For almost all irrational $\theta\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\theta$, then one has
$$\lim_{k\to \infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}=\frac{16}{\pi^3}\sum_{n\ge 1 \textrm{ odd}}\frac{1}{n^3\sin(n\pi\theta)}.$$

\begin{Remark}
  {\em
The $\mathrm{SU}_2$-TQFT $\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\ell=p-2$. Similar $\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\ref{DM1}.  But we won't consider even level $\mathrm{SU}_2$-TQFTs in this paper.
}\end{Remark}

\section{A trigonometric formula for the signature in genus 2}\label{genre2}
Let us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.

\begin{Theorem}\label{sigma2}
  For $0<q<p$ coprime odd integers, we have
\begin{equation}\nonumber \sigma_2({{\textstyle\frac q p}})= \frac {1-p^2}{6p^2} +
  \frac 1 {4p^2}\sum_{n=1,\, \mathrm{ odd}}^{p-2}\frac{f(n,p,q)}{\sin^3(\frac{n\pi}{2p})\sin^2(\frac{qn\pi}{p})}
\end{equation}
where
\begin{eqnarray*} f(n,p,q)&=(3p-3)\sin(\frac{(2q-1)n\pi}{2p})+(p+1)\sin(\frac{(2q-3)n\pi}{2p})\\
&+(p-1)\sin(\frac{(2q+3)n\pi}{2p})+(3p+3)\sin(\frac{(2q+1)n\pi}{2p}).
\end{eqnarray*}
\end{Theorem}

\begin{proof}
By a finite Fourier transform over $\Z/2p\Z$, we can write for $m\in\Z$ :
\begin{equation}\label{FFT}
  (-1)^{\lfloor m/p\rfloor}=
  \frac 1 p
  \sum_{n=1,\text{ odd}}^{2p-1}\frac{\sin(\pi n(2m+1)/2p)}{\sin(\pi n/2p)}
  =\frac 1 p  \sum_{n=1,\text{ odd}}^{2p-1} \frac {t^{2m+1}-t^{-2m-1}}{t-t^{-1}}
    \Big|_{t=e^{\frac {i\pi n}{2p}}~.
      }
\end{equation}
In the sequel, we set $\O_p=\{1,3,\ldots,2p-1\}$ and $\O_p^+=\{1,3,\ldots,p\}$. Using Formula \eqref{FFT} for each sign $\epsilon_{j+1}=(-1)^{\lfloor q(j+1)/p\rfloor}$ and plugging it into the formula of Lemma \ref{triplepsilon}, we get
\begin{equation}\nonumber\sigma_2({\textstyle{\frac q p}})\,\,
  =\sum_{(j,k,\ell)\in T_p} \epsilon_{j+1}\epsilon_{k+1}\epsilon_{\ell+1}
  =\frac1 {p^3} \sum_{(a,b,c)\in \O_p^3}\psi(a,b,c)\end{equation}
where
\begin{equation}
\psi(a,b,c)=\!\!\!\!\sum\limits_{(j,k,l)\in T_p}\!\!\!\!
\frac{(x^{2q(j+1)+1}-x^{-2q(j+1)-1})(y^{2q(k+1)+1}-y^{-2q(k+1)-1})(z^{2q(\ell+1)+1}-z^{-2q(\ell+1)-1})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\nonumber
\end{equation}
with $x=e^{\frac{i\pi a}{2p}}$,  $y=e^{\frac{i\pi b}{2p}}$ and $z=e^{\frac{i\pi c}{2p}}$.

The function $\psi$ is clearly symmetric in its three variables and satisfies also $\psi(2p-a,b,c)=\psi(a,b,c)$. We can hence rewrite $\sigma_2(\frac q p)$  as
\begin{equation}\label{eq4n}\sigma_2({\textstyle{\frac q p}})=
  \frac1 {p^3}
  \sum_{a,b,c\in \O_p^+} 2^{m_a+m_b+m_c}\psi(a,b,c)\end{equation}
where $m_a=1$ if $a\ne p$ and $m_p=0$.

In order to compute $\psi(a,b,c)$, we rewrite it as follows:
\begin{equation} \psi(a,b,c)= \sum\limits_{\epsilon,\mu,\nu=\pm 1}\epsilon\mu\nu \frac{ x^{\epsilon}y^{\mu}z^{\nu}F(x^{2q\epsilon},y^{2q\mu},z^{2q\nu})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\label{eq5n}
\end{equation} where $F$ is the polynomial defined by $F(X,Y,Z)=\sum_{(j,k,l)\in T_p}X^{j+1}Y^{k+1}Z^{\ell +1}$.

The computation of $F(x^{2q\epsilon},y^{2q\mu},z^{2q\nu})$ is simplified by the following observation:  we have $$F(X,Y,Z)=X^pY^p F(X^{-1}, Y^{-1},Z)~,$$ since the involution $(j,k,\ell ) \mapsto (p-2-j,p-2-k,\ell)$ preserves the set $T_p$. Therefore, since $F$ is symmetric in its three variables,  whenever $F(X,Y,Z)$ is evaluated at numbers $X=x^{2q}$, $Y=y^{2q}$, and $Z=z^{2q}$
satisfying $X^p=Y^p=Z^p=-1$, then for  any $\epsilon,\mu,\nu=\pm 1$, we have that $F(X^\epsilon, Y^\mu,Z^\nu)$ is equal to either $F(X,Y,Z)$ (when $\epsilon\mu\nu =1$) or to  $F(X^{-1}, Y^{-1},Z^{-1})= \overline {F(X,Y,Z)}$ (when $\epsilon\mu\nu =-1$).
\begin{Remark}\label{rk33}{\em For later use, we note the following special case: when $F(x^{2q},y^{2q},z^{2q})$ happens to be real, then $F(x^{2q\epsilon},y^{2q\mu},z^{2q\nu})$
    does not depend at all on $\epsilon,\mu$, or $\nu$, and formula \eqref{eq5n} for $\psi(a,b,c)$ simply becomes $\psi(a,b,c)=F(x^{2q},y^{2q},z^{2q})$.}
    \end{Remark}

Let $p$ be an odd positive integer and $\mathbb{Q}(\zeta)$ be the cyclotomic field generated by a primitive root of unity $\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\mathbb{Q}(\zeta)$-vector space $\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\mathbb{Q}(\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.

The dimension of $\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\mathbb{Q}(\zeta)$ into $\mathbb{C}$ by sending $\zeta$ to $e^{i\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by
$$\sigma_g(\textstyle{\frac{q}{p}})=\Sign\Big(\boV_p(S_g)\!\!\underset{\zeta=e^{i\pi q/p}}{\otimes}\!\!\mathbb{C},\,h_g\Big)$$
and which has been very little studied. It was shown in \cite{DM} that for a fixed $\zeta=e^{i\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\frac q p}$. Later it was proved in \cite{S2B} that $V_{\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\sigma_{g}(\frac{q}{p})$ when $\frac{q}{p}\to \theta$ for a fixed $\theta\in [0,1]$.

If we take $q=1$, the Hermitian form $h_g$ becomes positive definite,
so that $\sigma_g(\frac 1 p)$ coincides with the dimension of $\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\to \infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\mathcal{M}_g=\Hom(\pi_1(S_g),\SU_2)/\SU_2$. This led Witten \cite{Witten} to show that for $g>1$,
$$\lim_{p\to \infty}\frac{\dim \boV_p(S_g)}{p^{3g-3}}=\operatorname{Vol}(\mathcal{M}_g)=2(2\pi^2)^{1-g}\zeta(2g-2).$$

In genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\sigma_2(\frac q p)$:
\begin{equation}\label{defsigma2}
 \sigma_2({\textstyle{\frac{q}{p}}})=\sum_{(j,k,\ell)\in \Delta_p}
 (-1)^{\lfloor {jq}/{p}\rfloor +\lfloor {kq}/{p}\rfloor+\lfloor {\ell q}/{p}\rfloor}
\end{equation}
 where the sum runs over the set $\Delta_p$ of triples of integers satisfying
$$\begin{cases}0<j,k,\ell<p, \\
j<k+\ell,k<j+\ell,\ell<j+k,\\
j+k+\ell<2p,\textrm{ and } j+k+\ell\textrm{ odd}.
\end{cases}$$

Our main result is the following

Our main result is the following

Figure \ref{fig:sigma2} suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\sigma_2(q/p)/p^2)$ for $0<q<p<31$.

We briefly review the properties of the convergents $q_k/p_k$ of $\theta$ that we will need. See {\em e.g.} Perron \cite{P} or Khinchin \cite{Kh1} for more details\footnote{Unfortunately, \cite{Kh1} writes $p_k$ for our $q_k$ and {\em vice versa.}}. Each irrational $\theta\in {\mathbb R}$ has a unique regular continued fraction expansion $$\theta = [a_0; a_1, a_2, \ldots ] = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\dots}} $$ where regular means that $a_0\in \Z$ and $a_i\in \Z_{\geq 1}$ for $i\geq 1$. Note that in our case $a_0=0$ since
$\theta \in [0,1]\setminus \Q$.
The convergents
$$\frac{q_k}{p_k} = [a_0; a_1, a_2, \ldots, a_k] = a_0+\cfrac{1}{a_1+\cfrac{1}{\dots+ \frac{1}{a_k}}} $$ can also be defined recursively by
\begin{eqnarray} q_k&=a_k q_{k-1} + q_{k-2}, \ \ \ \ q_0=a_0, \ \ \ \ q_{-1}=1 \label{rec5}\\
 p_k&=a_k p_{k-1} + p_{k-2}, \ \ \ \ p_0=1, \ \ \ \ p_{-1}=0 \label{rec6}
\end{eqnarray}
The following classical facts are well-known.
\begin{Proposition}\label{qpfacts}
 For all irrational $\theta$, the convergents $q_k/p_k$ of $\theta$ satisfy the following properties:
 \begin{enumerate}
 \item[(i)] One has $1=p_0\leq p_1<p_2<\ldots$, {\em i.e.} the denominators $p_k$ are positive integers and moreover strictly increasing from $p_1$ onwards.
 \item[(ii)] For all $k\geq 1$, $q_k$ and $p_k$ are coprime.
 \item[(iii)] $\lim_{k\rightarrow\infty} q_k/p_k = \theta$.
 \item[(iv)]
 For all $i\geq 1$, we have
 \begin{equation}\nonumber
 \frac 1 {p_i(p_i+p_{i+1})} <
 \Big|\theta - \frac {q_i}{p_i} \Big| < \frac 1 {p_ip_{i+1}} < \frac 1 {p_i^2}~.
 \end{equation}
 \item[(v)] {\em [Lagrange]}\footnote{For a proof, see Perron \cite[\S 15, Satz 2.17]{P}.} For all $1\leq i<k$, and any $n,\ell \in\Z$ with $1\leq n<p_{i+1}$, we have
 \begin{equation}\nonumber
 \Big| n\frac {q_k}{p_k} -\ell \Big|
 \geq \Big|p_i\frac {q_k}{p_k} - q_i\Big| \geq \frac{1}{p_i + p_{i+1}} ~.
 \end{equation}
 \end{enumerate}
 \end{Proposition}

\begin{proof}[\bf{Proof of Theorem~\ref{asympto}}] We start from the trigonometric formula for $\sigma_2(\frac{q}{p})$ given in Theorem~\ref{sigma2}. The first observation is that the term $(1-p^2)/(6p^2)$ in this formula stays bounded as $p\rightarrow \infty$ and so does not contribute to the limit of $\sigma_2(\frac{q_k}{p_k})/ p_k^2$. Thus it suffices to show
 \begin{equation}\label{eq4}
 \lim_{k\rightarrow \infty} \frac{1}{4p_k^4} \sum_{n=1,\textrm{ odd}}^{p_k-2}\frac{f(n,p_k,q_k)}{\sin^3({n\pi}/{2p_k})\sin^2({q_kn\pi}/{p_k})} =\frac{16}{\pi^3}\sum_{n\ge 1, \textrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}
 \end{equation} with $f(n,p,q)$ given as in Theorem~\ref{sigma2}.

