting.

We give necessary notions and prove this corollary in Section 3. Based on this type of limit theorem, we can define a new generalized notion of scalar curvature lower bound via the existence of certain type of approximate sequences as follows.

Definition 1.1.

Let $M^{n}$ be a smooth closed $n$ -manifold and $\kappa$ a constant. For any $W^{1,p}~{}(p>n^{2}/2)$ metric $g$ on $M,$ $g$ is of $R(g)\geq\kappa$ in the approximate distributional sense if for any nonnegative continuous function $\phi$ there is an approximate sequence $(\phi_{i})$ satisfying $(1),(2)$ in Corollary 1.2, and there exists a $W^{1,p}~{}(p>n^{2}/2)$ -approximate sequence of $C^{2}$ -metrics $(g_{i})$ satisfying $(3)$ - $(5)$ in Corollary 1.2.

Suppose now that a metric $g$ in Definition 1.1 is actually $C^{2}.$ Then, from Corollary 1.2, $R(g)\geq\kappa$ in the approximate distributional sense on $M$ implies that the same bound $R(g)\geq\kappa$ holds in the distributional sense on $M.$ As a result, $R(g)\geq\kappa$ in the conventional sense from Remark 3.3. Note that $R(g)\geq 0$ in the conventional sense in this case due to $(3)$ of Corollary 1.2 and Theorem 1.1.

For example, if $(M,g_{i})$ is a complete gradient shrinking or steady Ricci soliton, then $R(g_{i})\geq 0$ on $M$ (see [6, Corollary 2.5], [22, Theorem 1.3]). Hence, from Main Theorem 1, if furthermore each total scalar curvature is bounded from below by some (nonnegative) constant, then such a lower bound is preserved under the $W^{1,p}~{}(p>n)$ convergence of metrics. On the other hand, if we assume that each metric is Ricci soliton (with certain additional assumption), then we can obtain a similar statement under weaker $C^{0}$ convergence of metrics. Our third main theorem is the following.

Main Theorem 3.

Let $M^{n}$ be a closed $n$ -manifold and $g$ a $C^{2}$ Riemannian metric on $M.$ Let $(g_{i})$ be a sequence of Ricci solitons on $M$ (i.e., $-2\,\mathrm{Ric}(g_{i})=\mathcal{L}_{Y_{i}}g_{i}-2\lambda_{i}\,g_{i}$ for some constant $\lambda_{i}\in\mathbb{R}$ and a vector field $Y_{i}\in\Gamma(TM)$ ) such that $g_{i}$ converges to $g$ on $M$ in the $C^{0}$ -sense as $i\rightarrow\infty.$ Assume

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq\kappa~{}~{}~{}\mathrm{for~{}some~{}constant}~{}\kappa\in\mathbb{R}.

Moreover, assume that $\lambda_{i}\leq C_{+}$ for all $i$ and some constant $C_{+}\in\mathbb{R}$ if $\kappa\geq 0$ (resp. $\lambda_{i}\geq C_{-}$ for some $C_{-}\in\mathbb{R}$ if $\kappa<0$ ). Then

\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa.

On a closed manifold, every Ricci soliton is a self-similar solution of the Ricci flow equation, and vice versa. This self-similarity is one of the reasons why the assumption of convergence in Main Theorem 3 can be weaker than $W^{1,p}.$

The rest of the paper will be arranged as follows. In Section 2, we will prove our Main theorems after preparing some lemmas. In Secrion 3, we will prove some corollaries of Main Theorems. In Section 4, we will give several (partial) counterexamples to Main theorem 1 and 2.

2 Proof of Main Theorems

Firstly, we need the following stability result for the Ricci-DeTurck flow like what is proved in [2, Lemma 2]. More precisely, we need the following.

Lemma 2.1 ([11, 19]).

Let $(M,h)$ be a closed Riemannian manifold endowed with a $C^{2}$ -Riemannian metric $h.$ Then there are constants $\tau,\varepsilon>0,~{}C<\infty$ such that the following is true: Consider a $C^{2}$ -Riemannian metric $g$ that is $(1+\epsilon)$ -bilipschitz close to $h.$ Then there is a continuous family of Riemannian metrics $(g_{t})_{t\in[0,\tau]}$ on $M$ such that the following holds:

$(a)$

For all $t\in[0,\tau],$ the metric $g_{t}$ is $1.1$ -bilipschitz to $h.$
$(b)$

$(g_{t})$ is smooth on $M\times(0,\tau]$ and the map $[0,\tau]\rightarrow C^{2}(M,S^{2}M),~{}t\mapsto g_{t}$ is continuous. In particular, $t\mapsto R_{g_{t}}$ is continuous on $[0,\tau]$ .
$(c)$

$g_{0}=g$ and $(g_{t})_{t\in[0,\tau]}$ is a solution to the Ricci DeTurck flow equation

$\mathrm{(RDE)}~{}~{}\frac{\partial}{\partial t}g_{t}=-2~{}\mathrm{Ric}(g_{t})-\mathcal{L}_{X_{h}(g_{t})}g_{t},$

where $\mathcal{L}_{X_{h}(g_{t})}g_{t}$ denotes the Lie derivative of $g_{t}$ with respect to the time-dependent vector field $X_{h}(g_{t})$ defined in Remark 2.1 below.
$(d)$

For any $t\in(0,\tau]$ and any $m=0,1,2$ we have

$|\nabla_{h}^{m}g_{t}|_{h}<\frac{C}{t^{m/2}}$

where $|\nabla_{h}^{m}g_{t}|_{h}$ denotes the norm with respect to $h$ of the covariant derivatives of $g(t)$ by the Levi-Civita connection of $h.$
$(e)$

If $(g_{i,t})_{t\in[0,\tau]}$ is a sequence of solutions to (RDE) that are continuous on $M\times[0,\tau]$ and smooth on $M\times(0,\tau]$ and if $g_{i,0}$ converges to some metric $g_{0}$ in the $C^{0}$ -sense, then there is a subsequence of $(g_{i,t})$ that converges to $(g_{t})$ in the $C^{0}$ -sense on $M\times[0,\tau]$ and in the locally smooth sense on $M\times(0,\tau]$ with respect to $h.$

Proof.

The items $(a),(c)$ and $(d)$ are the result of [19, Theorem 1.1]. $(e)$ follows from the standard argument using the derivative estimates in [19, Theorem 1.1] and Arzela-Ascoli’s theorem. $(b)$ follows from the same argument as in the proof of [19, Theorem 5.2]. But we use the equations which is obtained from differentiating the equation $(66)$ in [18] and $(2.15)$ in [19] twice with respect to $\nabla_{h}$ instead of [18, (66) Section 2] and [19, (2.15)] respectively. In particular, we use the continuity of the second derivatives of $g$ to derive the estimate corresponding to $(5.2)$ in the proof of Theorem 5.2 in [19]. ∎

Thanks to [11, Theorem 3.11], if we assume $g_{i}$ converges to $g$ in the $W^{1,p}~{}(p>n)$ -sense, we obtain the following as well.

Lemma 2.2 (c.f. [19, Theorem 4.3]).

Let $(M,g)$ be a closed Riemannian manifold endowed with a $C^{2}$ -Riemannian metric $g.$ Then there are constants $\varepsilon,~{}\tau>0$ and $C<\infty$ such that the following is true: Consider a $C^{2}$ -Riemannian metric $g^{\prime}$ on $M$ which is $\varepsilon$ -close to $g$ in the $W^{1,p}~{}(p>n)$ -sense with respect to $g$ (i.e., $|g-g^{\prime}|_{W^{1,p}(M,g)}<\varepsilon,$ where $|g-g^{\prime}|_{W^{1,p}(g)}$ denotes the $W^{1,p}$ -norm of the $(0,2)$ -tensor $g-g^{\prime}$ with respect to $g$ ). Then there is a continuous family of Riemannian metrics $(g^{\prime}_{t})_{t\in[0,\tau]}$ on $M$ such that $(a)$ - $(c)$ and $(e)$ in Lemma 2.1 hold. Moreover, the following $(d^{\prime})$ holds instead of $(d)$ in Lemma 2.1.

(d^{\prime})

For any $t\in(0,\tau]$ we have

|\nabla_{g}g^{\prime}_{t}|_{g}<\frac{C}{t^{n/2p}}~{}~{}\mathrm{and}~{}~{}|\nabla_{g}^{2}g^{\prime}_{t}|_{g}<\frac{C}{t^{\frac{n}{4p}+\frac{3}{4}}}

Proof.

The statements except for the item $(d^{\prime})$ are the results of Lemma 2.1 $(a)$ - $(c),(e)$ respectively. The item $(d^{\prime})$ is the result of [11, Theorem 3.11]. Indeed, since $g^{\prime}$ is $W^{1,p}$ -close to $g,$ the $W^{1,p}$ -bounded assumption in [11, Theorem 3.11]:

\int_{M}|\nabla_{g}g^{\prime}|^{p}\,d\mathrm{vol}_{g}\leq A=A(\varepsilon,g)

is satisfied. Therefore we obtain the assertion from [11, Theorem 3.11]. ∎

Remark 2.1.

Let $(g_{t})_{t\in[0,T)}~{}(0<T)$ be a solution of the Ricci-DeTurck flow equation with $g_{0}=g.$ Choose a background metric $\bar{g}$ on $M$ and define the Bianchi operator

X^{i}_{\bar{g}}(h)=(\bar{g}+h)^{ij}(\bar{g}+h)^{pq}\left(-\nabla^{\bar{g}}_{p}h_{qj}+\frac{1}{2}\nabla^{\bar{g}}_{j}h_{pq}\right),

(1)

which assigns a vector field to every symmetric 2-form $h$ on $M.$ Let $(\Phi_{t})_{t\in I}$ be the flow (we call this flow the Ricci-DeTurck deiffeomorphism below) generated by the time-dependent family of vector fields $X_{\bar{g}}(g_{t}),$ i.e.,

\frac{\partial}{\partial t}\Phi_{t}=X_{\bar{g}}(g_{t})\circ\Phi_{t}~{}~{}\mathrm{with}~{}~{}\Phi_{0}=\mathrm{id}_{M}.

(2)

Then $\tilde{g}_{t}:=\Phi^{*}_{t}g_{t}$ satisfies the Ricci flow equation:

\frac{\partial}{\partial t}\tilde{g}_{t}=-2\,\mathrm{Ric}(\tilde{g}_{t})~{}~{}\mathrm{with}~{}~{}\tilde{g}_{0}=g_{0}=g.

For each $g_{i}$ , from Lemma 2.2, we have a Ricci-DeTurck flow $g_{i}(t)$ defined on $[0,\tau]$ for some positive time $\tau$ , which is independent of $i$ . We also have the corresponding Ricci flow $\tilde{g}_{i}(t)$ and the Ricci-DeTurck diffeomorphism $\Phi_{i,t}$ both defined on the same interval $[0,\tau]$ . Indeed, we assume that there is the largest time $0<t_{0}<\tau$ so that the Ricci-DeTurck diffeomorphism exists on $[0,t_{0})$ . On the other hand, since the Ricci-DeTurck flow $g_{i}(t)$ is defined on the interval $[0,\tau]$ , from the Shi-type estimates for $g_{i}(t)$ , we can show that $\Phi_{i,t}$ converges to a diffeomorphism $\Phi_{i,t_{0}}$ as $t\nearrow t_{0}$ by the similar argument in the following proof of Lemma 2.3. This contradicts the definition of the time $t_{0}$ . Moreover, under the condition of Lemma 2.1 $(e)$ , we can see that for each $t_{0}\in(0,\tau),$ $\{\tilde{g}_{i,t}\}_{t\in[t_{0},\tau]}$ smoothly subconverges to $\{\tilde{g}_{t}\}_{t\in[t_{0},\tau]}.$

Let $M$ be a smooth manifold and $g$ a $C^{2}$ -Riemannian metric on $M.$ Assume a sequence of $C^{2}$ -Riemannian metrics $(g_{i})$ on $M$ converges to $g$ on $M$ in the $W^{1,p}~{}(p>n^{2}/2)$ -sense. Then, from Lemma 2.2, there are a positive time $\tau=\tau(M,g)>0$ and a positive constant $C=C(M,g)<\infty$ such that for sufficiently large $i,$ Lemma 2.2 holds for $g$ and $g_{i}.$ Thus, let $(\Phi_{i,t})_{t\in[0,\tau)}$ be the corresponding Ricci-DeTurck diffeomorphism of $g_{i}$ with background metric $g$ defined as in (2). The next lemma will be used in the proof of Main Theorem 2. In particular, the assumption $p>n^{2}/2$ is needed for this lemma.

Lemma 2.3 (Subconvergence of the Ricci-DeTurck diffeomorphisms).

In the above setting, there is a subsequence of $(\Phi_{i,t})_{t\in[0,\tau/2]}$ that converges to a time-dependent $C^{1}$ -diffeomorphism $(\Phi_{t})_{t\in[0,\tau/2]}$ (i.e., for all $t\in[0,\tau/2],~{}\Phi_{t}$ is a homeomorphism, $\Phi_{t},\Phi_{t}^{-1}$ are continuously differentiable and $\mathrm{d}\Phi_{t}$ and $\mathrm{d}\Phi^{-1}_{t}$ are mutually inverse. And, there is a subsequence $(\Phi_{i_{k},t})_{t\in[0,\tau/2]}$ such that for all $t\in[0,\tau/2],~{}\Phi_{i_{k},t},\Phi_{i_{k},t}^{-1},\mathrm{d}\Phi_{i_{k},t}$ and $\mathrm{d}\Phi_{i_{k},t}^{-1}$ are converges to $\Phi_{t},\Phi_{t}^{-1},\mathrm{d}\Phi_{t}$ and $\mathrm{d}\Phi_{t}^{-1}$ respectively) with $\Phi_{0}=\mathrm{id}_{M},$ where $\mathrm{id}_{M}:M\rightarrow M$ denotes the identity map.