Let $S^{(1)}(q_k,p_k,\theta)$ denote the sum $S^{(1)}(q_k,p_k)$ but with $\sin(n\pi\theta)$ in the denominator in place of $\sin({q_kn\pi}/{p_k})$. Using (\ref{sin-ineq}), we see that the $n$-th summand of $S^{(1)}(q_k,p_k,\theta)$ is bounded from above by $1/(n^3\sin(n\pi\theta))$. So we can apply the Lebesgue dominated convergence theorem to this sum. We get
 $$\lim_{k\rightarrow \infty}S^{(1)}(q_k,p_k,\theta) =\frac{8}{\pi^3}\sum_{n\ge 1, \textrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}~.$$
 Thus to prove Lemma~\ref{3.3} it suffices to show that the difference $d_k=S^{(1)}(q_k,p_k) -S^{(1)}(q_k,p_k,\theta)$ goes to zero as $k\rightarrow \infty$. We have
 $$d_k= \frac{1}{p_k^3} \sum_{n=1,\textrm{ odd}}^{p_k-2}\frac{\cos (n\pi/2p_k)}{\sin^3({n\pi}/{2p_k}) }\Biggl (\frac {1} {\sin({q_kn\pi}/{p_k})} - \frac {1} {\sin(n\pi\theta)}\Biggr)~.$$ We can bound the denominators from below using (\ref{sin-ineq}), (\ref{sinn-inequ}), and the trivial lower bound $|\sin({q_kn\pi}/{p_k})| \geq \sin (\pi/p_k) \geq 2/p_k$. Thus for all $\epsilon >0$ we have

\begin{proof}[\bf{Proof of Lemma~\ref{3.4}}]
 We assume that $\mu(\theta)=2$ and that $\theta$ satisfies the conditions
 in Theorem~\ref{ThKL}.
 Recall $S^{(2)}(q_k,p_k)$, defined in (\ref{sum2}), is a sum from $n=1$ to $n=p_k-2$. We cut
it into two parts, as follows. Put
\begin{equation}\label{floor} N(k)= \lfloor 3k/4 \rfloor
\end{equation} and write $$S^{(2)}(q_k,p_k) = S^{(2.1)}(q_k,p_k)
+S^{(2.2)}(q_k,p_k)~,$$ where $S^{(2.1)}(q_k,p_k)$ is the sum from
$n=1$ to $n=p_{N(k)+1}-1$ and $S^{(2.2)}(q_k,p_k)$ is the sum from
$n=p_{N(k)+1}$ to $p_k-2$. The following two lemmas provide upper bounds for these two sums.
\begin{Lemma}\label{3.9} We have $$|S^{(2.1)}(q_k,p_k) |\leq \frac
{p_{N(k)}} {p_k} F_7(k)$$ where $F_7(k) ={\mathcal O}(k^7)$.
\end{Lemma}
\begin{Lemma}\label{3.10} We have $$|S^{(2.2)}(q_k,p_k)| \leq \frac{1}{2} \zeta(2)
\frac {p_{k}} {p_{N(k)+1}^2}~.$$
\end{Lemma}
Assuming these two lemmas for the moment, since $\lim_{m\rightarrow\infty} p_m^{1/m}= L>1$ by the Khinchin-L{\'e}vy Theorem~\ref{ThKL}(ii), $p_k$ grows like $L^k$ as $k\rightarrow \infty$. It is not hard to deduce from this that both $ S^{(2.1)}(q_k,p_k)$ and $ S^{(2.2)}(q_k,p_k)$ converge to zero.~\footnote{Here is how to see this for $ S^{(2.1)}(q_k,p_k)$. Theorem~\ref{ThKL}(ii) gives that for all $\epsilon>0$ there exist $m_0(\epsilon)$ so that $L^{1-\epsilon} \leq p_m^{1/m} \leq L^{1+\epsilon}$ holds for for all $m\geq m_0(\epsilon)$. Thus also $L^{m-m\epsilon} \leq p_m\leq L^{m+m\epsilon}$ for all $m\geq m_0(\epsilon)$. Applying now the first inequality with $m=k$ and the second one with $m=N(k)=\lfloor 3k/4\rfloor$ and taking $\epsilon$ small enough, it follows that $p_{N(k)}/p_k \leq L^{-\delta k}$ for some $\delta>0$ as soon as $k\geq 2 m_0(\epsilon)$. Thus $p_{N(k)} F_7(k)/ {p_k} \leq k^7 L^{-\delta k}$ for $k$ big enough, proving that the limit as $k\rightarrow\infty$ is zero, as asserted. }
This proves Lemma~\ref{3.4}.
\end{proof}
It remains to prove Lemmas~\ref{3.9} and~\ref{3.10}. Let us first deal
with Lemma~\ref{3.10} whose proof is easy.
\begin{proof}[\bf{Proof of Lemma~\ref{3.10}}] Since $n\geq p_{N(k)+1}$
 in this sum,
 we can bound the
$\sin^2(n\pi/2p_k)$ term in the denominator from below by
$p_{N(k)+1}^2/p_k^2$. Thus
\begin{equation}\label{sum3}|S^{(2.2)}(q_k,p_k)| \leq \frac 1 {p_k
p_{N(k)+1}^2}\sum_{n=1,\textrm{ odd}}^{p_k-2} \frac 1
{\sin^2(q_kn\pi/p_k)}~.\end{equation}

\section{The higher genus case}\label{sec5}
In this section, we briefly discuss how the asymptotic behaviour of the genus $2$ signature might generalize to higher genus. Numerical experiments (see below) indicate that
 $\sigma_{g,n}(\frac{q}{p};\lambda)$ might grow like $p^{\max\{g,2g-2\}}$ when $\frac q p$ goes to an irrational $\theta\in[0,1]$. So we ask the following question.
\begin{Question}\label{51}
 Given $g,n\in \Z_{\geq 0}$ and $n$-tuples of integers $\lambda=(\lambda_1,\ldots,\lambda_n)\in \Z^n_{\geq 0}$,
 is there a function $F_{g;\lambda}:
 [0,1]
 \to \R$ defined almost everywhere satisfying the following:
 For almost all irrational $\theta\in[0,1]$,
 if we write $\frac{q_k}{p_k}$ for the convergents of $\theta$, one has
$$ \lim_{k\to\infty}\frac{\sigma_{g,n}(\frac{q_k}{p_k};\lambda)}{p_k^{\max\{g,2g-2\}}}=F_{g;\lambda}(\theta)$$
\end{Question}
Theorem \ref{asympto} shows that the answer is yes for $g=2$ and $n=0$. Let us
compare this to the asymptotics of the dimension
of the TQFT vector spaces
which
is well-known: it has a semi-classical interpretation for which we refer to \cite[Section 3]{Witten}. Precisely:
$$ \lim_{p\to\infty}\frac{\dim \boV_p(S_g)}{p^{\max\{g,3g-3\}}}=\operatorname{Vol}(\mathcal{M}_g)=2(2\pi^2)^{1-g}\zeta(2g-2)$$
where the last equality holds only for $g>0$ and $\mathcal{M}_g$ is the character variety of representations of $\pi_1(S_g)$ into SU$_2$, endowed with its Liouville measure. We have no conceptual explanation why the order of growth $3g-3$
for the dimension
should be replaced by $2g-2$
for the signature.

Figure \ref{fig:sigma2}  suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\sigma_2(q/p)/p^2)$ for $0<q<p<31$.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm]{genre2.png}
\caption{Graph of the signature function $\sigma_2$}\label{fig:sigma2}
\end{center}
\end{figure}

The proof of this theorem involves several steps and occupies Sections~\ref{genre2} and~\ref{sec2}. We first apply in Section~\ref{genre2} a finite Fourier transform in Equation \eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\Delta_p$. We then apply Brion's theorem to get an explicit trigonometric sum computing $\sigma_2(q/p)$ (see Theorem \ref{sigma2} for the precise statement): this step already presents miraculous cancellations for which we do not have any conceptual explanation.

Then in Section~\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper. In order to control its denominators, we need to impose that the coefficients of the continued fraction expansion of $\theta$ do not grow too quickly. We use for that  some classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.

In higher genus one can also write down a formula for
  $\sigma_g(q/p)$ as the sum of signs indexed by lattice points inside some polytope (see \cite[Remark 4.12]{BHMV}~\footnote{Our $\boV_p$ corresponds to the $V_{2p}$ of \cite{BHMV}.}.) But the signs are much more complicated than in Formula \eqref{defsigma2} which prevents us from generalizing our theorem. We briefly discuss in Section~\ref{sec5} how the asymptotics might look like for general surfaces including marked points, and illustrate this by some numerical experiments.

In the last section of the paper, we discuss some modular properties of the signature. We observe that the function
$$\Lambda(\theta)=\frac{16}{\pi^3}\sum_{n\ge 1\textrm{ odd}}\frac{1}{n^3\sin(n\pi\theta)}$$
is the boundary value of an Eichler integral of a modular form of weight $4$ for the level $2$ group $\Gamma(2)$. In addition to the obvious property $\Lambda(\theta+1)=-\Lambda(\theta)$, it satisfies the following transformation law:
\begin{equation}\label{Lambdamod}
\Lambda\Big(\frac{\theta}{2\theta+1}\Big)(2\theta+1)^2-\Lambda(\theta)=2\theta^2+2\theta+1.
\end{equation}
This functional equation suggests that a similar integral version holds for signatures as stated in the following conjecture, checked by computer for all $0<q<p<100$.
\begin{Conjecturestar}
For any odd coprime integers $0<q<p$ one has
$$\sigma_2\Big(\frac{q}{2q+p}\Big)-\sigma_2\Big(\frac{q}{p}\Big)=2q^2+2qp+p^2-1.$$
\end{Conjecturestar}

This formula is reminiscent of the reciprocity formula of Dedekind sums: in fact we will make this analogy more precise in Section~\ref{SectionS}. We prove that the signature at $\zeta=e^{i\pi q/p}$ of  the TQFT-vector space associated to the 4 times punctured sphere (with  colors $(p-1)/2$  at the marked points)  is actually equal to a 2-smoothed version, $S(q/p)$,  of the Dedekind sum $s(q,p)$. It is then classical that  this function $S(q/p)$ is the boundary value of a modular form (this time of weight $1/2$) and that it satisfies the following transformation law

$$S\Big(\frac{q}{2q+p}\Big)-S\Big(\frac{q}{p}\Big)=1.$$
We provide an elementary proof of these facts for the benefit of the non-specialist reader.

This very satisfactory state of things would hopefully generalize to higher genus as well. However, even in genus 2, a precise (non asymptotic) relation between the signature $\sigma_g(q/p)$
and the Eichler integral is still lacking. Such a relation would presumably lead to a proof  of the conjectured transformation law for $\sigma_2$.

To end this introduction, we would like to mention the parallel paper \cite{MY} in which the first author and S. Yoon compute the asymptotics of the function
$\sigma_g(q/p)$, but for sequences of the form $\frac{q_n}{p_n}=\frac{a+bn}{c+dn}$ where $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\in \Gamma(2)$. This regime is somewhat orthogonal to the one described in the present paper. The techniques used and the results are also very different. A synthesis of the two papers has yet to be made.
\vskip 8pt

{\bf Acknowledgements:} It is our pleasure to thank Pierre Charollois for early discussions on many aspects of this paper, Gaëtan Chenevier for his expertise on modular forms,
and Don Zagier for very helpful explanations about Eichler integrals and quantum modular forms.

The proof “involves several steps and occupies Sections~\ref{genre2} and~\ref{sec2}.” First, “we apply in Section~\ref{genre2} a finite Fourier transform in Equation \eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\Delta_p$.” Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\sigma_2(q/p)$ (see Theorem \ref{sigma2} for the precise statement),” a step that “already presents miraculous cancellations.” Then, “in Section~\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”

The proof “involves several steps and occupies” the part starting at

\begin{Remark}
  {\em
The $\mathrm{SU}_2$-TQFT $\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\ell=p-2$. Similar $\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\ref{DM1}.  But we won't consider even level $\mathrm{SU}_2$-TQFTs in this paper.
}\end{Remark}

\section{A trigonometric formula for the signature in genus 2}\label{genre2}
Let us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.

\begin{Theorem}\label{sigma2}
  For $0<q<p$ coprime odd integers, we have
\begin{equation}\nonumber \sigma_2({{\textstyle\frac q p}})= \frac {1-p^2}{6p^2} +
  \frac 1 {4p^2}\sum_{n=1,\, \mathrm{ odd}}^{p-2}\frac{f(n,p,q)}{\sin^3(\frac{n\pi}{2p})\sin^2(\frac{qn\pi}{p})}
\end{equation}
where
\begin{eqnarray*} f(n,p,q)&=(3p-3)\sin(\frac{(2q-1)n\pi}{2p})+(p+1)\sin(\frac{(2q-3)n\pi}{2p})\\
&+(p-1)\sin(\frac{(2q+3)n\pi}{2p})+(3p+3)\sin(\frac{(2q+1)n\pi}{2p}).
\end{eqnarray*}
\end{Theorem}

and the later section \ref{sec2}.

First, “we apply” a finite Fourier transform in the defining expression
\begin{equation}\label{defsigma2}
  \sigma_2({\textstyle{\frac{q}{p}}})=\sum_{(j,k,\ell)\in \Delta_p}
 (-1)^{\lfloor {jq}/{p}\rfloor +\lfloor {kq}/{p}\rfloor+\lfloor {\ell q}/{p}\rfloor}
\end{equation}
to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\Delta_p$.

Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\sigma_2(q/p)$,” namely the formula stated in
\begin{Theorem}\label{sigma2}
  For $0<q<p$ coprime odd integers, we have
\begin{equation}\nonumber \sigma_2({{\textstyle\frac q p}})= \frac {1-p^2}{6p^2} +
  \frac 1 {4p^2}\sum_{n=1,\, \mathrm{ odd}}^{p-2}\frac{f(n,p,q)}{\sin^3(\frac{n\pi}{2p})\sin^2(\frac{qn\pi}{p})}
\end{equation}
where
\begin{eqnarray*} f(n,p,q)&=(3p-3)\sin(\frac{(2q-1)n\pi}{2p})+(p+1)\sin(\frac{(2q-3)n\pi}{2p})\\
&+(p-1)\sin(\frac{(2q+3)n\pi}{2p})+(3p+3)\sin(\frac{(2q+1)n\pi}{2p}).
\end{eqnarray*}
\end{Theorem}

a step that “already presents miraculous cancellations.” Then, “we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”

Question:
For a reduced rational number \(q/p\in(0,1)\), define
\[
\sigma_2\!\left(\frac qp\right)=\sum_{(a,b,c)\in \Delta_p}(-1)^{\lfloor aq/p\rfloor+\lfloor bq/p\rfloor+\lfloor cq/p\rfloor},
\]
where \(\Delta_p\) is the set of triples \((a,b,c)\) of integers such that
\[
0<a,b,c<p,\qquad a<b+c,\quad b<a+c,\quad c<a+b,
\]
\[
a+b+c<2p,\qquad a+b+c\text{ is odd}.
\]
Now let \(\theta\in[0,1]\) be irrational, and let \(q_k/p_k\) be the convergents of the regular continued-fraction expansion of \(\theta\). Which statement holds for Lebesgue-almost every irrational \(\theta\in[0,1]\)?

The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}.
\]

Choices:
B.
The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1}\frac{1}{n^3\sin(n\pi\theta)}.
\]
C.
For Lebesgue-almost every irrational \(\theta\in[0,1]\), the limit
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
\]
exists.
D.
The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\,|\sin(n\pi\theta)|}.
\]
E.
For every irrational \(\theta\in[0,1]\), if \(q_k/p_k\) are the convergents of its regular continued-fraction expansion, then
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}.
\]

odd-index restriction in the trigonometric expansion

explicit identification of the limiting value by the odd-series formula

signed denominator \(\sin(n\pi\theta)\) replaced by absolute value, destroying the exact Fourier-cancellation structure

almost-everywhere metric Diophantine hypothesis strengthened to all irrationals

The stem does not explicitly reveal the correct limit formula. It defines the quantity and asks for the asymptotic statement; the correct answer is not directly embedded in the wording.

The item is close to a theorem-recall question: one option is essentially the exact stated limiting theorem, with distractors formed by small perturbations. It is not a pure tautology, but it is only a mild reformulation of a known result.

Some reasoning is needed to compare quantifiers, normalization, and the odd-index restriction, but the question mostly tests recognition/recall of the theorem rather than genuine derivation or generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: removing the odd restriction, weakening to mere existence, overstrengthening 'almost all' to 'every,' and changing the normalization to p_k^3 are all distinct and aligned with common failure modes.

A reasonably well-constructed theorem-recognition MCQ with strong distractors and little direct answer leakage, but it only moderately avoids tautology and does not strongly test generative reasoning.


Question:
For a reduced rational number \(q/p\in(0,1)\), define
\[
\sigma_2\!\left(\frac qp\right)=\sum_{(a,b,c)\in \Delta_p}(-1)^{\lfloor aq/p\rfloor+\lfloor bq/p\rfloor+\lfloor cq/p\rfloor},
\]
where \(\Delta_p\) is the set of triples \((a,b,c)\) of integers such that
\[
0<a,b,c<p,\qquad a<b+c,\quad b<a+c,\quad c<a+b,
\]
\[
a+b+c<2p,\qquad a+b+c\text{ is odd}.
\]
Now let \(\theta\in[0,1]\) be irrational, and let \(q_k/p_k\) be the convergents of the regular continued-fraction expansion of \(\theta\). Which statement holds for Lebesgue-almost every irrational \(\theta\in[0,1]\)?

Choices:
B.
The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1}\frac{1}{n^3\sin(n\pi\theta)}.
\]
C.
For Lebesgue-almost every irrational \(\theta\in[0,1]\), the limit
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
\]
exists.
D.
The limit exists and is given by
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\,|\sin(n\pi\theta)|}.
\]
E.
For every irrational \(\theta\in[0,1]\), if \(q_k/p_k\) are the convergents of its regular continued-fraction expansion, then
\[
\lim_{k\to\infty}\frac{\sigma_2(q_k/p_k)}{p_k^2}
=\frac{16}{\pi^3}\sum_{n\ge 1\,\mathrm{odd}}\frac{1}{n^3\sin(n\pi\theta)}.
\]

The stem does not explicitly reveal the correct formula or uniquely point to choice A. It sets up the objects and asks for the valid almost-everywhere statement, so there is no direct answer leakage.

This is essentially a recognition item for a specific theorem: the correct option is the theorem’s full conclusion, while the others are small perturbations of it. It mainly restates known content rather than asking for an independently derived conclusion.

Some reasoning is needed to distinguish subtle variants such as odd-only summation, signed versus absolute value, and 'almost every' versus 'every' irrational. However, the item primarily tests exact recall of the theorem statement rather than substantial generative reasoning from the definitions.

The distractors are mathematically plausible and target realistic failure modes: dropping the odd-index restriction, weakening to mere existence, inserting an absolute value, and overstrengthening an almost-everywhere claim to a universal one.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests genuine reasoning.


\begin{proof}[Proof of Theorem \ref{A}]
When the Yamabe metric is Einstein, Proposition \ref{ncsc} applies to complete the proof.
The remaining cases are handled by Theorem \ref{LCF} and Theorem \ref{local}.
\end{proof}

\section{The topology of  \(4\)-manifolds with PSC}\label{3}

Within the known classifications of closed, oriented \(3\)-manifolds admitting Riemannian metrics of positive scalar curvature and of locally conformally flat \(4\)-manifolds with positive scalar curvature (up to diffeomorphism type), one may further ask to classify closed, oriented Riemannian \(4\)-manifolds with positive scalar curvature up to homeomorphism type.

However, Carr \cite{MR936805} shows that for any nontrivial finitely presented group \(\pi\), there exists a closed Riemannian \(4\)-manifold \(M\) with positive scalar curvature such that \(\pi_1(M)=\pi\). By Markov’s theorem, it follows that closed \(4\)-manifolds with positive scalar curvature cannot be classified up to homeomorphism. In fact, Carr’s result extends to all dimensions \(n>4\): for every finitely presented group \(\pi\) and \(n\ge4\), there exists a closed smooth stably parallelizable Riemannian \(n\)-manifold with positive scalar curvature; moreover, if \(\pi\) contains a subgroup of index two, there also exists a closed smooth non-orientable Riemannian \(n\)-manifold with positive scalar curvature \cite[Theorems 6 and 7]{zbMATH07432160}.

To circumvent this group-theoretic obstruction, one may instead aim to classify those Riemannian \(n\)-manifolds with a fixed fundamental group. The simplest case is the simply connected one. For \(n\ge5\), it is known that a simply connected smooth manifold \(M^n\) admits a metric of positive scalar curvature if and only if \(M^n\) is non-spin, or it is spin and its \(\alpha\)-invariant vanishes in \(\mathrm{KO}_n\). In particular, since \(\mathrm{KO}_n = 0\) for \(n \equiv 3,5,6,7 \pmod{8}\) by Bott periodicity for the $\mathrm{KO}_n$-theory of a point, every simply connected smooth \(n\)-manifold in these dimensions admits a metric of positive scalar curvature. Consequently, in these cases, the geometric classification problem reduces entirely to a topological one.

The classical results of Freedman and Donaldson in  dimension 4 imply that if
\( M^4 \) is a smoothable, closed, simply connected, topological \(4\)-manifold,
then \( M^4 \) is homeomorphic to one of the following:
$S^4$, $\#^m \mathbb{C}P^2 \,\#^n \overline{\mathbb{C}P^2}$, or $  \#^{\pm m} M_{E_8} \,\#^n (S^2 \times S^2),$
for some integers \( m,n \geq 0 \), where \(M_{E_8}\) denotes the topological $4$–manifold whose intersection form is given by the $E_8$ lattice.