Proof.

Step 1( $C^{0}$ -convergence): From Lemma 2.2 $(d^{\prime})$ and the definition (1), the norm (with respect to $g$ ) of the time-dependent vector field defined in (1) is bounded by some positive constant $C=C(M,g).$ Thus, applying Gronwall’s lemma to (2), by the same argument in the proof of Lemma 2.1 in [5], one can obtain that

d_{g}(\Phi_{i,t}(p),\Phi_{i,s}(p))\leq C|t^{1-\frac{n}{2p}}-s^{1-\frac{n}{2p}}|,~{}~{}~{}~{}t,s\in[0,\tau/2],~{}p\in M.

(3)

Here, $C$ depends only on $M,g$ and $\tau.$ Unlike that in [5, Lemma 2.1], one can get the above type of estimate (in particular, the right-hand side is not $|\sqrt{t}-\sqrt{s}|$ but $|t^{1-\frac{n}{2p}}-s^{1-\frac{n}{2p}}|$ ). This follows from the first-derivative estimate (Lemma 2.2 $(d^{\prime})$ and hence $|X_{g}(g_{i,t})|_{g}$ is bounded by $Ct^{\frac{n}{2p}}$ where $C=C(M,g)$ is a positive constant. On the other hand, taking the derivative of both sides of (2), and using the estimate of the second derivatives (Lemma 2.2 $(d^{\prime})$ ), we obtain that

\frac{\partial}{\partial t}\,|\mathrm{d}\Phi_{i,t}|_{g}\leq\frac{C}{t^{\frac{n}{4p}+\frac{3}{4}}}|\mathrm{d}\Phi_{i,t}|_{g},~{}~{}~{}~{}t\in(0,\tau/2],

where $|\mathrm{d}\Phi_{i,t}|_{g}$ denotes the maximum of the operator norm of $\mathrm{d}\Phi_{i,t}:TM\rightarrow TM$ with respect to $g$ on $M.$ Hence, for all $t\in(0,\tau/2],$ we have

\frac{d}{dt}\left(\frac{u(t)}{v(t)}\right)\leq 0,

where $u(t)=|\mathrm{d}\Phi_{i,t}|_{g}$ and $v(t)=e^{Ct^{1-\frac{n}{4p}-\frac{3}{4}}}$ . Then, since $u(t)$ is continuous on $[0,\tau/2],$ from the mean value theorem and this estimate of time-derivative, we have

|\mathrm{d}\Phi_{i,t}|_{g}\leq C\exp{t^{1-\frac{n}{4p}-\frac{3}{4}}},~{}~{}~{}~{}t\in[0,\tau/2].

(4)

Since the pullback metric $\Phi^{*}_{i,t}\,g_{i,t}$ satisfies the Ricci flow equation:

\frac{\partial}{\partial t}(\Phi^{*}_{i,t}\,g_{i,t})=-2\,\mathrm{Ric}(\Phi^{*}_{i,t}\,g_{i,t}),

by Lemma 2.2 $(d^{\prime})$ and the above estimate, there is a constant $C=C(M,g,\tau)$ such that for all points $(p,t)\in M\times(0,\tau/2]$ and all vectors $v\in T_{p}M,$

\left|\frac{d}{dt}|v|^{2}_{\Phi^{*}_{i,t}g_{i,t}}\right|\leq\frac{C}{t^{\frac{n}{4p}+\frac{3}{4}}}|v|^{2}_{\Phi^{*}_{i,t}g_{i,t}}.

From this estimate, Lemma 2.2 $(b)$ and the mean value theorem, we obtain

e^{-C\tau^{1-\frac{n}{4p}-\frac{3}{4}}}|v|^{2}_{g_{i}}\leq|v|^{2}_{\Phi^{*}_{i,t}g_{i,t}}\leq e^{C\tau^{1-\frac{n}{4p}-\frac{3}{4}}}|v|^{2}_{g_{i}}.

Since $g_{i}$ converges to $g$ in the $C^{0}$ -sense, for all sufficiently large $i,$ we have

e^{-\tilde{C}\tau^{1-\frac{n}{4p}-\frac{3}{4}}}|v|^{2}_{g}\leq|v|^{2}_{\Phi^{*}_{i,t}g_{i,t}}\leq e^{\tilde{C}\tau^{1-\frac{n}{4p}-\frac{3}{4}}}|v|^{2}_{g}

for some constant $\tilde{C}=\tilde{C}(M,g,\tau).$ Therefore, from Lemma 2.2 $(a),$ the $C^{0}$ -closedness of $g_{i}$ and $g,$ and the previous estimate, there is a positive constant $C=C(M,g,\tau)$ such that for all sufficiently large $i,$

d_{g}(\Phi_{i,t}(p),\Phi_{i,t}(q))\leq Cd_{g_{i,t}}(\Phi_{i,t}(p),\Phi_{i,t}(q))\leq C\,d_{g}(p,q).

(5)

Combining the inequalities (3) and (5), we have

d_{g}(\Phi_{i,t}(p),\Phi_{i,s}(q))\leq C\left(|t^{1-\frac{n}{2p}}-s^{1-\frac{n}{2p}}|+d_{g}(p,q)\right),~{}~{}~{}t,s\in[0,\tau/2],~{}p,q\in M.

Then, by Arzela-Ascoli’s theorem, a subsequence $(\Phi_{i_{k},t})_{t\in[0,\tau/2]}$ of $(\Phi_{i,t})_{t\in[0,\tau/2]}$ converges to a time-dependent map $\Phi_{t}:M\rightarrow M~{}(t\in[0,\tau/2])$ as $i_{k}\rightarrow\infty.$ In exactly the same way, one can prove that there is a subsequence $(\Phi^{-1}_{i_{k_{l}},t})_{t\in[0,\tau/2]}$ of $(\Phi^{-1}_{i_{k},t})_{t\in[0,\tau/2]}$ that converges to some time-dependent map $\tilde{\Phi}_{t}:M\rightarrow M~{}(t\in[0,\tau/2])$ as $i_{k_{l}}\rightarrow\infty.$ But, since $\Phi^{-1}_{i_{k_{l}},t}\circ\Phi_{i_{k_{l}},t}=\mathrm{id}_{M},$ $\Phi^{-1}_{t}$ exists and $\Phi^{-1}_{t}=\tilde{\Phi}_{t}$ for each $t\in[0,\tau/2].$ To prevent complicated in expression, we will simply write this converging subsequence $(\Phi_{i_{k_{l}},t})_{t\in[0,\tau/2]}$ as $(\Phi_{i,t})_{t\in[0,\tau/2]}.$

Step 2( $C^{1}$ -convergence): Next, we will show that the first derivatives of $\Phi_{i,t}$ subconverges as $i\rightarrow\infty.$ Let $0<a<1.$ Since $\nabla_{g}g^{\prime}_{t}$ satisfies a parabolic type PDE (see [18, Section 4, Equation (4)]), from the derivative estimate Lemma 2.2 we obtain that

\frac{\partial}{\partial t}\left(\nabla_{g}g^{\prime}_{t}\cdot t^{a}\right)-\Delta_{g}\left(\nabla_{g}g^{\prime}_{t}\cdot t^{a}\right)\leq aCt^{a-1-\frac{n}{2p}}+Ct^{a-\frac{n}{2p}}+Ct^{a-\frac{n}{2p}-\frac{n}{4p}-\frac{3}{4}}+Ct^{a-\frac{3n}{2p}}.

Then, by the parabolic $W^{2,1}_{q}$ -estimate ([12, Ch. 4, Section 3, Theorem 7]), we have

||\nabla_{g}g^{\prime}_{t}\cdot t^{a}||_{W^{2,1}_{q}(M\times(0,\tau/2],g)}\leq C

for all $q$ satisfying $\left(1+\frac{n}{2p}-a\right)q<1$ and $q\geq 2.$ Note that if $\left(1+\frac{n}{2p}-a\right)q<1$ then it holds that

\left(\frac{3n+3p}{4p}-a\right)q,~{}\left(\frac{n}{2p}-a\right)q,~{}\left(\frac{3n}{2p}-a\right)q<1.

(Recall our assumption $p>n^{2}/2\geq n$ .) In particular, we have

||\nabla_{g}g^{\prime}_{t}||_{W^{2}_{q}(M,g)}\leq Ct^{-a}

for all such $a,q$ and $t\in(0,\tau/2].$ Since, $p>n^{2}/2,$ we can choose $0<a<1$ sufficiently close to $1$ so that

n<\frac{1}{1+\frac{n}{2p}-a}.

(6)

Hence, as noted above, the weaker relations:

n<\frac{1}{\frac{3n+3p}{4p}-a},~{}\frac{1}{\frac{n}{2p}-a},~{}\frac{1}{\frac{3n}{2p}-a}

are satisfied as well under the above condition (6). Thus, from Morrey’s embedding theorem,

||\nabla^{2}_{g}g^{\prime}_{t}||_{C^{\alpha}(M,g)}\leq Ct^{-a},~{}~{}~{}~{}t\in(0,\tau/2]

for some $0<\alpha<1.$ Therefore, by the same arguments that derived (4) above, we obtain that

|\mathrm{d}\Phi_{i,t}|_{C^{\alpha}(M,g)}\leq C\exp{t^{1-a}},~{}~{}~{}~{}t\in[0,\tau/2].

(7)

Then, by the inequalities (4) and (7), we can apply the Arzela-Ascoli’s theorem and obtain that there is a subsequence $(\Phi_{i_{k},t})_{t\in[0,\tau/2]}$ such that as $i_{k}\rightarrow\infty,$ $(\Phi_{i_{k},t})_{t\in[0,\tau/2]}\rightarrow(\Phi_{t})_{t\in[0,\tau/2]}$ for some time-dependent map $(\Phi_{t})_{t\in[0,\tau/2]}.$ Moreover, $(\Phi_{t})_{t\in[0,\tau/2]}$ are differentiable on $M$ and $(\mathrm{d}\Phi_{i_{k},t})_{t\in[0,\tau/2]}\rightarrow(\mathrm{d}\Phi_{t})_{t\in[0,\tau/2]}$ as $i_{k}\rightarrow\infty.$ Similarly, there is a subsequence $(\Phi_{i_{k_{l}},t})_{t\in[0,\tau/2]}$ of $(\Phi_{i_{k},t})_{t\in[0,\tau/2]}$ such that

(\Phi^{-1}_{i_{k_{l}},t})_{t\in[0,\tau/2]}\rightarrow(\tilde{\Phi}_{t})_{t\in[0,\tau/2]}~{}~{}\mathrm{and}~{}~{}(\mathrm{d}\Phi^{-1}_{i_{k_{l}},t})_{t\in[0,\tau/2]}\rightarrow(\mathrm{d}\tilde{\Phi}_{t})_{t\in[0,\tau/2]}

as $i_{k_{l}}\rightarrow\infty.$ Since $\Phi_{i_{k_{l}},t}\circ\Phi^{-1}_{i_{k_{l}},t}=\Phi^{-1}_{i_{k_{l}},t}\circ\Phi_{i_{k_{l}},t}=\mathrm{id}_{M}$ and $\mathrm{d}\Phi_{i_{k_{l}},t}\circ\mathrm{d}\Phi^{-1}_{i_{k_{l}},t}=\mathrm{d}\Phi^{-1}_{i_{k_{l}},t}\circ\mathrm{d}\Phi_{i_{k_{l}},t}=\mathrm{id}_{TM},$ $\mathrm{d}\Phi_{t}$ are invertible and $\mathrm{d}\Phi_{t}^{-1}=\mathrm{d}\tilde{\Phi}_{t}$ for all $t\in[0,\tau/2].$ Finally, $\Phi_{0}=\mathrm{id}_{M}$ easily follows from the definition (2) and the above construction of $(\Phi_{t})_{t\in[0,\tau/2]}.$ ∎

Remark 2.2.

In Lemma 2.3, it is not known that the limit $(\Phi_{t})_{t\in[0,\tau/2]}$ is the Ricci-DeTurck diffeomorphism of the Ricci flow starting at $g$ with background metric $g.$ Therefore, let $(g_{t})_{t\in[0.\tau/2]}$ be the Ricci-DeTurck flow starting at $g$ with background metric $g,$ then we don’t know whether or not $(\Phi_{t}^{*}g_{t})_{t\in[0,\tau/2]}$ is the solution of the Ricci flow equation starting at $g.$

Next, we need the following stability of the heat flow with the Ricci flow background which is proved by Lee and Tam [14, Theorem 3.1].

Lemma 2.4 ([14, Theorem 3.1]).

Let $(M^{n},g_{0})$ be a closed manifold of dimension $n\geq 2$ and $\{f_{i}\}$ a family of functions on $M$ satisfying the assumptions $(1)$ and $(2)$ in Main Theorem 2. Suppose $g(t)~{}(t\in[0,\tau])$ be a solution of the Ricci flow equation starting at $g_{0}$ such that $|\mathrm{Rm}(g(t))|\leq a\cdot t^{-1}$ for some $a>0$ on $(0,\tau].$ Then for all $i_{0}\in\mathbb{N},$ there are positive constants $\tau_{0}=\tau_{0}(n,\Lambda,i_{0})>0$ and $C_{0}=C_{0}(n,a,\Lambda,i_{0})>0$ such that the following holds. For all $i\geq i_{0},$ there exists $F_{i}(t)\in C^{\infty}(M)~{}(t\in(0,\min\{\tau,\tau_{0}\})$ satisfies the heat flow equation:

\frac{\partial}{\partial t}F_{i}(t)=\Delta_{g_{t}}F_{i}(t)

such that

(A)

$\left(\Lambda^{-2}-2(n-1)t\right)(F_{i})^{*}g_{Eucl}\leq g(t),$
(B)

$\sup_{x\in M}d_{g_{Eucl}}(F_{i}(x,t),f_{i}(x))\leq C_{0}\sqrt{t}.$

Moreover, for any integer $l\geq 0,$ there is a constant $C=C(n,l,a,\Lambda,i_{0})>0$ such that for all $t\in(0,\min\{\tau,\tau_{0}\}],$

|\nabla^{l}dF_{i}|\leq C\cdot t^{-l/2}.

Here, $g_{Eucl}$ denotes the Euclidean metric on $\mathbb{R}.$

Proof.

Since $M$ is compact, the image of $f$ is compact in the target space $\mathbb{R}.$ From the assumption $(2)$ in Main Theorem 2, there is a compact neighbourhood $N(\subset\mathbb{R})$ of the image of $f$ such that the image of $f_{i}$ is contained in $N$ for all $i\geq i_{0}.$ Then, from the assumption $(1)$ of Main Theorem 2, one can apply the proof of Lemma 3.1 and Theorem 3.1 in [14] to $(M,g(t)),~{}(N,h:=g_{Eucl}|_{N})$ and $f_{i}:M\rightarrow N\subset\mathbb{R}~{}(\mathrm{for~{}all}~{}i\geq i_{0}).$ Hence we obtain the desired assertions from Theoem 3.1 in [14]. ∎

The following lemma is a key to prove our main theorems. Note that we only need $p>n$ for this lemma.

Lemma 2.5 (cf. [2, Lemma 4]).

Let $M^{n}$ be a closed $n$ -manifold ( $n\geq 2$ ) and $g$ a $C^{2}$ -Riemannian metric on $M.$ Suppose $f:M\rightarrow\mathbb{R}$ be a $\Lambda$ -Lipschitz function for some $\Lambda>0.$ Let $g^{\prime}$ be a $C^{2}$ -Riemannian metric which is sufficiently $W^{1,p}~{}(p>n)$ -close to $g$ as in Lemma 2.2. Then for any given positive constant $\delta>0,$ there is a constant $\tau=\tau\left(M,g,|g-g^{\prime}|_{W^{1,p}(M,g)},\delta,\Lambda\right)>0$ such that the following holds: Assume that $\int_{M}R(g^{\prime})\,dm>a,$ where $dm:=e^{-f}d\mathrm{vol}_{g^{\prime}}$ for some $a\in\mathbb{R}$ and $R(g^{\prime})\geq 0$ on $M.$ Then there is a solution $(g^{\prime}_{t})_{t\in[0,\tau]}$ to the Ricci flow equation with initial metric $g^{\prime}$ and a solution $(f_{t})_{t\in(0,\tau]}$ of the heat flow equation with the Ricci flow background such that $\int_{M}R(g^{\prime}_{t})\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}(t)}>a-\delta$ for all $t\in[0,\tau].$

Proof.

From Lemma 2.2 and 2.4, there is a sufficiently small $1>\tau^{\prime}>0$ such that there is a Ricci flow $(g^{\prime}(t))_{t\in[0,\tau^{\prime}]}$ starting at $g^{\prime}$ and there is a heat flow $(f_{t})_{t\in(0,\tau^{\prime}]}$ with $f_{t}\rightarrow f$ as $t\searrow 0$ uniformly. Along these $(g^{\prime}_{t})$ and $(f_{t}),$ for all $t\in(0,\tau^{\prime}],$

\begin{split}&\frac{d}{dt}\left(\int_{M}R(g^{\prime}_{t})\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}\right)\\ &=\int_{M}\left(\Delta_{g(t)}R(g^{\prime}_{t})+2|\mathrm{Ric}(g^{\prime}_{t})|^{2}_{g^{\prime}_{t}}-R(g^{\prime}_{t})^{2}\right)\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}+\int_{M}\left(\frac{\partial}{\partial t}e^{-f_{t}}\right)R(g^{\prime}_{t})\,d\mathrm{vol}_{g^{\prime}_{t}}\\ &=\int_{M}\left(2|\mathrm{Ric}(g^{\prime}_{t})|^{2}_{g^{\prime}_{t}}-R(g^{\prime}_{t})^{2}\right)\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}-2\int_{M}\left(\Delta_{g^{\prime}_{t}}f_{t}-|\nabla f_{t}|^{2}\right)R(g^{\prime}_{t})\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}\\ &\geq\int_{M}\left(\frac{2}{n}R(g^{\prime}_{t})^{2}-R(g^{\prime}_{t})^{2}\right)\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}-2\int_{M}\left(\Delta_{g^{\prime}_{t}}f_{t}-|\nabla f_{t}|^{2}\right)R(g^{\prime}_{t})\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}\\ &=\left(\frac{2}{n}-1\right)\int_{M}R(g^{\prime}_{t})^{2}\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}-2\int_{M}\left(\Delta_{g^{\prime}_{t}}f_{t}-|\nabla f_{t}|^{2}\right)R(g^{\prime}_{t})\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}\\ &\geq-Ct^{-\frac{n}{4p}-\frac{3}{4}}\int_{M}R(g^{\prime}_{t})\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}_{t}}.\end{split}

Here, $C$ is a positive constant depending on $M,\Lambda$ and $|g-g^{\prime}|_{W^{1,p}(M,g)}.$ The first equality follows from the evolution of the scalar curvature and the volume form under the Ricci flow, and $\partial_{t}f_{t}=\Delta_{g_{t}}f_{t}~{}(t\in(0,\tau^{\prime}]).$ The second equality follows from applying the divergence formula to the term $\int_{M}\Delta_{g^{\prime}(t)}R(g^{\prime}(t))\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}(t)}.$ The first inequality follows from the Cauchy-Schwarz inequality $n|\mathrm{Ric}(g^{\prime}_{t})|^{2}\geq R(g^{\prime}_{t})^{2}.$ Finally, we have obtained the last inequality as follows. From the assumption $R(g^{\prime})\geq 0,$ by the maximum principle under the Ricci flow, we have $R(g^{\prime}_{t})\geq 0.$ Since the scalar curvature is invariant under the pullback action by a diffeomorphism, from the $C^{1}$ -closedness assumption, one can apply the same derivative estimate $(d^{\prime})$ in Lemma 2.2 to one $R(g^{\prime}_{t})$ in the integrand of the left-hand side of the inequality. Moreover, applying the derivative estimate for the heat flow as in Lemma 2.4 to the second integrand of the left-hand side of the last inequality, we obtain the desired estimate.

Set $u(t):=\int_{M}R(g^{\prime}_{t})e^{-f_{t}}\,d\mathrm{vol}_{g^{\prime}_{t}}$ and $v(t):=\exp\left(C\left(1-\frac{n}{4p}-\frac{3}{4}\right)^{-1}t^{1-\frac{n}{4p}-\frac{3}{4}}\right).$ Then, by the above calculation, we have

\begin{split}\frac{d}{dt}\left(\frac{u(t)}{v(t)}\right)&=\frac{u^{\prime}(t)v(t)-u(t)v^{\prime}(t)}{v^{2}(t)}\\ &\geq\frac{-Ct^{-\frac{n}{4p}-\frac{3}{4}}u(t)v(t)-u(t)\left(-Ct^{-\frac{n}{4p}-\frac{3}{4}}\right)v(t)}{v^{2}(t)}=0\end{split}

for all $t\in(0,\tau^{\prime}].$ Since $u(t)$ is continuous on $[0,\tau^{\prime}]$ from Lemma 2.1 $(b)$ and Lemma 2.4 $(B),$ by the mean value theorem and the previous derivative estimate, we finally obtain that

\int_{M}R(g^{\prime}(t))\,e^{-f_{t}}d\mathrm{vol}_{g^{\prime}(t)}\geq v(t)\cdot\int_{M}R(g^{\prime})\,e^{-f}d\mathrm{vol}_{g^{\prime}},~{}~{}~{}~{}t\in[0,\tau^{\prime}].

Therefore, if we take sufficiently small $0<\tau<<\tau^{\prime},$ the desired assertion holds for such a $\tau.$ ∎

We prove Main Theorems as follows. First, we prove Main Theorem 2 because we will use almost the same method in the proof of Main theorems 1, 3. The idea of proof here is the same as that of Bamler [2]. We will show the assertion by contradiction. In order to do it, we suppose that the weighted total scalar curvature of the limiting metric is less than $\kappa.$ Then, we can also suppose that it is less than or equal to $\kappa-\delta$ for a positive small constant $\delta.$ On the other hand, from Lemma 2.2, we can flow each metric $g_{i}$ and potential function $f_{i}$ respectively along the Ricci–DeTurck flow and the heat flow with the Ricci flow background, up to a uniform (i.e., independent of $i$ ) positive time $\tau$ . From Lemma 2.5, if we retake $\tau$ sufficiently small (independent of $i$ ), the weighted total scalar curvature is bounded from below by $\kappa-\delta/2$ along the Ricci flow and the heat flow for each $i.$ Thus, combining Lemma 2.2, 2.3 and 2.4, we can deduce that the weighted total scalar curvature of the limiting metric is greater than or equal to $\kappa-\delta/2.$ But this contradicts the supposition that the weighted total scalar curvature of the limiting metric is less than $\kappa.$ We prescribe these arguments more precisely below.

Proof of Main Theorem 2.

We show the assertion by contradiction. Suppose that

\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}<\kappa.

Then there is a positive constant $\delta>0$ such that

\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}\leq\kappa-\delta<\kappa.

On the other hand, since $\int_{M}R(g_{i})\,e^{-f_{i}}\,d\mathrm{vol}_{g_{i}}\geq\kappa$ , from Lemma 2.2, Lemma 2.5 and Lemma 2.4, there are a $\tau=\tau(\delta),$ a Ricci flow $(\tilde{g}_{i,t})_{t\in[0,\tau]}$ and a heat flow $(f_{i,t})_{t\in(0,\tau]}$ such that

\int_{M}R(\tilde{g}_{i,t})\,e^{-f_{i,t}}d\mathrm{vol}_{\tilde{g}_{i,t}}\geq\kappa-\frac{1}{2}\delta

and there also exists a heat flow $(f_{t})_{t\in(0,\tau]}$ with $f_{t}\rightarrow f$ as $t\searrow 0$ uniformly on $M.$ By Lemma 2.3, as $i\rightarrow\infty,$ we have a solution of the Ricci–DeTurck flow equation $(g_{t})_{t\in[0,\tau]}$ (retake a sufficiently small $\tau$ if necessary) starting at $g$ and a time-dependent $C^{1}$ -diffeomorphism $(\Phi_{t})_{t\in[0,\tau]}$ with $\Phi_{0}=\mathrm{id}_{M}$ such that

\int_{M}R(\Phi_{t}^{*}g_{t})\,e^{-f_{t}}d\mathrm{vol}_{\Phi_{t}^{*}g_{t}}=\int_{M}R(g_{t})\,e^{-f_{t}\circ\Phi^{-1}_{t}}d\mathrm{vol}_{g_{t}}\geq\kappa-\frac{1}{2}\delta

for all $t\in(0,\tau].$ On the other hand, by Lemma 2.2 $(b),~{}\Phi_{0}=\mathrm{id}_{M}$ and $f_{t}\rightarrow f~{}(t\searrow 0),$ we have

\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}=\lim_{t\rightarrow 0}\int_{M}R(g_{t})\,e^{-f_{t}\circ\Phi^{-1}_{t}}d\mathrm{vol}_{g_{t}}\geq\kappa-\frac{1}{2}\delta>\kappa-\delta.

This contradicts our supposition

\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}\leq\kappa-\delta

and concludes the proof. ∎

Next, we give a proof of Main Theorem 1. This is simpler than the proof of Main Theorem 2 because the unweighted total scalar curvature $\int_{M}R(g)\,d\mathrm{vol}_{g}$ is diffeomorphism invariant, hence it does not need to use Lemma 2.3.

Proof of Main Theorem 1.

We firstly prove the first half of the theorem. Then we can prove this part in the same way as in the proof of Main Theorem 2 since we can use Lemma 2.5 with $f\equiv 0$ (hence $f_{t}\equiv 0$ ). Note that since the total scalar curvature is diffeomorphism invariant, from Remark 2.1, we have

\int_{M}R(\tilde{g}_{i,t})\,d\mathrm{vol}_{\tilde{g}_{i,t}}=\int_{M}R(g_{i,t})\,d\mathrm{vol}_{g_{i,t}},

where $\tilde{g}_{i,t},g_{i,t}$ are the Ricci-DeTurck flow and the Ricci flow starting at $g_{i}$ respectively. Note also that the condition $p>n$ is sufficient to apply Lemma 2.5. Therefore, under the assumption of Main Theorem 1, arguing in the same way in the proof of Main Theorem 2 (and using the same notations), we obtain

\int_{M}R(g_{t})\,d\mathrm{vol}_{g_{t}}\geq\kappa-\frac{1}{2}\delta

for all $t\in(0,\tau],$ where $g_{t}$ is the Ricci-DeTurck flow starting at $g.$ Then we can deduce a contradiction as $t\rightarrow 0$ as in the same way in the proof of Main Theorem 2.