 For a simply connected closed \(4\)-manifold \(M^4\), it admits a spin structure if and only if its intersection form is even. The intersection form of $\#^{\pm m} M_{E_8} \,\#^n (S^2 \times S^2)$
has rank \(8m+n\), signature \(\pm 8m\), and is even. Furthermore, by the Hirzebruch signature theorem, the \(\hat{A}\)-genus of a \(4\)-manifold equals \(-\tfrac{1}{8}\) times its signature. Hence,
\(\#^{\pm m} M_{E_8} \,\#^n (S^2 \times S^2)\) does not admit a metric of positive scalar curvature.

Therefore, a closed simply connected \(4\)-manifold admitting a metric of positive scalar curvature is homeomorphic to one of the following:
\[
S^4, \quad
\#^m \mathbb{C}P^2 \,\#^n \overline{\mathbb{C}P^2}, \quad
\#^n (S^2 \times S^2),
\]
or a connected sum of these manifolds (see \cite[Theorem 7.8]{MR3450199}).  This classification cannot, in general, be improved to the diffeomorphism type.  Indeed, for some \(k\), Teicher \cite[Theorem~5.8]{MR1720873} constructs examples of simply connected complex surfaces of general type which are spin and have vanishing signature.  These manifolds are homeomorphic to \( \#^k (S^2 \times S^2) \), but, being of general type, they do not admit metrics of positive scalar curvature.

The next step is to analyze \(4\)-manifolds which are not simply connected. If its fundamental group is finite, we have the following result.

\begin{proposition}\label{finite}
Let $M^4$ be a connected, closed, non-simply connected Riemannian $4$-manifold with positive scalar curvature and finite fundamental group. Then:
\begin{enumerate}
\item[(i)] If $M$ is spin, then its universal cover $\tilde{M}$ is homeomorphic to
$\#^{k} (S^2 \times S^2)$ for some integer $k \ge 1$, where $k = \tfrac{1}{2} b_2(\tilde{M})$.

\item[(ii)] If $M$ is non-spin, then $\tilde{M}$ is homeomorphic to one of the following:
\begin{enumerate}
\item[(a)] $\#^{k} \mathbb{CP}^2 \,\#^{l} \overline{\mathbb{CP}}^{\,2}$,
 for some non-negative integers $k+l \ge 1$, if $\tilde{M}$ is non-spin;
\item[(b)] $\#^{m} (S^2 \times S^2)$, for some integer $m \ge 0$, if $\tilde{M}$ is spin.
\end{enumerate}
\end{enumerate}
\end{proposition}

\begin{proof}
If $M$ is spin, then its universal cover $\tilde{M}$ is also spin  and the intersection form of $\tilde{M}$ is even. The lifted positive scalar curvature metric implies that the signature of $\tilde{M}$ vanishes. Hence, $\tilde{M}$ is homeomorphic to $\#^{k} (S^2 \times S^2)$.
If $k = 0$, then $M$ would be homeomorphic to $\mathbb{RP}^4$. However, $\mathbb{RP}^4$ is non-spin, which leads to a contradiction. Therefore, $k \ge 1$.

Now suppose $M$ is non-spin.
If $\tilde{M}$ is spin, then as above $\tilde{M}$ is homeomorphic to $\#^{m} (S^2 \times S^2)$ for some $m \ge 0$.
If $\tilde{M}$ is non-spin, then the intersection form of $\tilde{M}$ is odd. By the classification of odd unimodular forms and Freedman's theorem, $\tilde{M}$ must be homeomorphic to a connected sum of copies of the complex projective plane $\mathbb{CP}^2$ and its orientation-reversed version $\overline{\mathbb{CP}}^{\,2}$ for some non-negative integers $k+l \ge 1$.
\end{proof}

\label{2}

The Bishop–Gromov volume comparison theorem for Ricci curvature bounded below has many applications.
It is natural to seek analogous volume comparison results for scalar curvature bounded

\begin{remark}
For \( \alpha = 2 \) and \( \beta = \tfrac{m+1}{m} \) with \( m > 0 \), Case defines the weighted Yamabe constant \( \Lambda(g, e^{-f} \, d\mathrm{Vol}_g, m) \) and solves the corresponding weighted Yamabe problem in \cite{zbMATH06537658}. However, the weighted Yamabe invariant for general parameters \( \alpha \neq 0 \) and \( \beta \neq 0 \) has not yet been defined or systematically studied. It is also natural to ask whether weighted analogues of the results in this section can be formulated and established. In particular, if the manifold admits a hyperbolic metric, one may ask whether the inequality
\[
\Lambda(g, e^{-f} \, d\mathrm{Vol}_g, m) \leq \Lambda(g_H, e^{-f} \, d\mathrm{Vol}_{g_H}, m)
\]
holds for a fixed function \( f \in C^{\infty}(M^n) \) and \( m > 0 \).

\end{remark}

\section{LCF  $4$-manifolds with NSC}\label{5}

Let \( (M^n, g) \) be a closed, oriented, smooth, locally conformally flat Riemannian manifold with \( n \geq 4 \) and positive scalar curvature. Schoen and Yau~\cite[Theorem~4.5]{zbMATH04075988} prove that the  developing (conformal) map of \( (\widetilde{M^n}, \tilde{g}) \) is injective, and that \( \pi_1(M^n) \) admits a faithful holonomy representation into a discrete subgroup of \( \operatorname{Conf}(S^n) \). Let \( \Gamma \) denote the image of \( \pi_1(M^n) \) under the representation induced by the developing map. The image of the developing map is an open subset \( \Omega \subset S^n \) on which \( \Gamma \) acts properly discontinuously, and \( M^n \) is diffeomorphic to the quotient \( \Omega / \Gamma \).

Moreover, \( \Omega \) coincides with the domain of discontinuity of \( \Gamma \), and can be written as \( \Omega = S^n \setminus \Lambda \), where \( \Lambda \subset S^n \) is the limit set of \( \Gamma \). In particular, \( \Lambda \) is the minimal closed \( \Gamma \)-invariant subset of \( S^n \), and \( \Omega \) serves both as the image of the developing map and as the maximal open set on which the action of \( \Gamma \) is properly discontinuous.

The above results of Schoen and Yau~\cite{zbMATH04075988}   hold in dimensions not less than four under the assumption \( \mathrm{Sc}_g \geq c > 0 \).
  In that paper, they also remarked that their result \cite[Proposition 4.4$^{\prime}$]{zbMATH04075988}  would remain valid under the assumption of nonnegative scalar curvature, provided the positive energy theorem can be extended to the case of complete manifolds—namely, when \( M \) has one asymptotically flat end and other ends that are merely complete.
Recently, Lesourd, Unger, and Yau~\cite[Theorem~1.2]{MR4836036} used minimal hypersurfaces to prove that there does not exist a complete Riemannian metric with positive scalar curvature on \( T^3 \# X \), where \( X \) is an arbitrary (possibly noncompact) manifold.
Furthermore, they show that the nonexistence of a complete smooth metric with positive scalar curvature on \( T^n \# X \) $(n\geq 3)$ implies the following Liouville theorem~\cite[Theorem~1.7]{MR4836036}, see also~\cite[Corollary~4]{zbMATH07817078}:

\begin{theorem}[Liouville theorem]\label{Liouville theorem}
Let \( (M^n, g) \), \( n \ge 3 \), be a complete, locally conformally flat manifold with nonnegative scalar curvature.
If \( \Phi : M^n \to S^n \) is a conformal map, then \( \Phi \) is injective and the boundary \( \partial \Phi(M) \) has zero Newtonian capacity (i.e.,  $2$-capacity).
\end{theorem}

In another paper, Lesourd, Unger, and Yau~\cite[Theorem~1.2]{MR4773185} use $\nu$-buddle to prove the above Schoen–Yau conjecture and obtain a new proof of Liouville theorem, which is completely in the spirit of the approach outlined in~\cite{zbMATH04075988}.

 A Kleinian group $\Gamma$ is called \textit{elementary} if its limit set $\Lambda$ is empty or consists of at most two points; otherwise, it is called \textit{non-elementary}. If $\Lambda$ is empty, then $\Gamma$ is finite. If $\Lambda$ consists of a single point, then $\Gamma$ contains an abelian subgroup of finite index of rank $k$ with $k \leq n$. If $\Lambda$ consists of two points, then $\Gamma$ contains an infinite cyclic subgroup of finite index. If the limit set $\Lambda(\Gamma)$ does not consist of exactly two points, $\Lambda$ can be characterized as the minimal closed \( \Gamma \)-invariant subset of \( S^n \).

 Let \( B^{n+1} := \{ x \in \mathbb{R}^{n+1} \mid |x| < 1 \} \) be the Poincaré ball model endowed with the hyperbolic metric  $
g_{H} = 4(1 - |x|^2)^{-2} \sum_{i=1}^{n+1} (dx^i)^2.
$
Every element of the conformal group \( \mathrm{Conf}(S^n) \) extends to a diffeomorphism of the closed ball \( \overline{B^{n+1}} := B^{n+1} \cup S^n \), and restricts to an isometry of \( B^{n+1} \) with respect to the hyperbolic metric. Conversely, every isometry in \( \mathrm{Isom}(B^{n+1}) \) extends continuously to the boundary \( S^n \), acting as a conformal transformation of the standard sphere \( (S^n, g_{st}) \). Therefore, there is a natural group isomorphism: $\mathrm{Isom}(B^{n+1}) \cong \mathrm{Conf}(S^n).$

For an infinite Kleinian group \( \Gamma \), the \emph{critical exponent} \( \delta(\Gamma) \) is defined by
\[
\delta(\Gamma) := \inf\left\{ s > 0 \,\middle|\, \sum_{\gamma \in \Gamma} \exp\left( -s \, \mathrm{dist}(x, \gamma y) \right) < \infty \right\},
\]
where  $x, y \in B^{n+1}$ and \( \mathrm{dist}(\cdot, \cdot) \) denotes the hyperbolic distance function on the Poincaré ball model \( B^{n+1} \). The value of \( \delta(\Gamma) \) is independent of the particular choice of \( x \) and \( y \). If \( \Gamma \) is non-elementary, then \( 0 < \delta(\Gamma) \leq n \). If \( \Gamma'  \leq \Gamma \) is a subgroup, then   $\delta(\Gamma') \leq \delta(\Gamma).$

A Kleinian group \( \Gamma \) is said to be \emph{convex cocompact} if the quotient
$(\Omega(\Gamma) \cup B^{n+1}) / \Gamma$ is compact. Equivalently, \( \Gamma \) is convex cocompact if the hyperbolic convex hull \( \mathrm{CH}(\Gamma) \subset B^{n+1} \) of the limit set \( \Lambda(\Gamma) \) satisfies that \( \mathrm{CH}(\Gamma)/\Gamma \) is nonempty and compact. Here, \( \mathrm{CH}(\Gamma) \) denotes the minimal convex subset of \( B^{n+1} \) whose closure in the compactified ball \( \overline{B^{n+1}} := B^{n+1} \cup S^n \) contains \( \Lambda(\Gamma) \).

A Kleinian group \( \Gamma \) is said to be \emph{geometrically finite} if there exists a uniform bound on the orders of its finite subgroups and the \( \epsilon \)-neighborhood of \( \mathrm{CH}(\Gamma)/\Gamma \) in \( B^{n+1}/\Gamma \) has finite volume for some \( \epsilon > 0 \). Equivalently, \( \Gamma \) is geometrically finite if it admits a fundamental polyhedron in \( B^{n+1} \) with finitely many faces.