Next, we prove the theorem for $n=3$ without assuming $R(g_{i})\geq 0.$ This statement follows from the following more general claim.

Claim 2.1 (cf. [4, Lecture Series 2, Proposition 2.3.1]).

Let $(M^{n},g)$ be a closed Riemannian manifold of dimension $n\geq 2.$ Suppose that a sequence of $C^{2}$ -Riemannian metrics $(g_{i})$ on $M$ converges to $g$ in the $W^{1,2}$ -sense. Suppose also that $M$ is parallelizable i.e., the tangent bundle of $M$ is trivial. Then

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\rightarrow\int_{M}R(g)\,d\mathrm{vol}_{g}~{}~{}~{}~{}\mathrm{as}~{}~{}i\rightarrow\infty.

Proof of Claim 2.1.

We will use the similar technique in the proof of [4, Lecture Series 2, Proposition 2.3.1]. Since $M$ is parallelizable, we can take a global section $\{v_{1}(x),\cdots,v_{n}(x)\}_{x\in M}$ of the orthonomal frame bundle of $M.$ Define the 1-form $\omega_{k}(\cdot):=g(v_{k},\cdot)~{}(k=1,2,\cdots,n)$ and the vector field $v^{i}_{k}~{}(i=1,2,\cdots,~{}k=1,2,\cdots,n)$ be the dual of $\omega_{k}$ with respect to $g_{i}.$ Next, we will use the Bochner identity for 1-forms:

\Delta_{g}=\nabla_{g}^{*}\nabla_{g}+\mathrm{Ric}(g)

(\mathrm{resp.}~{}~{}\Delta_{g_{i}}=\nabla_{g_{i}}^{*}\nabla_{g_{i}}+\mathrm{Ric}(g_{i})).

Here $\Delta_{g}=D^{*}_{g}D_{g}~{}(\mathrm{resp.}~{}~{}\Delta_{g_{i}}=D^{*}_{g_{i}}D_{g_{i}})$ for the Dirac operator $D_{g}~{}(\mathrm{resp.}~{}~{}D_{g_{i}})$ related to the deRham complex. Thus, we have for $g_{i}$ and $v^{i}_{k}$ using this Bochner identity and the integration by parts,

\int_{M}\left(||D_{g_{i}}\omega^{i}_{k}||_{g_{i}}^{2}-||\nabla_{g_{i}}\omega^{i}_{k}||^{2}_{g_{i}}\right)d\mathrm{vol}_{g_{i}}=\int_{M}\mathrm{Ric}(g_{i})(v^{i}_{k},v^{i}_{k})\,d\mathrm{vol}_{g_{i}}.

Snice $g_{i}\rightarrow g$ in the $W^{1,2}$ -sense on $M,$ the left-hand side of the above equality converges to the following quantity

\int_{M}\left(||D_{g}\omega_{k}||_{g}^{2}-||\nabla_{g}\omega_{k}||^{2}_{g}\right)d\mathrm{vol}_{g}=\int_{M}\mathrm{Ric}(g)(v_{k},v_{k})\,d\mathrm{vol}_{g}.

Taking the sum from 1 to $n$ for $k,$ we obtain that

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\rightarrow\int_{M}R(g)\,d\mathrm{vol}_{g}~{}~{}~{}~{}\mathrm{as}~{}~{}i\rightarrow\infty.

This completes the proof of the claim. ∎

Since every oriented closed three manifold is parallelizable, using this claim we have

\int_{M^{3}}R(g_{i})\,d\mathrm{vol}_{g_{i}}\rightarrow\int_{M^{3}}R(g)\,d\mathrm{vol}_{g}~{}~{}\mathrm{as}~{}i\rightarrow\infty.

(Of course, if necessary, we discuss similarly after taking its orientable double cover.) In particular, if $\int_{M^{3}}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq\kappa$ for all $i,$ then $\int_{M^{3}}R(g)\,d\mathrm{vol}_{g}\geq\kappa$ . This completes the proof of Main Theorem 1. ∎

Example 2.1.

For example, the following manifolds are known to be parallelizable:

(1)

Every orientable closed three manifold.
(2)

$n$ -dimensional sphere $S^{n}$ where $n=0,1,3$ or $7.$
(4)

Every Lie group.
(3)

The product of parallelizable manifolds.

Question 2.1.

•

In Claim 2.1, is the parallelizability of $M$ necessary?
•

What is the relation between parallelizability of a closed manifold and $W^{1,2}$ -convergence of metric tensors on it?

Next, we give a proof of Main Theorem 3.

Proof of Main Theorem 3.

The proof is similar to the one of Main Theorem 2. However, since each metric is Ricci soliton, the Ricci flow starting from such a metric is homothetic. Hence Lemma 2.1, 2.3 and 2.5 are hold “more explicitly” in this situation. That is, the solution $(\tilde{g}_{i,t})_{t\in[0,\tau)}$ of the Ricci flow equation starting from $g_{i}$ constructed in Lemma 2.1 is

\tilde{g}_{i,t}=(1-2\lambda_{i}t)\Phi^{*}_{i,t}(g_{i})~{}~{}~{}~{}t\in[0,\tau),

where $\Phi_{i,t}$ is the family of diffeomorphisms generated by the time-dependent vector field $X_{i}(t):=(1-2\lambda_{i}t)^{-1}Y_{i}$ with the initial condition $\Phi_{i,0}=\mathrm{id}_{M},$ and $\tau>0$ is the uniform existence time of the flows guaranteed in Lemma 2.1. Thus, we have

\int_{M}R(\tilde{g}_{i,t})\,d\mathrm{vol}_{\tilde{g}_{i,t}}=(1-2\lambda_{i}t)^{n/2-1}\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}.

(8)

for all $t\in[0,\tau).$ We will firstly consider the case of $\kappa\geq 0.$ From (8), for any $\delta>0,$ there is a positive time $\tau=\tau(\delta,C_{+})>0$ such that for any diffeomorphism $\Phi:M\rightarrow M$ and $t\in[0,\tau],$

\int_{M}R(\Phi^{*}\tilde{g}_{i,t})\,d\mathrm{vol}_{\Phi^{*}g_{i,t}}=\int_{M}R(\tilde{g}_{i,t})\,d\mathrm{vol}_{\tilde{g}_{i,t}}\geq\kappa-\delta.

Similarly, when $\kappa<0,$ we see that for any $\delta>0,$ there is a positive time $\tau=\tau(\delta,C_{-})>0$ such that for any diffeomorphism $\Phi:M\rightarrow M$ and $t\in[0,\tau],$

\int_{M}R(\Phi^{*}\tilde{g}_{i,t})\,d\mathrm{vol}_{\Phi^{*}\tilde{g}_{i,t}}=\int_{M}R(\tilde{g}_{i,t})\,d\mathrm{vol}_{\tilde{g}_{i,t}}\geq\kappa-\delta.

In particular, we take the inverse of the Ricci-DeTurck diffeomorphism (see Remark 2.1) of $\tilde{g}_{i,t}$ as $\Phi$ here, then we have

\int_{M}R(g_{i,t})\,d\mathrm{vol}_{g_{i,t}}\geq\kappa-\delta~{}~{}~{}\mathrm{for~{}all}~{}t\in[0,\tau],

where $g_{i,t}$ is the Ricci-DeTurck flow starting at $g_{i}.$ Hence we can use this claim instead of Lemma 2.5 in the proof of Main Theorem 2. Therefore, using Lemma 2.1 instead of Lemma 2.2, we can prove the assertion in the same way as in the proof of Main Theorem 2. ∎

Remark 2.3.

If $g_{i}$ is a shrinking Ricci soliton (i.e., $\lambda_{i}>0$ in the above situation), then the maximal existence time of the corresponding Ricci flow is $(2\lambda_{i})^{-1}.$ But, since we assume $g_{i}$ converges to $g$ in the $C^{0}$ -sense, Lemma 2.1 implicitly prevents $\lambda_{i}$ being $\lambda_{i}\rightarrow\infty$ as $i\rightarrow\infty.$

3 Some corollaries

In the assumption in Main Theorem 1, the nonnegativity of each scalar curvature can be replaced with general lower bound of each scalar curvature provided that a suitable condition about the volumes is additionally assumed.

Corollary 3.1 (for Main Theorem 1).

Let $M^{n}$ be a closed $n$ -manifold and $g$ a $C^{2}$ -Riemannian metric on $M.$ Let $(g_{i})$ be a sequence of $C^{2}$ -Riemannian metrics on $M$ that converges to $g$ on $M$ in the $W^{1,p}$ -sense $(p>n)$ . Assume that there are a constant $\kappa\in\mathbb{R}$ and a continuous function $\sigma\in C^{0}(M)$ such that for all $i,$

•

$\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq\kappa,$
•

$R(g_{i})\geq\sigma$ on $M$ , and
•

$\mathrm{Vol}(M,g_{i})\geq\mathrm{Vol}(M,g),$

Here $\mathrm{Vol}(M,g)$ is the volume of $M$ with respect to $g.$ Then

\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa.

Remark 3.1.

If the limiting metric $g$ here is a hyperbolic metric (i.e., its sectional curvature is constant $-1$ ) and each metric $g_{i}$ has its volume entropy $h(g_{i})\leq n-1~{}(n\geq 3),$ then $\mathrm{Vol}(M,g_{i})\geq\mathrm{Vol}(M,g)$ for all $i.$ This is the result of [3]. Here, the volume entropy of a closed Riemannian manifold $(M,G)$ is given by the limit

h(G)=\lim_{r\rightarrow\infty}\frac{\log\left(\mathrm{Vol}\left(B_{\tilde{G}}(p_{0},r),\tilde{G}\right)\right)}{r},

where $\mathrm{Vol}\left(B_{\tilde{G}}(p_{0},r),\tilde{G}\right)$ is the volume of a ball of radius $r$ in the universal cover $(\tilde{M},\tilde{G}).$ (In particular, Bishop-Gromov’s volume comparison implies that a manifold with Ricci curvature lower bound $\mathrm{Ric}_{G}\geq-(n-1)G$ satisfies $h(G)\leq n-1.$ ) Moreover, as a deep corollary of Perelman’s resolution of the Geometrization conjecture of W. Thurston, a very strong generalization of this result [3] in dimension 3 holds (see [1]): if $(M,g_{0})$ is a closed hyperbolic 3-manifold, for any metric $g$ on $M,$

\mathrm{if}~{}~{}R(g)\geq-6~{}~{}\mathrm{then}~{}~{}\mathrm{Vol}(M,g)\geq\mathrm{Vol}(M,g_{0}).

Proof of Corollary 3.1.

There is a sufficient large natural number $n\in\mathbb{N}$ such that the Riemannian product manifold $(N:=M\times S^{n},g^{N}_{i}:=g_{i}\times g_{\mathrm{std}})$ satisfies $R(g^{N}_{i})\geq 0,$ where $(S^{n},g_{\mathrm{std}})$ is the standard Riemannian $n$ -sphere of constant scalar curvature $n(n-1).$ Indeed, since

R(g^{N}_{i})=R(g_{i})+R(g_{\mathrm{std}})=R(g_{i})+n(n-1)\geq\min_{M}\sigma+n(n-1),

we see that $R(g^{N}_{i})\geq 0$ for sufficiently large $n.$ Therefore we have

\begin{split}\int_{N}R(g^{N}_{i})\,d\mathrm{vol}_{g^{N}_{i}}&=\int_{M\times S^{n}}\left(R(g_{i})+R(g_{\mathrm{std}})\right)\,d\mathrm{vol}_{g_{i}\times g_{\mathrm{std}}}\\ &=\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\\ &~{}~{}~{}~{}+\mathrm{Vol}(M,g_{i})\,\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,n(n-1).\end{split}

On the other hand, from our assumption, $g_{i}^{N}=g_{i}\times g_{\mathrm{std}}$ converges to $g\times g_{\mathrm{std}}$ in the $W^{1,p}$ -sense and

\int_{N}R(g^{N}_{i})\,d\mathrm{vol}_{g^{N}_{i}}\geq\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,\kappa+\mathrm{Vol}(M,g)\,\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,n(n-1).

Therefore, from Main Theorem 1,

\begin{split}\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,&\int_{M}R(g)\,d\mathrm{vol}_{g}+\mathrm{Vol}(M,g)\,\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,n(n-1)\\ &=\int_{N}R(g\times g_{\mathrm{std}})\,d\mathrm{vol}_{g\times g_{\mathrm{std}}}\\ &\geq\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,\kappa+\mathrm{Vol}(M,g)\,\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,n(n-1).\end{split}

Then, subtracting $\mathrm{Vol}(M,g)\,\mathrm{Vol}(S^{n},g_{\mathrm{std}})\,n(n-1)$ from both sides and dividing both sides by $\mathrm{Vol}(S^{n},g_{\mathrm{std}})>0,$ we obtain $\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa.$ ∎

We also present another corollary of Corollary 3.1 here. In order to do this, we need to recall the definition of the Yamabe constant:

Definition 3.1 (Yamabe constant).