Convex cocompact Kleinian groups are geometrically finite. Conversely, geometrically finite Kleinian groups without parabolic elements are precisely the convex cocompact ones.

If \( \Gamma \) is a non-elementary geometrically finite Kleinian group, then the Patterson–Sullivan theorem~\cite[Theorem~1]{zbMATH03903608} states that
\begin{equation*}\label{Sullivan}
\delta(\Gamma) = \dim_{\mathcal{H}}(\Lambda(\Gamma)),
\end{equation*}
where \( \dim_{\mathcal{H}} \) denotes the Hausdorff dimension of the limit set \( \Lambda(\Gamma) \).

In the following, we  study the topological types of locally conformally flat \(4\)-manifolds with scalar-flat metrics. One reason this is interesting is that the Riemannian connection associated with a locally conformally flat scalar-flat metric achieves an absolute minimum of the Yang–Mills functional on a closed oriented \(4\)-manifold. Therefore, it is natural to study the topological types of closed \(4\)-manifolds \(M^4\) admitting locally conformally flat scalar-flat metrics.
Based on the topological classification of orientable, simply connected, closed \(4\)-manifolds, LeBrun and Maskit~\cite[Corollary~1.2]{zbMATH05319680} characterized the manifolds that admit scalar-flat anti-self-dual metrics as follows: A compact, simply connected topological \(4\)-manifold \(M\) admits a smooth structure supporting a scalar-flat anti-self-dual metric \(g\) if and only if \(M\) is homeomorphic to one of the following:
$K3$,  $\mathbb{C}P^2 \# k\, \overline{\mathbb{C}P}^2 \ (k \ge 10)$, or $ k\, \overline{\mathbb{C}P}^2 \ (k \ge 5).$

We now consider the case where the manifold \( M^4 \) is not necessarily simply connected.  By the Gauss–Bonnet–Chern formula, one has \(\chi(M^4) \le 0\), with equality if and only if \( M^4 \) admits a flat metric. In 1978, Brown, Bülow, Neubüser, Wondratschek, and Zassenhaus complete a computer-assisted classification of all isomorphism classes of \(4\)-dimensional crystallographic groups, thereby classifying closed flat \(4\)-manifolds \cite{MR484179}. They obtained \(74\) homeomorphism equivalence classes of closed flat \(4\)-manifolds, consisting of \(27\) orientable classes and \(47\) non-orientable classes.

From now on, suppose that \(\chi(M^4) < 0\), and let \(g\) be a locally conformally flat scalar-flat metric on \(M^4\). In this case, the Ricci tensor of \(g\) must be nonzero, since otherwise \(\chi(M^4) = 0\).
Thus, the metric \(g\) can be deformed to a metric \(g'\) with positive scalar curvature.
However, the metric \(g'\) may not have vanishing Weyl tensor,
because the existence of a locally conformally flat metric with positive scalar curvature implies that the second Betti number \(b_2(M)\) vanishes.

Moreover, even when \(b_2(M) = 0\), the metric \(g\) cannot, in general, be deformed to a locally conformally flat metric with positive scalar curvature, as the following example shows.

\begin{example}\label{example scalar flat}
Let \((\Sigma_g, g_H)\) be a hyperbolic surface that admits an orientation-reversing isometry \(r: \Sigma_g \to \Sigma_g\). Consider \((X, g) = (S^2 \times \Sigma_g, g_{st} \oplus g_H)\), and let \(A: (S^2, g_{st}) \to (S^2, g_{st})\) be the antipodal map,
i.e., the orientation-reversing fixed-point-free isometry.
Then \(F := A \times r\) acts freely and isometrically on
\((X, g) = (S^2 \times \Sigma_g, g_{st} \oplus g_H)\).
Since \(\mathrm{deg}(F)=\mathrm{deg}(A)\mathrm{deg}(r) = (-1)\cdot(-1) = +1\), the map \(F\) is orientation-preserving.

Let \(M := X / \langle F \rangle\).
Because \(F\) is an isometry, the metric \(g\) descends to a smooth metric \(\bar{g}\) on \(M\); and since both the locally conformally flat and scalar-flat properties are local, \((M, \bar{g})\) is still locally conformally flat and scalar-flat. For a free finite group action one has $H^2(M; \mathbb{Q}) \cong H^2(X; \mathbb{Q})^{\langle F \rangle}$ (the $G$-invariants of the $H^2(X; \mathbb{Q})$).
Now, $H^2(X; \mathbb{Q}) \cong H^2(S^2; \mathbb{Q}) \oplus H^2(\Sigma_g; \mathbb{Q}),$
which is generated by the area classes of the two factors.
Since both \(A\) and \(r\) reverse orientation on their respective factors,
\(F_*\) acts by \(-1\) on each summand.
Hence, the \(\langle F \rangle\)-invariant subspace of \(H^2(X; \mathbb{Q})\) is trivial, and therefore $b_2(M) = \dim H^2(M; \mathbb{Q}) = 0.$ Thus, \((M, \bar{g})\) is a closed, nonflat, locally conformally flat, scalar-flat \(4\)-manifold with \(b_2(M) = 0\).

The manifold \( M \) does not admit a locally conformally flat metric with positive scalar curvature.
Otherwise, a finite cover of \( M^4 \) would be diffeomorphic to \( k\#(S^1 \times S^3) \) for some \( k \ge 2 \).
Its universal cover \( \tilde{M} \) would then be homeomorphic to the complement of a Cantor set in \( S^4 \),
so its second homotopy group would vanish.
However, \( \tilde{M} \) is homeomorphic to \( S^2 \times \mathbb{H}^2 \), whose second homotopy group does not vanish — a contradiction.
\end{example}

For a closed, orientable, locally conformally flat \(2n\)-manifold $N^{2n}$ \((n \ge 2)\) with nonnegative scalar curvature,
Noronha~\cite[Theorem~2]{MR1235219} analyzes the possible holonomy groups case by case and shows that either \(b_n(N^{2n}) = 0\), or \(N^{2n}\) is covered by \(\mathbb{E}^{2n}\) or by \(S^n \times \mathbb{H}^n\).

Furthermore, by imposing the additional condition that, on a closed locally conformally flat scalar-flat \(4\)-manifold, the largest eigenvalue of the Ricci operator is not greater than the absolute value of its smallest eigenvalue, Noronha applies the Weitzenböck formula to \(2\)-forms to show that \(\nabla \mathrm{Rm} \equiv 0\) in that paper; that is, the manifold is locally symmetric, and is covered by either \(\mathbb{E}^4\) or \(S^2 \times \mathbb{H}^2\).

Noronha's results were motivated by the following question posed in her paper concerning compact \(4\)-manifolds:
\begin{quote}
If \( \pi_2(M^4) \neq 0 \) and \( M^4 \) admits a conformally flat metric with zero scalar curvature, is \( M^4 \) covered by \( S^2 \times \mathbb{H}^2 \)?
\end{quote}

We will give a positive answer to her question in the following theorem.

\begin{theorem} \label{the universal cover}
Let \( (M^4, g) \) be a closed, locally conformally flat, and scalar–flat \(4\)-manifold with \( \pi_2(M^4) \neq 0 \).  Then its Riemannian universal cover \( (\widetilde{M}, \tilde{g}) \) is isometric to $(\mathbb{H}^2  \times S^2 ,  g_{H} \oplus g_{st})$ up to homothety.
\end{theorem}

\begin{proof}
The manifold \( M^4 \) admits neither a flat metric nor a locally conformally flat metric with positive scalar curvature, since \( \pi_2(M^4) \neq 0 \). Identifying \( \pi_1(M) \) with its holonomy image \( \Gamma \), We have the diffeomorphisms
$M \cong \Omega / \Gamma$, $\widetilde{M} \cong \Omega = S^4 \setminus \Lambda(\Gamma).$

The group \( \Gamma \) is non-elementary.  Otherwise, \(\Lambda(\Gamma)\) would have at most two points, which would imply \(\pi_2(S^4 \setminus \Lambda(\Gamma)) = 0\).
However, by assumption, \(\pi_2(\widetilde{M}) \cong \pi_2(\Omega) = \pi_2(S^4 \setminus \Lambda(\Gamma)) \neq 0\), leading to a contradiction.

The group \( \Gamma \) is geometrically finite.
Since \( M \cong \Omega / \Gamma \) is compact and admits a locally conformally flat, scalar–flat metric, we have
\[
\dim_{\mathcal{H}}(\Lambda(\Gamma)) \le 1 < 4.
\]
Chang, Qing, and Yang~\cite[Theorem 0.1]{MR2070141} show that if \( \Gamma \) is a nonelementary, finitely generated, conformally finite  subgroup of \( \mathrm{Conf}(S^n) \), then \( \Gamma \) is geometrically finite if and only if \( \dim_\mathcal{H}(\Lambda(\Gamma)) < n \). Here,  conformally finite means that
$\Omega(\Gamma)/\Gamma$ is the disjoint union of a compact set and finitely many standard conformal cusp ends.  Since \( M \cong \Omega / \Gamma \) is the closed manifold, the group \( \Gamma \) is finitely generated and conformally finite. Combining this with \( \dim_{\mathcal{H}}(\Lambda(\Gamma)) \le 1 < 4 \), we conclude that \( \Gamma \) is geometrically finite.

Then, by the Patterson–Sullivan theorem, we have
$\dim_{\mathcal{H}}(\Lambda(\Gamma)) = \delta(\Lambda),$ where \( \delta(\Lambda) \) is the critical exponent of \( \Gamma \).
The existence of the metric \( g \) implies \( \delta(\Lambda) = 1 \) by Nayatani’s theorem~\cite[Corollary~3.4]{zbMATH01028179}.

The Čech–Alexander duality theorem states that for any nonempty compact subset
\( K \subset S^n \),
\[
\widetilde{H}_k(S^n \setminus K; G)
\cong
\check{\widetilde{H}}^{\,n-k-1}(K; G),
\]
that is, the reduced singular homology of the complement is isomorphic to the reduced Čech cohomology of the set,
for every abelian coefficient group \( G \) and every \( k \ge 0 \).

Since \( \widetilde{M} \) is simply connected and \( \pi_2(M) \neq 0 \),
by Hurewicz theorem, one has
\[
0\neq \pi_2(\widetilde{M}) \cong H_2(\widetilde{M}; \mathbb{Z}) \cong \widetilde{H}_2(S^4 \setminus \Lambda(\Gamma); \mathbb{Z})  \cong \check{\widetilde{H}}^{\,1}(\Lambda; \mathbb{Z})\cong \check{H}^{\,1}(\Lambda; \mathbb{Z}).
\]
Hence \( H_2(\widetilde{M}; \mathbb{Z}) \neq 0 \) implies $\check{H}^{\,1}(\Lambda; \mathbb{Z}) \neq 0.$

Recall the cohomological dimension \( \dim_G(X) \) of a topological space \( X \) with respect to an abelian group \( G \) as the largest integer \( n \) such that there exists a closed subset \( A \subset X \) with $\check{H}^{\,n}(X, A; G) \neq 0.$ That is,
\[
\dim_G X := \sup \{\, n : \exists\, A \subset X \text{ such that } \check{H}^{\,n}(X, A; G) \neq 0 \,\}.
\]

For any abelian group \( G \) and any compact space \( X \), one has
\[
\dim_G X \leq \dim_{\mathbb{Z}} X \leq \dim X,
\]
where \( \dim X \) denotes the topological (Lebesgue covering) dimension of \( X \).