The Yamabe constant $Y(M,g)$ of a closed Riemannian manifold $(M,g)$ is defined as

Y(M,g):=\inf\left\{\int_{M}R(\tilde{g})\,d\mathrm{vol}_{\tilde{g}}~{}\middle|~{}\tilde{g}\in[g]~{}\mathrm{and}~{}\mathrm{Vol}(M,\tilde{g})=1\right\},

where $[g]:=\left\{\tilde{g}=u^{\frac{4}{n-2}}g~{}|~{}u\in C^{\infty}(M),~{}u>0~{}\mathrm{on}~{}M\right\}$ is the conformal class of the metric $g.$ By the definition, $Y(M,g)$ depends only on the conformal class $[g]$ of $g.$ A Riemannian metric $\tilde{g}\in[g]$ with $\mathrm{Vol}(M,\tilde{g})=1$ is called Yamabe metric of $[g]$ if

Y(M,g)=\int_{M}R(\tilde{g})\,d\mathrm{vol}_{\tilde{g}}.

Corollary 3.2 (for Corollary 3.1).

Let $M^{n}$ be a closed $n$ -manifold and $g$ a $C^{2}$ -Riemannian metric on $M.$ Let $(g_{i})$ be a sequence of $C^{2}$ -Riemannian metrics on $M$ that converges to $g$ on $M$ in the $W^{1,p}$ -sense $(p>n)$ , and $\mathrm{Vol}(M,g_{i})=1.$ Assume that $g$ is a Yamabe metric of $[g],$ and there are a constant $\kappa\in\mathbb{R}$ and a continuous function $\sigma\in C^{0}(M)$ such that for all $i,$ $Y(M,g_{i})\geq\kappa$ and $R(g_{i})\geq\sigma$ on $M.$ Then $Y(M,g)\geq\kappa.$

Proof.

From the definition of $Y(M,g_{i})$ and $\mathrm{Vol}(M,g_{i})=1,$ we have

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq\kappa.

Since $g_{i}\rightarrow g$ in the $W^{1,p}$ -sense $(p>n)$ and $\mathrm{Vol}(M,g_{i})=1,$ we also have $\mathrm{Vol}(M,g)=1.$ Hence, from Corollary 3.1, we have

\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa.

Therefore, since $g$ is a Yamabe metric of $[g]$ and $\mathrm{Vol}(M,g)=1,$ we obtain

Y(M,g)=\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\kappa.

∎

Remark 3.2.

If $\sigma\geq\kappa,$ this corollary directly follows from Gromov’s theorem (Theorem 1.1 in this paper). Indeed, since $g_{i}\rightarrow g$ in the $W^{1,p}$ -sense for $p>n$ (in particular, in the $C^{0}$ -sense), from Theorem 1.1, we have $R(g)\geq\sigma.$ On the other hand, since $g$ is a Yamabe metric of $[g]$ (hence, its scalar curvature $R(g)$ is constant) and $\mathrm{Vol}(M,g)=1,$ we have

Y(M,g)=\int_{M}R(g)\,d\mathrm{vol}_{g}\geq\sigma\geq\kappa.

However, if $\sigma\leq\kappa,$ this corollary does not follow from Gromov’s theorem.

Up to this point, we have only dealt with case that the underlying manifold is closed. Under a special situation, we can prove a similar statement as Claim 2.1 for a non-compact manifold.

Proposition 3.1 (Conformal deformations on an open manifold).

Let $(M^{n},g)$ be a non-compact Riemannian $n$ -manifold ( $n\geq 3$ ) with $\int_{M}R(g)\,d\mathrm{vol}_{g}<+\infty$ and $u_{i}:M\rightarrow\mathbb{R}$ a sequence of positive $C^{2}$ -functions. Assume that each $u_{i}$ is equal to 1 outside a compact set and

u_{i}\rightarrow 1~{}\mathrm{uniformly}~{}C^{1}\mathrm{-sense~{}on}~{}M.

Set $g_{i}:=u_{i}^{\frac{4}{n-2}}g.$ Then

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\overset{i\rightarrow\infty}{\longrightarrow}\int_{M}R(g)\,d\mathrm{vol}_{g}.

Proof.

From the formula for the scalar curvature and the volume form under this conformal change:

R(g_{i})=-4\frac{n-1}{n-2}u_{i}^{-\frac{n+2}{n-2}}\Delta_{g}u_{i}+u_{i}^{-\frac{4}{n-2}}R(g),~{}~{}~{}~{}\mathrm{vol}_{g_{i}}=u_{i}^{\frac{2n}{n-2}}\mathrm{vol}_{g},

we have

\begin{split}\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}&=-4\frac{n-1}{n-2}\int_{M}u_{i}^{-\frac{n+2}{n-2}}\Delta_{g}u_{i}\left(u_{i}^{\frac{2n}{n-2}}\,d\mathrm{vol}_{g}\right)+\int_{M}u_{i}^{\frac{2n-4}{n-2}}R(g)\,d\mathrm{vol}_{g}\\ &=-4\frac{n-1}{n-2}\int_{M}u_{i}\Delta_{g}u_{i}\,d\mathrm{vol}_{g}+\int_{M}u_{i}^{\frac{2n-4}{n-2}}R(g)\,d\mathrm{vol}_{g}\\ &=4\frac{n-1}{n-2}\int_{M}|\nabla_{g}u_{i}|_{g}^{2}d\mathrm{vol}_{g}+\int_{M}u_{i}^{\frac{2n-4}{n-2}}R(g)\,d\mathrm{vol}_{g}.\end{split}

We have used the divergence formula and the fact that $u_{i}$ is equal to 1 outside a compact set in the third equality. Since $u_{i}\overset{i\rightarrow\infty}{\longrightarrow}1$ $C^{1}$ -uniformly on $M,$ we have

\int_{M}|\nabla_{g}u_{i}|_{g}^{2}d\mathrm{vol}_{g}\rightarrow 0~{}~{}\mathrm{as}~{}i\rightarrow\infty

and

\int_{M}u_{i}^{\frac{2n-4}{n-2}}R(g)\,d\mathrm{vol}_{g}\rightarrow\int_{M}R(g)\,d\mathrm{vol}_{g}~{}~{}\mathrm{as}~{}i\rightarrow\infty

Therefore we obtain the desired assertion from the above equality. ∎

The proof of Proposition 3.1 also provides the following.

Corollary 3.3.

In Main Theorem 1, if $p=\infty,$ and if for each $i,$ $g_{i}=u_{i}\,g$ for some positive $C^{2}$ -functions $u_{i}:M\rightarrow\mathbb{R}_{+},$ then the assumption $R(g_{i})\geq 0~{}(i=1,2,\cdots)$ is not needed.

Lee and LeFloch [13] defined a notion of distributional scalar curvature for smooth manifolds that have a metric tensor which has only certain lower regularity.

Definition 3.2 (Distributional scalar curvature ([13, Definition 2.1], [11, Section 2])).

Let $M$ be a smooth manifold endowed with a smooth background metric $h.$ Given any Riemannian metric $g$ with $L^{\infty}_{loc}(M)\cap W^{1,2}_{loc}(M)$ regularity and locally bounded inverse $g^{-1}\in L^{\infty}_{loc}(M),$ the scalar curvature distribution $R_{g}$ is defined, for every compactly supported smooth test function $u:M\rightarrow\mathbb{R}$ by

\langle R_{g},u\rangle:=\int_{M}\left(-V\cdot\overline{\nabla}\left(u\frac{d\mathrm{vol}_{g}}{d\mathrm{vol}_{h}}\right)+Fu\frac{d\mathrm{vol}_{g}}{d\mathrm{vol}_{h}}\right)\,d\mathrm{vol}_{h},

where $V=(V^{k})\in\Gamma(M)$ is given by $V^{k}:=g^{ij}\Gamma^{k}_{ij}-g^{ik}\Gamma^{j}_{ji},$ $F$ is a function as

F:=R_{h}-\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 $\Gamma^{k}_{ij}:=\frac{1}{2}g^{kl}\left(\overline{\nabla}_{i}g_{jl}+\overline{\nabla}_{j}g_{il}-\overline{\nabla}_{l}g_{ij}\right).$ Here, $\overline{\nabla}$ denotes the Levi-Civita connection of $h.$

Let $\kappa$ be a continuous function on $M.$ We say that $R_{g}\geq\kappa$ in the distributional sense if $\langle R_{g},u\rangle-\int_{M}\kappa u\,d\mathrm{vol}_{g}\geq 0$ for any nonnegative compactly supported test function $u\in C^{\infty}_{+}(M)\cap C^{\infty}_{0}(M).$

Remark 3.3.

If a metric $g$ is $C^{2},$ then the scalar curvature distribution $\langle R_{g},u\rangle$ coincides with $\int_{M}R(g)u\,d\mathrm{vol}_{g}.$

For more details about the distributional scalar curvature and related results, see [11, 13, 20, 21]. From Gromov’s $C^{0}$ -limit theorem (Theorem 1.1), there has already been a definition of scalar curvature lower bounds for $C^{0}$ metrics (see [14, Definition 1.2] for example). Namely, a $C^{0}$ metric $g$ on a smooth manifold $M$ is of $R(g)\geq\kappa$ on $M$ in the Gromov’s sense if and only if there exists a sequence of $C^{2}$ metrics $(g_{i})$ such that $g_{i}$ converge $C^{0}$ -locally to $g$ and satisfy $R(g_{i})\geq\kappa$ on $M.$ Note that Burkhardt-Guim [5] pointed out that her definition (via the Ricci–DeTurck flow) and this Gromov’s definition are actually equivalent on a closed manifold. For example, on tori, there is no metric $g$ which is of $R(g)\geq\kappa>0$ in the Gromov’s sense (or equivalently in the sense of [5, Definition 1.2]) from the resolution of Geroch’s conjecture [7, 16, 17]. In contrast, a metric $g$ which is of $R(g)\geq\kappa>0$ in the sense of the definition 3.2 might exist on a torus. At least on a manifold whose Yamabe invariant is nonpositive, the question of how different these definitions are is related to Schoen’s conjecture (cf. [11], see also the preprint, arXiv:2111.05582v2 by Lee and Tam). As a corollary of Main Theorem 2, we can obtain the following. This is the same as Corollary 1.2 in Section 1.

Corollary 3.4.

(1)

a sequence $(\phi_{i})$ of nonnegative smooth functions on $M$ satisfying: for any positive constant $a>0$ there is a positive constant $\Lambda>0$ such that $\log(\phi_{i}+a)$ is $\Lambda$ -Lipschitz on $M$ for all $i,$
(2)

$(\phi_{i})$ converges to some nonnegative continuous function $\phi$ in the uniformly $C^{0}$ -sense on $M,$
(3)

$R(g_{i})\geq 0$ on $M$ for each $i,$
(4)

$\int_{M}R(g_{i})\phi_{i}\,d\mathrm{vol}_{g_{i}}\geq\kappa\int_{M}\phi_{i}\,d\mathrm{vol}_{g_{i}},$
(5)

$g_{i}$ converges to $g$ in the $W^{1,p}$ -sense.

Then

\int_{M}R(g)\phi\,d\mathrm{vol}_{g}\geq\kappa\int_{M}\phi\,d\mathrm{vol}_{g}.

Proof.

From $(3)$ and $(4),$ for any (small) positive constant $a>0,$

\begin{split}\int_{M}R(g_{i})(\phi_{i}+a)\,d\mathrm{vol}_{g_{i}}&\geq\kappa\int_{M}\phi_{i}\,d\mathrm{vol}_{g_{i}}\\ &=\kappa\int_{M}\phi\,d\mathrm{vol}_{g}+\kappa\left(\int_{M}\phi_{i}\,d\mathrm{vol}_{g_{i}}-\int_{M}\phi\,d\mathrm{vol}_{g}\right).\end{split}

Then applying Main Theorem 2 to $f_{i}=-\log(\phi_{i}+a)$ and $f=-\log(\phi+a),$ we obtain that

\int_{M}R(g)(\phi+a)\,d\mathrm{vol}_{g_{i}}\geq\kappa\int_{M}\phi\,d\mathrm{vol}_{g}+\kappa\delta

for all $0<\delta<<1.$ Here $a,\delta>0$ can be taken to be arbitrarily small, hence we obtain the desired inequality:

\int_{M}R(g)\phi\,d\mathrm{vol}_{g_{i}}\geq\kappa\int_{M}\phi\,d\mathrm{vol}_{g}.

∎

Question 3.1.

Can we replace the condition $(3)$ and $(4)$ with “ $R(g_{i})\geq\kappa$ in the distributional sense for some nonnegative constant $\kappa\geq 0$ ”?

If we make the regularity of convergence much stronger $W^{1,p}~{}(p>n^{2}/2)$ in [11, Theorem 3.2 (1)], then it can be proven that the above question is positively true.

4 Counterexamples

If $M$ is a compact smooth manifold and $C^{2}$ -Riemannian metrics $\{g_{i}\}$ converges to a $C^{2}$ -Riemannian metric $g$ on $M$ in the $C^{2}$ sense as $i\rightarrow\infty$ , then $\max_{M}R(g_{i})\rightarrow\max_{M}R(g).$ So, by Lebesgue’s dominated convergence theorem, we have

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\rightarrow\int_{M}R(g)\,d\mathrm{vol}_{g}~{}~{}~{}~{}\mathrm{as}~{}~{}i\rightarrow\infty.

However, if $M$ is non-compact, it is not known that there is a Lebesgue integrable function $f:M\rightarrow\mathbb{R}$ such that $|R(g_{i})|\leq f$ a.e. on $M$ for all $i.$ Hence in this situation, Lebesgue’s dominated convergence theorem cannot be applied in general. Indeed, the following example implies that Main Theorem 1 (without the assumption that each $g_{i}$ has nonnegative scalar curvature) does not hold in general if $M$ is non-compact.

Example 4.1 ( $C^{\infty}$ locally uniformly convergence and incomplete limiting metric).

The limiting metric of the first example constructed below is incomplete on $\mathbb{R}^{n}~{}(n\geq 3)$ . Consider the smooth positive function $u_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined as

u_{i}=\phi\left(i^{-1+l}e^{-ir^{2}}\right)+1

and $\left(\mathbb{R}^{n},~{}g_{i}:=u_{i}^{\frac{4}{n-2}}\cdot g_{Eucl}\right)~{}(n\geq 3).$ Here $\phi:\mathbb{R}^{n}\rightarrow[0,1]$ is a smooth cut-off function such that $\phi\equiv 1$ on the closed ball $\overline{B_{r_{0}}}:=\{x\in\mathbb{R}^{n}|~{}r(x)\leq r_{0}\}$ and $\phi\equiv 0$ outside of the $\varepsilon/2$ -neighbourhood $\left(\overline{B_{r_{0}}}\right)_{\varepsilon/2}$ of $\overline{B_{r_{0}}}$ where $r_{0}>0$ is an arbitrarily fixed positive constant. Here, $g_{Eucl}$ denotes the Euclidean metric on $\mathbb{R}^{n},~{}r:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ is the Euclidean distance function from the origin $o\in\mathbb{R}^{n}$ , and

l:=\frac{n+2}{4}~{}\mathrm{when}\begin{cases}n=2m+1~{}(m\geq 1),\\ n=2m~{}(m\geq 2).\end{cases}

Then for each $i,$ $(\mathbb{R}^{n},g_{i})$ is a non-compact smooth Riemannian manifold with

\begin{split}R(g_{i})&=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}+R(g_{Eucl})u_{i}\right)\\ &=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}\right).\end{split}

(9)

Moreover, $g_{i}$ converges to $g_{Ecul}$ in the locally $C^{\infty}$ -sense in $\mathbb{R}^{n}\setminus\{o\}$ but not in the $C^{0}$ -sense on $\mathbb{R}^{n}.$ Note that $(\mathbb{R}^{n}\setminus\{o\},g_{Eucl})$ is incomplete. On $\overline{B_{r_{0}}},$

\begin{split}|\nabla u_{i}|^{2}=\sum^{n}_{j=1}\left|\frac{\partial}{\partial x^{j}}i^{-1+l}e^{-ir^{2}}\right|^{2}&=\sum^{n}_{j=1}\left|\frac{\partial r}{\partial x^{j}}\frac{\partial}{\partial r}i^{-1+l}e^{-ir^{2}}\right|^{2}\\ &=\sum^{n}_{j=1}\left|\frac{x^{j}}{r}(-2)i^{l}re^{-ir^{2}}\right|^{2}\\ &=4i^{2l}r^{2}e^{-2ir^{2}}.\end{split}

When $i\rightarrow\infty$ ,

R(g_{i})\rightarrow\begin{cases}0&\mathrm{on}~{}\overline{B_{r_{0}}}\setminus\{o\},\\ \infty&\mathrm{at}~{}o.\end{cases}

Indeed, we can observe such a behavior from the form of the scalar curvature on $\overline{B_{r_{0}}}$ as follows.

R(g_{i})=-\frac{4(n-1)}{n-2}u_{i}^{-\frac{n+2}{n-2}}\left(-2i^{l}n+4i^{l+1}r^{2}\right)e^{-ir^{2}}

(10)

From (9) and the divergence formula, we have

\begin{split}\int_{\mathbb{R}^{n}}R(g_{i})\,d\mathrm{vol}_{g_{i}}&=-4\frac{n-1}{n-2}\int_{\mathbb{R}^{n}}u_{i}^{-\frac{n+2}{n-2}}\Delta_{g_{Eucl}}u_{i}\left(u_{i}^{\frac{2n}{n-2}}\,d\mathrm{vol}_{g_{Eucl}}\right)\\ &=-4\frac{n-1}{n-2}\int_{\mathbb{R}^{n}}u_{i}\Delta_{g_{Eucl}}u_{i}\,d\mathrm{vol}_{g_{Eucl}}\\ &=4\frac{n-1}{n-2}\int_{\left(\overline{B_{r_{0}}}\right)_{\varepsilon}}|\nabla u_{i}|^{2}d\mathrm{vol}_{g_{Eucl}}\\ &\geq 16\frac{n-1}{n-2}\int_{\overline{B_{r_{0}}}}i^{2l}r^{2}e^{-2ir^{2}}d\mathrm{vol}_{g_{Eucl}}\\ &=16\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\int^{r_{0}}_{0}i^{2l}r^{2}e^{-2ir^{2}}r^{n-1}\,dr,\end{split}

(11)

where $\mathrm{Vol}(S^{n-1})$ denotes the volume of $(n-1)$ -sphere with respect to the standard metric. Here,

\begin{split}\int^{r_{0}}_{0}r^{n+1}e^{-2ir^{2}}dr&=\left[-\frac{1}{2i}r^{n}e^{-2ir^{2}}\right]^{r_{0}}_{0}+\frac{n}{2i}\int^{r_{0}}_{0}r^{n-1}e^{-2ir^{2}}dr\\ &=-\frac{1}{2i}r_{0}^{n}e^{-2ir_{0}^{2}}+\frac{n}{2i}\int^{r_{0}}_{0}r^{n-1}e^{-2ir^{2}}dr\end{split}

Set the left hand side of this equation as $I_{n+1}:=\int^{r_{0}}_{0}r^{n+1}e^{-2ir^{2}}dr.$ Then we have

\begin{cases}I_{n+1}=-\frac{e^{-2ir_{0}^{2}}}{2i}\sum^{m}_{k=0}\prod^{k}_{s=0}\left(\frac{n-2s}{2i}\right)+\prod^{m}_{s=0}\left(\frac{2s+1}{2i}\right)\,I_{0}&\mathrm{if}~{}n=2m+1~{}(m\geq 1),\\ I_{n+1}=-\frac{e^{-2ir_{0}^{2}}}{2i}\sum^{m-1}_{k=0}\prod^{k}_{s=0}\left(\frac{n-2s}{2i}\right)+\prod^{m}_{s=1}\left(\frac{2s}{2i}\right)\,I_{1}&\mathrm{if}~{}n=2m~{}(m\geq 2).\end{cases}

Moreover,

\begin{split}I_{0}=\int^{r_{0}}_{0}e^{-2ir^{2}}\,dr&\geq\left(\int^{r_{0}}_{0}\int^{\frac{\pi}{2}}_{0}e^{-2ir^{2}}rdrd\theta\right)^{1/2}\\ &=\sqrt{\frac{\pi}{2}\left(\frac{1}{2i}-\frac{e^{-2ir_{0}^{2}}}{2i}\right)},\end{split}

and

I_{1}=\int^{r_{0}}_{0}re^{-2ir^{2}}\,dr=\left[-\frac{1}{2i}e^{-2ir^{2}}\right]^{r_{0}}_{0}=\frac{1}{2i}-\frac{1}{2i}e^{-2ir^{2}_{0}}.

Combining these, as $i\rightarrow\infty,$ we can see that the rightmost term of (11) converges to

\begin{cases}16\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\frac{\sqrt{\pi}}{2}\prod_{s=0}^{m}\left(\frac{2s+1}{2}\right)\,(>0)&\mathrm{if}~{}n=2m+1~{}(m\geq 1),\\ 8\,\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\prod^{m}_{s=1}s\,(>0)&\mathrm{if}~{}n=2m~{}(m\geq 2).\end{cases}

Hence, for all sufficiently large $i,$

\begin{split}\int_{\mathbb{R}^{n}}&R(g_{i})\,d\mathrm{vol}_{g_{i}}\\ &\geq\begin{cases}8\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\frac{\sqrt{\pi}}{2}\prod_{s=0}^{m}\left(\frac{2s+1}{2}\right)\,(>0)&\mathrm{if}~{}n=2m+1~{}(m\geq 1),\\ 4\,\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\prod^{m}_{s=1}s\,(>0)&\mathrm{if}~{}n=2m~{}(m\geq 2).\end{cases}\end{split}

Note that $R(g_{i})$ cannot be non-negative on $\mathbb{R}^{n}$ by the positive mass theorem or the resolution of Geroch’s conjecture on tori [7, 16, 17]). Indeed, from (10),

R(g_{i})(o)=8\frac{n(n-1)}{n-2}i^{l}\left(i^{-1+l}+1\right)^{-\frac{n+2}{n-2}}~{}~{}>0,

and

R(g_{i})(x)=-4\frac{n-1}{n-2}i^{l}\left(i^{-1+l}e^{-\frac{ir^{2}_{0}}{4}}+1\right)^{-\frac{n+2}{n-2}}\left(-2n+ir^{2}_{0}\right)\,e^{-\frac{ir^{2}_{0}}{4}}~{}~{}<0

for any point $x\in\left\{x\in\mathbb{R}^{n}|~{}r(x)=\frac{r_{0}}{2}\right\}$ and sufficiently large $i.$

Next, we will construct a counterexample (to Main Theorem 1 without the assumption that each $g_{i}$ has nonnegative scalar curvature), in which each $g_{i}$ is complete.

Example 4.2 (Not $C^{1}$ but $C^{0}$ ).

Consider $\left(\mathbb{R}^{n},~{}g_{i}:=u_{i}^{\frac{4}{n-2}}\cdot g_{Eucl}\right)~{}(n\geq 3,~{}i=2,3,\cdots).$ Here the smooth positive function $u_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ has been defined as

u_{i}=\phi\left(i^{-1}\sin(ir^{2})\right)+1.

Here, $\phi:\mathbb{R}^{n}\rightarrow[0,1]$ is a smooth cut-off function such that $\phi\equiv 1$ on $\overline{B_{r_{0}}}:=\{x\in\mathbb{R}^{n}|~{}r(x)\leq r_{0}\}$ and $\phi\equiv 0$ outside of the $\varepsilon/2$ -neighbourhood $\overline{B_{r_{0}}}.$ where $r_{0}>0$ is an arbitrarily fixed positive constant. Here, $r:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ is the Euclidean distance function from the origin $o\in\mathbb{R}^{n}.$ Then, for each $i,$ $(\mathbb{R}^{n},g_{i})$ is a non-compact smooth Riemannian manifold with

\begin{split}R(g_{i})&=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}+R(g_{Eucl})u_{i}\right)\\ &=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}\right),\end{split}

(12)

and $g_{i}\rightarrow g_{Ecul}$ on $\mathbb{R}^{n}$ in the uniformly $C^{0}$ but not in the $C^{1}$ -sense. On $\overline{B_{r_{0}}},$

\begin{split}|\nabla u_{i}|^{2}=\sum^{n}_{j=1}\left|\frac{\partial}{\partial x^{j}}i^{-1}\sin(ir^{2})\right|^{2}&=\sum^{n}_{j=1}\left|\frac{\partial r}{\partial x^{j}}\frac{\partial}{\partial r}i^{-1}\sin(ir^{2})\right|^{2}\\ &=\sum^{n}_{j=1}\left|\frac{x^{j}}{r}(2r)\cos(ir^{2})\right|^{2}\\ &=4r^{2}\cos^{2}(ir^{2})\\ &=2r^{2}(1+\cos(2ir^{2})).\end{split}

Note that $R(g_{i})$ is not non-negative on $\mathbb{R}^{n}.$ Indeed, for sufficiently large $i$ and $k\in\mathbb{Z}$ such that $\overline{B_{r_{0}}}\cap\left\{x\in\mathbb{R}^{n}|~{}r(x)=\sqrt{\frac{(2k-1)}{2i}}\right\}\neq\emptyset,$ we can take a point $x_{i}\in\overline{B_{r_{0}}}\cap\left\{x\in\mathbb{R}^{n}|~{}r(x)=\sqrt{\frac{(2k-1)}{2i}}\right\}.$ Then

R(g_{i})(x_{i})\rightarrow\begin{cases}\frac{8(n-1)(2k-1)\pi}{n-2}&\mathrm{if}~{}k~{}\mathrm{is~{}odd},\\ -\frac{8(n-1)(2k-1)\pi}{n-2}&\mathrm{if}~{}k~{}\mathrm{is~{}even}.\end{cases}

This is checked as follows. For sufficiently large $i,$ such a point $x_{i}$ is contained in $\overline{B_{r_{0}}}.$ Hence, from the above formula and the choice of the point $x_{i},$ we have

\begin{split}R(g_{i})(x_{i})&=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}\right)\\ &=-4\frac{n-1}{n-2}\left(i^{-1}+1\right)^{-\frac{n+2}{n-2}}\left(2n\cos(ir(x_{i})^{2})-4ir^{2}\sin(ir(x_{i})^{2})\right)\\ &=(-1)^{k+1}\frac{8(n-1)(2k-1)\pi}{n-2}\left(i^{-1}+1\right)^{-\frac{n+2}{n-2}}.\end{split}