In fact, by the Alexandroff theorem, if \( X \) is a compact metric space of finite topological dimension, then $\dim_{\mathbb{Z}} X = \dim X.$

Thus, \( \check{H}^{\,1}(\Lambda; \mathbb{Z}) \neq 0 \) implies that
\( \dim_{\mathbb{Z}}(\Lambda(\Gamma)) \geq 1 \),
and hence the topological dimension of the limit set satisfies
\( \dim(\Lambda(\Gamma)) \geq 1 \). (It also follows that \( \Gamma \) is non-elementary.)
Therefore,
\[
1
\le \dim(\Lambda(\Gamma))
\le \dim_{\mathcal{H}}(\Lambda(\Gamma))
\le 1
\;\implies\;
\dim(\Lambda(\Gamma))
= \dim_{\mathcal{H}}(\Lambda(\Gamma)) = 1.
\]

Finally, we need the following theorem of Kapovich~\cite[Theorem~1.3]{MR2491697}:

Suppose that \( \Gamma \subset \mathrm{Isom}(\mathbb{H}^n) \) is a nonelementary, geometrically finite group such that the Hausdorff dimension of its limit set equals its topological dimension \(d\).
Then the limit set of \( \Gamma \) is a round \(d\)-sphere.

Thus, in our case, \( \Lambda(\Gamma) \) is a round circle.  Consequently, \( \widetilde{M},\tilde{g}$ is conformal to $  (\Omega = S^4 \setminus S^1, g_{st}|_{S^4 \setminus S^1}) \).  By the following Lemma~\ref{Stereographic}, it follows that \( \widetilde{M} \) is conformal to $(\mathbb{H}^2  \times S^2 ,  g_{H} \oplus g_{st}).$ Finally, Lemma~\ref{iso} implies that \( \widetilde{M} \) is isometric to
\( (\mathbb{H}^{2} \times S^{2},\, g_{H} \oplus g_{st}) \) up to homothety.
\end{proof}

\begin{lemma}\label{Stereographic}
Let $m\ge 2$ and $1\le k\le m-2$. Then
$$(S^{m}\setminus S^{k}, g_{st}|_{S^{m}\setminus S^{k}})\ \cong_{\mathrm{conf}}\ (\mathbb H^{k+1}\times S^{\,m-k-1}, g_{H}\oplus g_{st}).$$
In particular, for $m=2n$ and $k=n-1$,
$S^{2n}\setminus S^{n-1}\ \cong_{\mathrm{conf}}\ \mathbb H^{n}\times S^{n}.$
\end{lemma}

\begin{proof}

Fix $p\in S^{k}\subset S^{m}$ and let $\sigma_p:\ S^{m}\setminus\{p\}\longrightarrow \mathbb R^{m}$ be stereographic projection from $p$. If a round subsphere passes through $p$,
its image under $\sigma_p$ is an affine linear subspace. Hence
$\Pi:=\sigma_p(S^{k}\setminus\{p\})$ is an affine $k$--plane in $\mathbb R^{m}$ and
\[
\sigma_p:\ S^{m}\setminus S^{k}\xrightarrow{\ \cong\ }\mathbb R^{m}\setminus \Pi.
\]
Choose orthogonal coordinates $\mathbb R^{m}=\mathbb R^{k}\oplus\mathbb R^{m-k}$ so that
$\Pi=\mathbb R^{k}\times\{0\}$ and write $x=(u,v)$ with $u\in\mathbb R^{k}$, $v\in\mathbb R^{m-k}$.
Then $\mathbb R^{m}\setminus\Pi\ \cong\ \mathbb R^{k}\times\big(\mathbb R^{m-k}\setminus\{0\}\big),$ and since $m-k\ge 2$, the complement is connected.

For convenience in this proof, denote by \( g_H = g_{\mathbb{H}^{k+1}} \) the hyperbolic metric on \( \mathbb{H}^{k+1} \), and by \( g_{\mathrm{st}} = g_{S^{m-k-1}} \) the standard round metric on \( S^{m-k-1} \). Write $v=\rho\,\omega$ with $\rho:=|v|>0$ and $\omega\in S^{m-k-1}$. The Euclidean metric splits as
\[
ds_{\mathbb E^{m}}^{2}=|du|^{2}+d\rho^{2}+\rho^{2}\,d\omega^{2}.
\]
Rescaling by $\rho^{-2}$ yields
\begin{equation*}\label{eq:key}
\rho^{-2}\,ds_{\mathbb E^{m}}^{2}
=\frac{|du|^{2}+d\rho^{2}}{\rho^{2}}+d\omega^{2}.
\end{equation*}
Identify $\big(\mathbb R^{k}\times(0,\infty), (|du|^{2}+d\rho^{2})/\rho^{2}\big)$ with the upper half--space
model of $\mathbb H^{k+1}$ and $(S^{m-k-1},d\omega^{2})$ with the round sphere. Thus
\[
\rho^{-2}\,ds_{\mathbb E^{m}}^{2} \;=\; g_{\mathbb H^{k+1}}\oplus g_{S^{m-k-1}}.
\]
Define
\[
J:\ \mathbb R^{m}\setminus\Pi\longrightarrow \mathbb H^{k+1}\times S^{m-k-1},\qquad
J(u,\rho,\omega)=\big((u,\rho),\,\omega\big),
\]
so that
\begin{equation}\label{eq:pull}
J^{*}\big(g_{\mathbb H^{k+1}}\oplus g_{S^{m-k-1}}\big)=\rho^{-2}\,ds_{\mathbb E^{m}}^{2}.
\end{equation}

Stereographic projection is conformal with
\begin{equation}\label{eq:stereo}
\sigma_p^{*}\,ds_{\mathbb E^{m}}^{2}
=\Big(\frac{1+|x|^{2}}{2}\Big)^{2} g_{S^{m}},\qquad x=(u,\rho\omega)=\sigma_p(y).
\end{equation}
Let $F:=J\circ\sigma_p:\ S^{m}\setminus S^{k}\to \mathbb H^{k+1}\times S^{m-k-1}$. Combining
\eqref{eq:pull} and \eqref{eq:stereo},
\[
\begin{aligned}
F^{*}\big(g_{\mathbb H^{k+1}}\oplus g_{S^{m-k-1}}\big)
&=(\sigma_p)^{*}\big(\rho^{-2}\,ds_{\mathbb E^{m}}^{2}\big)
=\rho^{-2}\,\sigma_p^{*} ds_{\mathbb E^{m}}^{2}\\
&=\Big(\frac{1+|x|^{2}}{2\rho}\Big)^{2} g_{S^{m}},
\end{aligned}
\]
a smooth positive multiple of $g_{S^{m}}$. Hence $F$ is a conformal diffeomorphism, proving
\[
S^{m}\setminus S^{k}\ \cong_{\mathrm{conf}}\ \mathbb H^{k+1}\times S^{m-k-1}.
\]
\end{proof}

\begin{lemma}\label{iso}
Suppose the Riemannian universal cover \( (\widetilde{M}, \tilde{g}) \) of a closed manifold \( (M^{2n}, g) \) ($n\geq 2$) with \( \mathrm{Sc}_{\tilde{g}} = 0 \)
is conformal to
$
\bigl( S^{n} \times \mathbb{H}^{n},\;
g_{0} := g_{st} \oplus g_{H} \bigr).$
Then there exists a constant \( c > 0 \) such that
$(\widetilde{M}, \tilde{g})$  is isometric to
$\bigl( S^{n} \times \mathbb{H}^{n},\, c^{2} g_{0} \bigr).
$
\end{lemma}

\begin{proof}
Let $F : (\widetilde M, \tilde g) \longrightarrow S^{n} \times \mathbb{H}^{n}$
be a conformal diffeomorphism.
Then there exists a smooth function
$\varphi : \widetilde M \longrightarrow \mathbb{R}$
such that
\begin{equation*}
F^{*}(g_{0}) = e^{-2\varphi}\,\tilde g .
\end{equation*}
Equivalently,
$\tilde g = e^{2\varphi}\, F^{*}(g_{0}).$

Since $\mathrm{Scal}(g_{0}) = 0$ and $\mathrm{Scal}(\tilde g)=0$, the standard conformal--change formula for the scalar curvature implies that
\[
\Delta_{\tilde g}\varphi
= \frac{m-2}{2}\,|\nabla\varphi|^{2}_{\tilde g} \;\geq\; 0
\quad\text{on } \widetilde M.
\]

Let $\Gamma = \pi_{1}(M)$ denote the group of deck transformations of the universal covering
$\widetilde M \to M$.
Each $\gamma \in \Gamma$ acts on $\widetilde M$ by an isometry of $(\widetilde M,\tilde g)$.
Define
\[
f_{\gamma} := F \circ \gamma \circ F^{-1} :
S^{n} \times \mathbb H^{n} \longrightarrow S^{n} \times \mathbb H^{n}.
\]
We now compute the conformal factor of $f_{\gamma}$ with respect to $g_{0}$.

Let $\psi := \varphi \circ F^{-1} : S^{n} \times \mathbb H^{n} \to \mathbb{R}$.
Then we can rewrite
\[
\tilde g = e^{2\varphi}\,F^{*}(g_{0})
\qquad\Longrightarrow\qquad
(F^{-1})^{*}\tilde g = e^{2\psi}\,g_{0}.
\]
Since $\gamma$ is an isometry of $(\widetilde M,\tilde g)$, we have $\gamma^{*}\tilde g = \tilde g$, and hence
\[
\gamma^{*}F^{*}g_{0}
= \gamma^{*}\bigl(e^{-2\varphi}\tilde g\bigr)
= e^{-2\varphi\circ\gamma}\,\tilde g.
\]
Using $f_{\gamma} = F\circ\gamma\circ F^{-1}$ and pulling back by $F^{-1}$, we obtain
\begin{align*}
f_{\gamma}^{*}g_{0}
&= (F\gamma F^{-1})^{*}g_{0}
= (F^{-1})^{*}\bigl(\gamma^{*}F^{*}g_{0}\bigr) \\
&= (F^{-1})^{*}\bigl(e^{-2\varphi\circ\gamma}\tilde g\bigr) \\
&= e^{-2(\varphi\circ\gamma)\circ F^{-1}} \,(F^{-1})^{*}\tilde g \\
&= e^{-2\psi\circ f_{\gamma}} \, e^{2\psi} g_{0}
= e^{2(\psi - \psi\circ f_{\gamma})}\,g_{0}.
\end{align*}
Thus each $f_{\gamma}$ is a conformal diffeomorphism of $(S^{n} \times \mathbb H^{n},g_{0})$ with conformal factor $e^{2(\psi - \psi\circ f_{\gamma})}$.