Since $\left(i^{-1}+1\right)^{-\frac{n+2}{n-2}}\rightarrow 1~{}(i\rightarrow\infty),$ we can observe the desired behavior of the scalar curvature as above. Moreover, from (12) and the divergence formula, we have

\begin{split}\int_{\mathbb{R}^{n}}R(g_{i})\,d\mathrm{vol}_{g_{i}}&=-4\frac{n-1}{n-2}\int_{\mathbb{R}^{n}}u_{i}^{-\frac{n+2}{n-2}}\Delta_{g_{Eucl}}u_{i}\left(u_{i}^{\frac{2n}{n-2}}\,d\mathrm{vol}_{g_{Eucl}}\right)\\ &=-4\frac{n-1}{n-2}\int_{\mathbb{R}^{n}}u_{i}\Delta_{g_{Eucl}}u_{i}\,d\mathrm{vol}_{g_{Eucl}}\\ &=4\frac{n-1}{n-2}\int_{\left(\overline{B_{r_{0}}}\right)_{\varepsilon}}|\nabla u_{i}|^{2}d\mathrm{vol}_{g_{Eucl}}\\ &\geq 8\frac{n-1}{n-2}\int_{\overline{B_{r_{0}}}}r^{2}(1+\cos(2ir^{2}))d\mathrm{vol}_{g_{Eucl}}\\ &=8\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\int_{0}^{r_{0}}r^{2}(1+\cos(2ir^{2}))r^{n-1}\,dr.\end{split}

Here,

\begin{split}\int_{0}^{r_{0}}&r^{2}(1+\cos(2ir^{2}))r^{n-1}dr=\left[\frac{1}{n+2}r^{n+2}\right]_{0}^{r_{0}}+\int_{0}^{r_{0}}r^{n+1}\cos(2ir^{2})\,dr\\ &=\frac{1}{n+2}r^{n+2}_{0}+\left[\frac{r^{n}i^{-1}}{4}\sin(2ir^{2})\right]^{r_{0}}_{0}+\frac{ni^{-2}}{16}\left[r^{n-2}\cos(2ir^{2})\right]^{r_{0}}_{0}\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{n(n-2)i^{-2}}{16}\int^{r_{0}}_{0}r^{n-3}\cos(2ir^{2})\,dr\\ &\geq\frac{1}{n+2}r^{n+2}_{0}+\left[\frac{r^{n}i^{-1}}{4}\sin(2ir^{2})\right]^{r_{0}}_{0}+\frac{ni^{-2}}{16}\left[r^{n-2}\cos(2ir^{2})\right]^{r_{0}}_{0}\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{n(n-2)i^{-1}}{16}\int^{r_{0}}_{0}r^{n-3}\,dr\\ &=\frac{1}{n+2}r^{n+2}_{0}+\frac{r_{0}^{n}i^{-1}}{4}\sin(2ir_{0}^{2})+\frac{ni^{-2}}{16}\,r_{0}^{n-2}\cos(2ir_{0}^{2})-\frac{ni^{-2}}{16}r_{0}^{n-2}.\end{split}

Since

\frac{r_{0}^{n}i^{-1}}{4}\sin(2ir_{0}^{2})+\frac{ni^{-2}}{16}\,r_{0}^{n-2}\cos(2ir_{0}^{2})-\frac{ni^{-2}}{16}r_{0}^{n-2}\rightarrow 0~{}~{}\mathrm{as}~{}i\rightarrow\infty,

there is a sufficiently large $i_{0}=i_{0}(n,r_{0})$ such that for all $i\geq i_{0},$

\frac{r_{0}^{n}i^{-1}}{4}\sin(2ir_{0}^{2})+\frac{ni^{-2}}{16}\,r_{0}^{n-2}\cos(2ir_{0}^{2})-\frac{ni^{-2}}{16}r_{0}^{n-2}>-\frac{1}{2(n+2)}r^{n+2}_{0}.

Hence, for all $i\geq i_{0},$

\int_{\mathbb{R}^{n}}R(g_{i})\,d\mathrm{vol}_{g_{i}}>4\frac{n-1}{(n+2)(n-2)}\mathrm{Vol}(S^{n-1})\,r_{0}^{n+2}>0.

From the Morrey embedding, we have

C^{1}\hookrightarrow W^{1,p}\hookrightarrow C^{0,1-\frac{n}{p}}\hookrightarrow C^{0}~{}~{}~{}\mathrm{if}~{}p>n.

Therefore the same statement of Main Theorem 1 still holds even though one replace $W^{1,p}~{}(p>n)$ with $C^{0,\alpha}$ for all $\alpha\in(0,1].$ On the other hand, in Main Theorem 1, if we weaken the assumption from $W^{1,p}$ to $C^{0},$ then the same statement (without the assumption $R(g_{i})\geq 0$ ) does not hold in general. Indeed, using the same local construction as in the previous example in dimension $\geq 3,$ we can also construct a counterexample on a closed manifold to Main Theorem 1 (without the assumption that each $g_{i}$ has nonnegative scalar curvature) as follows. Note that all metrics $g_{i}$ in each such example has sign-changing scalar curvature, i.e., for each $i,$ there are some points $x_{i},y_{i}\in M$ s.t. $R(g_{i})(x_{i})<0<R(g_{i})(y_{i}).$

Example 4.3 (On every closed manifold).

Consider $\left(M^{n},~{}g_{i}:=u_{i}^{\frac{4}{n-2}}\cdot g_{0}\right)$ $(n\geq 3,~{}i=2,3,\cdots),$ where $M^{n}$ is a closed $n$ -manifold and $g_{0}$ is a Riemannian metric on $M.$ Here the smooth positive function $u_{i}:M\rightarrow\mathbb{R}$ has been defined as

u_{i}=\phi\left(i^{-1}\sin(ih^{2})\right)+1.

Here, $\phi:M\rightarrow[0,1]$ is a smooth cut-off function such that $\phi\equiv 1$ on $\overline{B_{r_{0}}}(p):=\{x\in M|~{}d_{g_{0}}(p,x)\leq r_{0}\}$ and $\phi\equiv 0$ outside of the $\varepsilon/2$ -neighbourhood $\overline{B_{r_{0}}}$ for some point $p\in M$ where $0<r_{0}<\mathrm{inj}(M,g_{0})$ is a sufficiently small positive constant. Here, $h:=d_{g_{0}}(\cdot,p):M\rightarrow\mathbb{R}_{\geq 0}$ is the distance function of $g_{0}$ from the point $p$ and $\mathrm{inj}(M,g_{0})$ is the injectivity radius of $(M,g_{0}).$ Then, for each $i,$ $(M,g_{i})$ is a smooth Riemannian manifold with

R(g_{i})=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{0}}u_{i}+R(g_{0})u_{i}\right)

and $g_{i}$ converges to $g_{0}$ on $M$ in the $C^{0}$ but not in the $C^{1}$ -sense. In the same calculation as in the previous example, we have

\begin{split}\int_{M}&R(g_{i})\,d\mathrm{vol}_{g_{i}}\\ &=-4\frac{n-1}{n-2}\int_{M}u_{i}^{-\frac{n+2}{n-2}}\left(\Delta_{g_{0}}u_{i}+R_{g_{0}}u_{i}\right)\,u_{i}^{\frac{2n}{n-2}}\,d\mathrm{vol}_{g_{0}}\\ &=-4\frac{n-1}{n-2}\int_{M}\left(u_{i}\Delta_{g_{0}}u_{i}+R_{g_{0}}u_{i}^{2}\right)\,d\mathrm{vol}_{g_{0}}\\ &=4\frac{n-1}{n-2}\int_{\left(\overline{B_{r_{0}}}\right)_{\varepsilon}}|\nabla u_{i}|_{g_{0}}^{2}d\mathrm{vol}_{g_{0}}+\int_{M}R(g_{0})u_{i}^{2}\,d\mathrm{vol}_{g_{0}}\\ &\geq 8\frac{n-1}{n-2}\int_{\overline{B_{r_{0}}}}h^{2}(1+\cos(2ih^{2}))d\mathrm{vol}_{g_{0}}+\int_{M}R(g_{0})u_{i}^{2}\,d\mathrm{vol}_{g_{0}}\\ &\geq 8\frac{n-1}{n-2}\widetilde{C}\int_{0}^{r_{0}}h^{2}(1+\cos(2ih^{2}))h^{n-1}\,dh+\int_{M}R(g_{0})u_{i}^{2}\,d\mathrm{vol}_{g_{0}}.\end{split}

Here, the constant $\widetilde{C}$ depends only on $n$ and $g_{0}.$ Thus, from the observation as in the previous example, there is $i_{0}\in\mathbb{N}$ and a positive constant $C=C(n,g_{0},r_{0})>0$ such that for all $i\geq i_{0},$

8\frac{n-1}{n-2}\widetilde{C}\int_{0}^{r_{0}}h^{2}(1+\cos(2ih^{2}))h^{n-1}\,dh\geq C>0.

Therefore, for all $i\geq i_{0},$

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq C+\int_{M}R(g_{0})u_{i}^{2}\,d\mathrm{vol}_{g_{0}}.

Moreover, by the definition of $u_{i},$

\left|\int_{M}R(g_{0})u^{2}_{i}\,d\mathrm{vol}_{g_{0}}-\int_{M}R(g_{0})\,d\mathrm{vol}_{g_{0}}\right|\leq\int_{M}\left|R(g_{0})\right|\left(2i^{-1}+i^{-2}\right)\,d\mathrm{vol}_{g_{0}}.

Hence there is a sufficiently large $i_{1}$ such that for all $i\geq i_{1},$

\left|\int_{M}R(g_{0})u^{2}_{i}\,d\mathrm{vol}_{g_{0}}-\int_{M}R(g_{0})\,d\mathrm{vol}_{g_{0}}\right|\leq\frac{C}{2}.

Thus, for all $i\geq\max\{i_{0},i_{1}\},$ we have

\int_{M}R(g_{i})\,d\mathrm{vol}_{g_{i}}\geq C+\int_{M}R(g_{0})u_{i}^{2}\,d\mathrm{vol}_{g_{0}}>\int_{M}R(g_{0})\,d\mathrm{vol}_{g_{0}}.

Here, we have a question about the regularity of convergence in the assumption of Main Theorem 2.

Question 4.1.

Are there any $C^{2}$ -metrics $(g_{i})$ and $\Lambda$ -Lipschitz ( $\Lambda>0$ ) functions $(f_{i})$ on a closed $n$ -manifold $M^{n}~{}(n\geq 3)$ satisfying the followings ?

•

$g_{i}$ converges to a $C^{2}$ -metric $g$ in the $W^{1,\frac{n^{2}}{2}}$ -sense,
•

$f_{i}$ converges to a $\Lambda$ -Lipschitz function $f$ in the uniformly $C^{0}$ -sense,
•

there is a constant $\kappa$ such that $\int_{M}R(g_{i})\,e^{-f_{i}}d\mathrm{vol}_{{g}_{i}}\geq\kappa>\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}.$

Or, additionally,

•

there is a point $p_{i}\in M$ for each $i$ such that $R(g_{i})(p_{i})\rightarrow-\infty$ as $i\rightarrow\infty.$

In the following Example 4.4, we give another counterexample which is similar to the one in Example 4.2 ( $n\geq 3$ ). However, in the following example of dimension $\geq 3,$ the support of $u_{i}-1$ ( $\subset(\mathbb{R}^{n},d_{g_{Eucl}})$ ) with the origin $o\in\mathbb{R}^{n}$ converges to $(\mathbb{R}^{n},g_{Eucl},o)$ in the pointed Gromov-Hausdorff sense as $i\rightarrow\infty.$ (In Example 4.2, the support of $u_{i}-1$ has been contained in a fixed compact subset.) Hence, unfortunately it is not possible to localize this construction directly and construct such a counterexample on a closed manifold as in Example 4.3. On the other hand, in the two-dimensional example of Example 4.4, the support of $e^{u_{i}}-1$ is not compact for each $i$ (see the last half of Example 4.4).

Example 4.4 (Not $C^{2}$ but $C^{1}$ ).

u_{i}=\phi_{i}\left(i^{-2}\sin(ir^{2})\right)+1.