By a result of Jimenez and Tojeiro \cite[Cor.~2]{MR4374774}, every conformal diffeomorphism of a product
$\mathbb H^{k} \times S^{\ell}$ is in fact an isometry.
Applying this to $f_{\gamma}$, we conclude that $f_{\gamma}$ is an isometry of $(S^{n} \times \mathbb H^{n},g_{0})$ for every $\gamma \in \Gamma$.
Hence its conformal factor must be identically $1$, so that
\[
\psi \circ f_{\gamma} = \psi
\qquad \text{for all } \gamma\in\Gamma.
\]

Returning to $\varphi$ on $\widetilde M$, we note that
\[
\psi \circ f_{\gamma}
= \psi \quad\Longleftrightarrow\quad
(\varphi \circ F^{-1}) \circ (F\circ\gamma\circ F^{-1}) = \varphi \circ F^{-1}
\quad\Longleftrightarrow\quad
\varphi \circ \gamma = \varphi.
\]
Thus
\begin{equation*}
\varphi \circ \gamma = \varphi
\qquad\text{for all } \gamma \in \Gamma,
\end{equation*}
and $\varphi$ is $\Gamma$--invariant.
Therefore $\varphi$ descends to a smooth function on the closed manifold $M$.
Since $\varphi$ is $\Gamma$--invariant, there exists a smooth function
$\tilde\varphi : M \to \mathbb R$ such that
$\varphi = \tilde\varphi \circ \pi.$
Because $\pi : (\widetilde M,\tilde g) \to (M,g)$ is a Riemannian covering, it is a local isometry and
$\tilde g = \pi^{*}g$. In particular, for any smooth function $\tilde\varphi$ on $M$ one has
\[
\nabla_{\tilde g}(\tilde\varphi \circ \pi) = (\nabla_{g}\tilde\varphi)\circ \pi,
\qquad
\Delta_{\tilde g}(\tilde\varphi \circ \pi) = (\Delta_{g}\tilde\varphi)\circ \pi,
\]
and therefore
\[
|\nabla\varphi|^{2}_{\tilde g}
= |\nabla(\tilde\varphi\circ\pi)|^{2}_{\tilde g}
= \bigl(|\nabla\tilde\varphi|^{2}_{g}\bigr)\circ\pi.
\]
Substituting $\varphi = \tilde\varphi\circ\pi$ into the identity
\[
\Delta_{\tilde g}\varphi = \frac{m-2}{2}\,|\nabla\varphi|^{2}_{\tilde g}
\quad\text{on } \widetilde M
\]
we obtain
\[
\bigl(\Delta_{g}\tilde\varphi\bigr)\circ\pi
= \frac{m-2}{2}\,\bigl(|\nabla\tilde\varphi|^{2}_{g}\bigr)\circ\pi.
\]
Since $\pi$ is surjective, it follows that
\begin{equation*}
\Delta_{g}\tilde\varphi = \frac{m-2}{2}\,|\nabla\tilde\varphi|^{2}_{g}
\qquad\text{on } M.
\end{equation*}

Since $\tilde\varphi$ is now globally defined on $M$, we may integrate the identity
\[
\Delta_{g}\tilde\varphi = \frac{m-2}{2}\,|\nabla\tilde\varphi|^{2}_{ g}
\]
over $M$:
\[
0 = \int_{M} \Delta_{ g}\tilde\varphi\, d\mathrm{vol}_{ g}
= \frac{m-2}{2} \int_{M} |\nabla\tilde\varphi|^{2}_{g}\, d\mathrm{vol}_{ g}.
\]
It follows that $|\nabla\tilde\varphi|_{ g} \equiv 0$ on $M$, and hence $\varphi$ is constant on $\widetilde M$ as well.
Let $c := e^{\varphi} > 0$ denote this constant.
Thus, we obtain
\[
\tilde g = e^{2\varphi} F^{*} g_{0} = c^{2} F^{*}g_{0},
\]
and, upon identifying $\widetilde M$ with $S^{n} \times \mathbb H^{n}$ via $F$, we conclude
$\tilde g = c^{2} g_{0}.$
Thus $(\widetilde M,\tilde g)$ is isometric to $\bigl(S^{n} \times \mathbb H^{n}, c^{2} g_{0}\bigr)$,
as claimed.
\end{proof}

\begin{remark}
The condition \(\pi_2(M^4) \neq 0\) in Theorem~\ref{the universal cover} cannot be replaced by the nonexistence of a flat metric.
For example, LeBrun and Maskit~\cite[Proposition~3.3]{zbMATH05319680} show that there exists a smooth family of metrics \(h_t\) on \((S^1 \times S^3) \# (S^1 \times S^3)\), \(t \in [-1,1]\), such that for each \(t\), the metric \(h_t\) is locally conformally flat, with $Y(M, [h_1]) > 0$ and  $Y(M, [h_{-1}]) < 0.$ On the other hand, Große and Nardmann~\cite[Theorem~1]{MR3192307} show that if \(N\) is compact, then the Yamabe constant \(Y(N,[g])\) is continuous at \(g\) with respect to the compact-open \(C^2\)-topology.  Therefore, there exists \(s \in (-1,1)\) such that \(Y(M, [h_s]) = 0\).
Thus, \(h_s\) is conformal to a locally conformally flat scalar-flat metric.  However, the universal cover of \((S^1 \times S^3) \# (S^1 \times S^3)\) is homeomorphic to the complement of a Cantor set in \(S^4\). Its second fundamental group is trivial, so it is not of the form \(S^2 \times \mathbb{H}^2\).

 LeBrun and Maskit’s examples also show that the Hausdorff dimension of a Kleinian group depends not only on the group’s algebraic structure but also on its specific representation in the isometry group of hyperbolic space. For instance, for
\[
(M, h_1) = \bigl((S^1 \times S^3) \# (S^1 \times S^3),\, h_1\bigr),
\quad \Gamma_1 = \rho_1(\pi_1(M)),
\]
one has \( \dim_{\mathcal{H}}(\Gamma_1) < 1 \),
whereas for
\[
(M, h_s) = \bigl((S^1 \times S^3) \# (S^1 \times S^3),\, h_s\bigr),
\quad \Gamma_s = \rho_s(\pi_1(M)),
\]
one obtains \( \dim_{\mathcal{H}}(\Gamma_s) = 1 \).

\end{remark}

\begin{proof}[Proof of Theorem \ref{B}]
If the bundle is trivial, Proposition \ref{split4} implies the conclusion.
If the fiber or the base is $S^2$, Proposition \ref{S^2} implies the conclusion.
If the fiber is $S^1$, Proposition \ref{S^1-fiber} implies the conclusion.

Any smooth fiber bundle $\pi: M \to S^1$ with fiber $N$ is isomorphic to the mapping torus $M_\varphi$ of a diffeomorphism $\varphi: N \to N$.
Thus, when the base is $S^1$, $M^4$ is the mapping torus of $N^3$, and Propositions \ref{base manifold} and \ref{mapping torus} complete the proof.
\end{proof}

It remains an open problem to classify the topological types of closed $4$–manifolds that admit scalar-flat metrics but no metrics of positive scalar curvature. In other words, the topological classification of Ricci-flat $4$–manifolds is still not fully understood. In particular, no examples are known of closed simply connected Ricci-flat manifolds with generic holonomy, and the existence of a Ricci-flat metric on $S^n$ for $n \ge 4$ remains a major open problem.

\section{Rigidity theorems about scalar curvature}\label{4}

Obata \cite[Proposition 6.1.]{zbMATH03374588}  shows that if $g \in [g_{st}]$ with $Sc_g\equiv n(n-1)$ on $S^n$, then the metric $g$ is isometric to  $g_{st}$. In fact, if $(M^n, g)$ is Einstein with positive scalar curvature, then $g$ is the unique constant scalar curvature metric in its conformal class (up to  a scaling factor). However, on $S^n$, if the conformal class of a metric does not contain  an Einstein metric, then the number of constant positive scalar curvature metrics within  the conformal class may  not be unique up to  a scaling factor.

The above result of Obata is a clue for Obata's solution of the conjecture on conformal transformation of Riemannian manifold and the proof of the sharp Sobolev inequality on ($S^n,g_{st}$) (see, \cite[P. 121, Theorem 5.1]{MR1688256}), which means that every smooth function $f \in C^{\infty}(S^n)$ satisfies  the following inequality:
\begin{equation*}
(\int_{S^n}|f|^{\frac{2n}{n-2}}dv_{g_{st}})^\frac{n-2}{n}\leq \frac{4}{n(n-2)\mathrm{Vol}_{g_{st}}(S^n)^{\frac{2}{n}}}\int_{S^n}|\nabla f|^2_{g_{st}}dv_{g_{st}}  + \frac{1}{\mathrm{Vol}_{g_{st}}(S^n)^{\frac{2}{n}}} \int_{S^n}f^2dv_{g_{st}}.
\end{equation*}
Equality holds if and only if $f$ is a constant. Conversely, the sharp Sobolev inequality can be used to establish the following weaker version of Obata's result.

\begin{theorem}\label{LCF n}
Let ($M^n,g$) ($n\geq 3$) be a oriented, closed, simply connected, locally conformally flat manifold  with $\mathrm{Sc}_g = n(n-1)$, then \( \mathrm{Vol}_g(M) \geq \mathrm{Vol}_{g_{st}}(S^n) \). If the equality holds, then ($M^n,g$) is isometric to ($S^n,g_{st}$).
\end{theorem}

\begin{proof}
Since a closed, simply connected, locally conformally flat manifold is conformally diffeomorphic to the round sphere ($S^n, g_{st}$), the above assumptions guarantee the existence of a positive smooth function \( u \) on \( S^n \) and a diffeomorphism \( \phi: (S^n, g_{st}) \to (M,g) \) such that    $\phi^*g=u^{4/(n-2)}g_{st}$.  We continue to denote \( \phi^*g \) by \( g \).   Moreover, the assumption \( \mathrm{Sc}_g \equiv n(n-1) \), together with the formula for scalar curvature under a conformal transformation, implies that the function \( u \) must satisfy the following equation:
\begin{equation}\label{n(n-1)}
\frac{4(n-1)}{(n-2)}\Delta_{g_{st}}u  +  n(n-1)u =n(n-1)u^{\frac{n+2}{n-2}}.
\end{equation}

On the other hand, the function  $u$ also satisfies the sharp Sobolev inequality,
\begin{equation*}
(\int_{S^n}u^{\frac{2n}{n-2}}dv_{g_{st}})^\frac{n-2}{n}\leq \frac{1}{n(n-1)\mathrm{Vol}_{g_{st}}(S^n)^{\frac{2}{n}}}\int_{S^n}(\frac{4(n-1)}{n-2}|\nabla u|^2_{g_{st}}+n(n-1)u^2)dv_{g_{st}}.
\end{equation*}
Equality holds if and only if $u$ is a constant.  Multiplying equation (\ref{n(n-1)}) by \( u \) and integrating, we obtain:
\begin{equation*}
\int_{S^n}(\frac{4(n-1)}{n-2}|\nabla u|^2_{g_{st}}+n(n-1)u^2)dv_{g_{st}}=n(n-1)\int_{S^n}u^{\frac{2n}{n-2}}dv_{g_{st}}.
\end{equation*}
Hence, we have
\begin{equation}\label{Sobolev}
(\int_{S^n}u^{\frac{2n}{n-2}}dv_{g_{st}})^\frac{n-2}{n}\leq \frac{1}{\mathrm{Vol}_{g_{st}}(S^n)^{\frac{2}{n}}}\int_{S^n}u^{\frac{2n}{n-2}}dv_{g_{st}}.
\end{equation}

Since
\begin{equation*}
  \mathrm{Vol}_g(M)=\int_{S^n}u^{\frac{2n}{n-2}}dv_{g_{st}},
\end{equation*}
thus, inequality  (\ref{Sobolev}) means
\begin{equation*}
 \mathrm{Vol}_g(M)^\frac{n-2}{n} \leq \frac{\mathrm{Vol}_g(M)}{\mathrm{Vol}_{g_{st}}(S^n)^{\frac{2}{n}}}
\end{equation*}
That means \( \mathrm{Vol}_g(M) \geq \mathrm{Vol}_{g_{st}}(S^n) \). If \( \mathrm{Vol}_g(M) = \mathrm{Vol}_{g_{st}}(S^n) \), then the sharp Sobolev inequality must achieve equality, which implies \( u \equiv 1 \), i.e., \( (M^n, g) \) is isometric to \( (S^n, g_{st}) \).