Here, $\phi_{i}:\mathbb{R}^{n}\rightarrow[0,1]$ is a smooth cut-off function such that $\phi_{i}\equiv 1$ on $\overline{B_{r_{i}}}:=\{x\in\mathbb{R}^{n}|~{}r(x)\leq r_{i}\}$ and $\phi_{i}\equiv 0$ outside of the $\varepsilon/2$ -neighbourhood $\overline{B_{r_{i}}}.$ where $r_{i}:=i^{\frac{2}{n+2}}.$ Note that $r_{i}\rightarrow\infty$ as $i\rightarrow\infty.$ Here, $r:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ is the Euclidean distance function from the origin $o.$ Then, for each $i,$ $(\mathbb{R}^{n},g_{i})$ is a non-compact smooth Riemannian manifold with

\begin{split}R(g_{i})&=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}+R(g_{Eucl})u_{i}\right)\\ &=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}\right).\end{split}

Moreover, $g_{i}$ converges to $g_{Ecul}$ on $\mathbb{R}^{n}$ in the uniformly $C^{1}$ but not in the $C^{2}$ -sense. On $\overline{B_{r_{i}}},$

\begin{split}|\nabla u_{i}|^{2}=\sum^{n}_{j=1}\left|\frac{\partial}{\partial x^{j}}i^{-2}\sin(ir^{2})\right|^{2}&=\sum^{n}_{j=1}\left|\frac{\partial r}{\partial x^{j}}\frac{\partial}{\partial r}i^{-2}\sin(ir^{2})\right|^{2}\\ &=\sum^{n}_{j=1}\left|\frac{x^{j}}{r}(2i^{-1}r)\cos(ir^{2})\right|^{2}\\ &=4i^{-2}r^{2}\cos^{2}(ir^{2})\\ &=2i^{-2}r^{2}(1+\cos(2ir^{2})).\end{split}

Note that $R(g_{i})$ is not non-negative on $\mathbb{R}^{n}.$ Indeed, when $i\rightarrow\infty,~{}R(x)$ oscillates for each $x\in\{x\in\mathbb{R}^{n}|~{}r(x)\neq 0\}.$ Indeed, for sufficiently large $i,$ such a point $x$ is contained in $\overline{B_{r_{i}}}.$ Hence, from the above formula,

\begin{split}R(g_{i})(x)&=u_{i}^{-\frac{n+2}{n-2}}\left(-4\frac{n-1}{n-2}\Delta_{g_{Eucl}}u_{i}\right)\\ &=-4\left(\frac{n-1}{n-2}\right)\frac{2ni^{-1}\cos(ir(x)^{2})-4r(x)^{2}\sin(ir(x)^{2})}{\left(i^{-2}\sin(ir(x)^{2})+1\right)^{\frac{n+2}{n-2}}}.\end{split}

Since $\left(i^{-1}+1\right)^{-\frac{n+2}{n-2}}\rightarrow 1$ and $2ni^{-1}\cos(ir(x)^{2})\rightarrow 0$ as $i\rightarrow\infty,$ we can easily observe the desired behavior of the scalar curvature. Moreover, by the divergence formula,

\begin{split}\int_{\mathbb{R}^{n}}R(g_{i})\,d\mathrm{vol}_{g_{i}}&=-4\frac{n-1}{n-2}\int_{\mathbb{R}^{n}}u_{i}^{-\frac{n+2}{n-2}}\Delta_{g_{Eucl}}u_{i}\left(u_{i}^{\frac{2n}{n-2}}\,d\mathrm{vol}_{g_{Eucl}}\right)\\ &=-4\frac{n-1}{n-2}\int_{\mathbb{R}^{n}}u_{i}\Delta_{g_{Eucl}}u_{i}\,d\mathrm{vol}_{g_{Eucl}}\\ &=4\frac{n-1}{n-2}\int_{\left(\overline{B_{r_{i}}}\right)_{\varepsilon}}|\nabla u_{i}|^{2}d\mathrm{vol}_{g_{Eucl}}\\ &\geq 8\frac{n-1}{n-2}\int_{\overline{B_{r_{i}}}}r^{2}i^{-2}(1+\cos(2ir^{2}))d\mathrm{vol}_{g_{Eucl}}\\ &=8\frac{n-1}{n-2}\mathrm{Vol}(S^{n-1})\int_{0}^{r_{i}}r^{2}i^{-2}(1+\cos(2ir^{2}))r^{n-1}\,dr.\end{split}

Here,

\begin{split}\int_{0}^{r_{i}}&i^{-2}r^{2}(1+\cos(2ir^{2}))r^{n-1}dr=\left[\frac{i^{-2}}{n+2}r^{n+2}\right]_{0}^{r_{i}}+\int_{0}^{r_{i}}i^{-2}r^{n+1}\cos(2ir^{2})\,dr\\ &=\frac{1}{n+2}+\left[\frac{r^{n}i^{-3}}{4}\sin(2ir^{2})\right]^{r_{i}}_{0}+\frac{ni^{-4}}{16}\left[r^{n-2}\cos(2ir^{2})\right]^{r_{i}}_{0}\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{n(n-2)i^{-4}}{16}\int^{r_{i}}_{0}r^{n-3}\cos(2ir^{2})\,dr\\ &\geq\frac{1}{n+2}+\left[\frac{r^{n}i^{-3}}{4}\sin(2ir^{2})\right]^{r_{i}}_{0}+\frac{ni^{-4}}{16}\left[r^{n-2}\cos(2ir^{2})\right]^{r_{i}}_{0}\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{n(n-2)i^{-4}}{16}\int^{r_{i}}_{0}r^{n-3}\,dr\\ &=\frac{1}{n+2}+\frac{r_{i}^{n}i^{-3}}{4}\sin(2ir_{i}^{2})+\frac{ni^{-4}}{16}\,r_{i}^{n-2}\cos(2ir_{i}^{2})-\frac{ni^{-4}}{16}r_{i}^{n-2}\\ &\geq\frac{1}{n+2}+\frac{i^{\frac{2n}{n+2}}i^{-3}}{4}\sin(2i^{1+\frac{4}{n+2}})+\frac{ni^{-4}}{16}\,i^{\frac{2(n-2)}{n+2}}\cos(2i^{1+\frac{4}{n+2}})-\frac{ni^{-4}}{16}i^{\frac{2(n-2)}{n+2}}.\end{split}

Since

\frac{i^{\frac{2n}{n+2}}i^{-3}}{4}\sin(2i^{1+\frac{4}{n+2}})+\frac{ni^{-4}}{16}\,i^{\frac{2(n-2)}{n+2}}\cos(2i^{1+\frac{4}{n+2}})-\frac{ni^{-4}}{16}i^{\frac{2(n-2)}{n+2}}\rightarrow 0~{}~{}\mathrm{as}~{}i\rightarrow\infty,

there is a sufficiently large $i_{0}=i_{0}(n)$ such that for all $i\geq i_{0},$

\frac{i^{\frac{2n}{n+2}}i^{-3}}{4}\sin(2i^{1+\frac{4}{n+2}})+\frac{ni^{-4}}{16}\,i^{\frac{2(n-2)}{n+2}}\cos(2i^{1+\frac{4}{n+2}})-\frac{ni^{-4}}{16}i^{\frac{2(n-)}{n+2}}>-\frac{1}{2(n+2)}.

Hence for all $i\geq i_{0},$

\int_{\mathbb{R}^{n}}R(g_{i})\,d\mathrm{vol}_{g_{i}}>4\frac{n-1}{(n+2)(n-2)}\mathrm{Vol}(S^{n-1})>0.

Next, we will construct a two-dimensional example. Consider the smooth function $u_{i}$ on $\mathbb{R}^{2}$ defined by

u_{i}:=e^{-ir^{2}}\sin\left(-\frac{i}{2}r^{2}\right)~{}~{}(i=1,2,\cdots),

where $r(\cdot):=|o-\cdot|$ denotes the Euclidean distance function from the origin $o\in\mathbb{R}^{2}.$ Then $u_{i}$ uniformly converges to the constant function $0$ in the $C^{1}$ topology on $\mathbb{R}^{2}$ , but $u_{i}$ does not converge to $0$ in the $C^{2}$ topology on $\mathbb{R}^{2}$ . Hence the sequence of complete metrics $(g_{i}:=e^{u_{i}}g_{Eucl})$ on $\mathbb{R}^{2}$ uniformly converges to $g_{Eucl}$ in the $C^{1}$ sense on $\mathbb{R}^{2},$ but $g_{i}$ does not converge to $g_{Eucl}$ in the $C^{2}$ sense on $\mathbb{R}^{2}.$ Set $a:=-i,b:=\frac{1}{2}a=-\frac{i}{2}.$ Then we can check that

\begin{split}\Delta u_{i}&=(4a+4a^{2}r^{2})e^{ar^{2}}\sin(br^{2})+(4b\cos r^{2}-4b^{2}r^{2}\sin br^{2})e^{ar^{2}}\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+8abr^{2}e^{ar^{2}}\cos br^{2},\end{split}

and

\int_{\mathbb{R}^{2}}R(g_{i})\,d\mathrm{vol}_{g_{i}}=-\int_{\mathbb{R}^{2}}\Delta_{g_{Eucl}}u_{i}\,d\mathrm{vol}_{g_{Ecul}}=-\int^{2\pi}_{0}\int_{0}^{\infty}r\Delta_{g_{Eucl}}u_{i}\,dr\,d\theta.

Moreover,

•

$I:=\int^{\infty}_{0}re^{ar^{2}}\cos br^{2}\,dr=-\frac{a}{2(a^{2}+b^{2})},$
•

$J:=\int^{\infty}_{0}re^{ar^{2}}\sin br^{2}\,dr=-\frac{b}{2(a^{2}+b^{2})},$
•

$K:=\int^{\infty}_{0}r^{3}e^{ar^{2}}\cos br^{2}\,dr=-\frac{aI+bJ}{a^{2}+b^{2}},$
•

$L:=\int^{\infty}_{0}r^{3}e^{ar^{2}}\sin br^{2}\,dr=-\frac{aJ-bI}{a^{2}+b^{2}}.$

Combining these, we obtain that

\begin{split}\int_{\mathbb{R}^{2}}R(g_{i})\,d\mathrm{vol}_{g_{i}}&=-2\pi\left(4bI+4aJ-4(a^{2}-b^{2})\frac{aJ-bI}{a^{2}+b^{2}}-8ab\frac{bJ-aI}{a^{2}+b^{2}}\right)\\ &=2\pi\frac{8a^{3}b}{(a^{2}+b^{2})^{2}}\\ &=\frac{128\pi}{25}>0=\int_{\mathbb{R}^{2}}R(g_{Eucl})\,d\mathrm{vol}_{g_{Ecul}}.\end{split}

Note that we have used $b=\frac{1}{2}a$ in the third equality.

Question 4.2.

As we have seen in the above examples, in Main Theorem 1 (without the assumption that the scalar curvature of each $g_{i}$ is nonnegative), we cannot weaken the assumptions that the manifold is closed and the convergence is in the sense of $W^{1,p}~{}(p>n)$ to that the manifold is open and the convergence is in the sense of $C^{0},$ respectively. Then, can we weaken the assumptions in Main Theorems in any sense?

Remark 4.1.

In the above examples, we have constructed these counterexamples by deforming the Euclidean metric locally in a conformal direction. Then, due to the factors from changes of the volume forms, the total scalar curvatures are uniformly bounded from below by a positive constant. On the other hand, if we try to investigate the similar examples for the weighted total scalar curvature $\int_{M}R(g)\,e^{-f}d\mathrm{vol}_{g}$ (i.e., counterexamples to Main Theorem 2), we cannot use the same method in the above examples since there is no contribution from the factors associated with the conformal changes of the volume forms in this situation.

Acknowledgements The author thanks Prof. Boris Botvinnik for suggesting the problem related to Main Theorem 2. The author also thanks Prof. Kazuo Akutagawa for giving him the opportunity to visit the University of Oregon from September 25 to October 10, 2022.

Author Contributions SH has written the manuscript.

Funding The author had been supported by the Foundation of Research Fellows of Mathematical Society of Japan (from April 2022 to March 2023), and by Mitsubishi Electric Corporation Advanced Technology R&D Center (from May 2022 to March 2024). The author was supported by JSPS KAKENHI Grant Number 24KJ0153.

Data availability Not applicable.

Declarations

Conflict of interest The author declares that there is no conflict of interest.

Ethics approval and consent to participate Not applicable.

Consent for publication The author declares the consent for publication.

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Limit theorems for the total scalar curvature

Abstract

keywords:

pacs:

1 Introduction

Theorem 1.1 ([8, p.1118] and [2]).

Example 1.1 ([4, Lecture Series 2, Counterexample 2.3.2]. cf. Example 4.2 and 4.3 in this paper).

Fact 1.1 (Conformal change of the scalar curvature).

Theorem 1.2 ([4, Lecture Series 1, Theorem 4.1]).

Main Theorem 1.

Remark 1.1.

Corollary 1.1.

Main Theorem 2.

Corollary 1.2 (== Corollary 3.4).

Remark 1.2.

Definition 1.1.

Main Theorem 3.

2 Proof of Main Theorems

Lemma 2.1 ([11, 19]).

Proof.

Lemma 2.2 (c.f. [19, Theorem 4.3]).

Proof.

Remark 2.1.

Lemma 2.3 (Subconvergence of the Ricci-DeTurck diffeomorphisms).

Proof.

Remark 2.2.

Lemma 2.4 ([14, Theorem 3.1]).

Proof.

Lemma 2.5 (cf. [2, Lemma 4]).

Proof.

Proof of Main Theorem 2.

Proof of Main Theorem 1.

Claim 2.1 (cf. [4, Lecture Series 2, Proposition 2.3.1]).

Proof of Claim 2.1.

Example 2.1.

Question 2.1.

Proof of Main Theorem 3.

Remark 2.3.

3 Some corollaries

Corollary 3.1 (for Main Theorem 1).

Remark 3.1.

Proof of Corollary 3.1.

Definition 3.1 (Yamabe constant).

Corollary 3.2 (for Corollary 3.1).

Proof.

Remark 3.2.

Proposition 3.1 (Conformal deformations on an open manifold).

Proof.

Corollary 3.3.

Definition 3.2 (Distributional scalar curvature ([13, Definition 2.1], [11, Section 2])).

Remark 3.3.

Corollary 3.4.

Proof.

Question 3.1.

4 Counterexamples

Example 4.1 (C∞C^{\infty} locally uniformly convergence and incomplete limiting metric).

Example 4.2 (Not C1C^{1} but C0C^{0}).

Example 4.3 (On every closed manifold).

Question 4.1.

Example 4.4 (Not C2C^{2} but C1C^{1}).

Question 4.2.

Remark 4.1.

Declarations

References

Corollary 1.2 ( $=$ Corollary 3.4).

Example 4.1 ( $C^{\infty}$ locally uniformly convergence and incomplete limiting metric).

Example 4.2 (Not $C^{1}$ but $C^{0}$ ).

Example 4.4 (Not $C^{2}$ but $C^{1}$ ).