\end{proof}

For metrics with negative scalar curvature, making the scalar curvature more negative tends to increase the volume locally. Globally, however, the behavior is more subtle.
In the study of the Yamabe problem, Schoen \cite{MR994021} conjectures that if \( g \) is any Riemannian metric on a closed hyperbolic \( n \)-manifold \((M^n, g_H)\) with \( n \ge 3 \) and its scalar curvature satisfies $\mathrm{Sc}_g \ge -n(n-1),$ then the total volume of \( M \) with respect to \( g \) is at least as large as the hyperbolic volume of \( M \). In dimension 3, Schoen's conjecture has been studied by Anderson \cite[page~132]{MR2213687}. In this paper, we give a partial confirmation of Schoen's conjecture in dimension 4 in the following cases.

\tableofcontents
\section{Introduction}
For metrics with negative scalar curvature, making the scalar curvature more negative tends to increase the volume locally. Globally, however, the behavior is more subtle.
In the study of the Yamabe problem, Schoen \cite{MR994021} conjectures that if \( g \) is any Riemannian metric on a closed hyperbolic \( n \)-manifold \((M^n, g_H)\) with \( n \ge 3 \) and its scalar curvature satisfies $\mathrm{Sc}_g \ge -n(n-1),$ then the total volume of \( M \) with respect to \( g \) is at least as large as the hyperbolic volume of \( M \). In dimension 3, Schoen's conjecture has been studied by Anderson \cite[page~132]{MR2213687}. In this paper, we give a partial confirmation of Schoen's conjecture in dimension 4 in the following cases.

The strategy of the proof is to reduce the problem to the Yamabe metric.
First, we establish a Schoen–conjecture–type volume inequality for the Yamabe metric of negative scalar curvature in the given conformal class.
Second, we apply the Gauss–Bonnet–Chern formula to the Yamabe metric to complete the argument.

\begin{theorem} \label{local}
Let \( g \) be a Riemannian metric with constant scalar curvature \( \mathrm{Sc}_g \equiv -12 \) on a closed, oriented hyperbolic 4-manifold \( (M^4, g_H) \). Assume that for every point \( x \in M^4 \), there exists a sufficiently small radius \( r(x) > 0 \) such that for all \( 0 < r \leq r(x) \), the volumes of the geodesic \( r \)-balls \( B_r(x) \) satisfy
\[
\mathrm{Vol}_g(B_r(x)) \geq \mathrm{Vol}_{g_H}(B_r(x)).
\]
Then it follows that
$ \mathrm{Vol}_g(M^4) \geq \mathrm{Vol}_{g_H}(M^4)$. Moreover, if equality holds, i.e., \( \mathrm{Vol}_g(M^4) = \mathrm{Vol}_{g_H}(M^4) \), then \( g \equiv g_H \).

\begin{corollary}
Let \( g \) be a Riemannian metric with constant scalar curvature \( \mathrm{Sc}_g \equiv 12 \) on a $(S^4,g_{st})$. Assume that for every point \( x \in S^4 \), there exists a sufficiently small radius \( r(x) > 0 \) such that for all \( 0 < r \leq r(x) \), the volumes of the geodesic \( r \)-balls \( B_r(x) \) satisfy
\[
\mathrm{Vol}_g(B_r(x)) \geq \mathrm{Vol}_{g_{st}}(B_r(x)).
\]
Then it follows that
$ \mathrm{Vol}_g(S^4) \geq \mathrm{Vol}_{g_{st}}(S^4)$. Moreover, if equality holds, i.e., \( \mathrm{Vol}_g(S^4) = \mathrm{Vol}_{g_{st}}(S^4) \), then \( g \) is isometric to $g_{st}$.

Motivated by Conjecture \ref{4D CSC}, we propose the following volume rigidity conjecture in higher dimensions:
\begin{conjecture}[Volume Rigidity Conjecture] \label{HYC}
 Assume a Riemannian metric $g$ on a closed hyperbolic $n$-manifold $(M^n,g_H)$ ($n\geq 5$) satisfies $\mathrm{Sc}_g=-n(n-1)$ and $\mathrm{Vol}_g(M^n)=\mathrm{Vol}_{g_H}(M^n)$, then the metric $g$ is isometric to the hyperbolic metric $g_H$.
\end{conjecture}

\begin{theorem}\label{Einstein}
Let $(N,g_N)$ be an orientable closed (Riemannian) Einstein $n$-manifold with scalar curvature $R^*<0$ and $(M,g_M)$ be an orientable closed Riemannian $n$-manifold with $\mathrm{Sc}_g\geq R^*$. Suppose there exists a smooth non-zero degree $1$-expansive map $f: (M,g_M) \to (N, g_N)$, i.e., $|f_*v|_{g_N}\geq |v|_{g_M}$ for $v\in TM$, and the map $f$ is harmonic with condition $C\leq 0$, then $f$ is a locally isometric map.

Inspired by Theorem \ref{Einstein}, we propose the following Llarull-type rigidity conjecture in the hyperbolic setting.
\begin{conjecture}[Hyperbolic Rigidity Conjecture]
Assume $(N^n,g)$ is an orientable closed $n$-manifold with $\mathrm{Sc}_g\geq -n(n-1)$ and $(M^n,g_H)$ be an orientable closed hyperbolic $n$-manifold. Suppose there exists a smooth non-zero degree $1$-expansive map $f: (N^n,g) \to (M^n, g_H)$, i.e., $|f_*v|_{g_H}\geq |v|_{g_N}$ for $v\in TN$, then $f$ is a isometric map.
\end{conjecture}

\begin{lemma}\label{iso}
Suppose the Riemannian universal cover \( (\widetilde{M}, \tilde{g}) \) of a closed manifold \( (M^{2n}, g) \) ($n\geq 2$) with \( \mathrm{Sc}_{\tilde{g}} = 0 \)
is conformal to
$
\bigl( S^{n} \times \mathbb{H}^{n},\;
g_{0} := g_{st} \oplus g_{H} \bigr).$
Then there exists a constant \( c > 0 \) such that
$(\widetilde{M}, \tilde{g})$ is isometric to
$\bigl( S^{n} \times \mathbb{H}^{n},\, c^{2} g_{0} \bigr).
$
\end{lemma}

The strategy of the proof is to reduce the problem to the Yamabe metric.
First, we establish a Schoen–conjecture–type volume inequality for the Yamabe metric of negative scalar curvature in the given conformal class.
Second, we apply the Gauss–Bonnet–Chern formula to the Yamabe metric to complete the argument.

Since any finitely presented group $\pi$ can be realized as the fundamental group of a closed Riemannian $4$--manifold with positive scalar curvature, additional assumptions are necessary in order to classify Riemannian $4$--manifolds admitting positive scalar curvature. For example, when the fundamental group is trivial, the combined results of Freedman and Donaldson lead to a classification, up to homeomorphism, of closed Riemannian $4$--manifolds that admit metrics of positive scalar curvature. Independently, by imposing the assumption that the manifold is the total space of a fiber bundle with fiber of positive dimension, one obtains the following classification.

\begin{theoremB}\label{B}
Let $M^4$ be a closed, oriented, smooth $4$–manifold which is the total space of a fiber bundle. Then:
\begin{enumerate}
\item[(1)] If either the fiber or the base is $S^2$, then $M^4$ admits a Riemannian metric of positive scalar curvature.
\item[(2)] If either the fiber or the base is a closed, oriented, connected $3$–manifold $N^3$, then $M^4$ admits a Riemannian metric of positive scalar curvature if and only if $N^3$ does.
\end{enumerate}
\end{theoremB}

When the fiber or the base is \( S^2 \), the proof relies on results concerning the group of orientation-preserving diffeomorphisms of oriented closed surfaces;
When the fiber is  \( N^3 \), the argument depends on the classification of \(3\)-manifolds admitting metrics of positive scalar curvature;
 When the base is $N^3$ and $N^3$ admits a metric of positive scalar curvature, we use the fact that the space of positive scalar curvature metrics on $N^3$ is path-connected in order to glue concordance metrics and thereby construct a metric of positive scalar curvature on $M^4$. The main result of Bamler and Kleiner \cite{2019arXiv190908710B} shows that the space of positive scalar curvature metrics on $N^3$ is contractible.
 Relying on this result, one can complete the proof.

On the other hand, closed locally conformally flat Riemannian $4$--manifolds with positive scalar curvature were classified by Hamilton, Chen, Tang, and Zhu \cite{zbMATH06081388} using the Ricci flow. A natural next step is to classify locally conformally flat $4$--manifolds admitting scalar-flat metrics. One motivation for this study is that the Riemannian connection associated with a locally conformally flat, scalar-flat metric attains an absolute minimum of the Yang--Mills functional on a closed oriented $4$--manifold.
 Note that for a locally conformally flat $4$-manifold $M^4$ with scalar-flat metric, its universal cover may not be the Euclidean space $\mathbb{E}^4$ or the product $(\mathbb{H}^2 \times S^2, g_{H} \oplus g_{st})$.

In 1993, Noronha \cite{MR1235219} asks whether, under the additional assumption that $\pi_2(M^4) \neq 0$, the universal cover of $M^4$ is $(\mathbb{H}^2 \times S^2, g_{H} \oplus g_{st})$.  She shows that if $b_2(M^4) > 0$, then $M^4$ is covered by $(\mathbb{H}^2 \times S^2, g_{H} \oplus g_{st})$ using Bochner-type formulas.  We provide a positive answer to Noronha's question in the following.

\begin{theoremC}[Theorem \ref{the universal cover}]\label{D}
Let \( (M^4, g) \) be a closed, locally conformally flat, and scalar–flat \(4\)-manifold with \( \pi_2(M^4) \neq 0 \).  Then its Riemannian universal cover \( (\widetilde{M}, \tilde{g}) \) is isometric to $(\mathbb{H}^2  \times S^2 ,  g_{H} \oplus g_{st})$ up to homothety.
\end{theoremC}

Using the Liouville theorem (Theorem \ref{Liouville theorem}) of Schoen and Yau \cite{MR4836036} together with the theorem of Chang–Qing–Yang \cite{MR2070141}, we show that $\pi_1(M^{4})$, viewed as a Kleinian group $\Gamma$, is geometrically finite.
It then follows from the Patterson–Sullivan theorem ~\cite{zbMATH03903608} and Nayatani's theorem \cite{zbMATH01028179} that the critical exponent satisfies
$\delta(\Gamma) = \dim_{\mathcal{H}}(\Lambda(\Gamma)) = 2.$
The condition \( \pi_2(M^4) \neq 0 \)  further implies, via Alexander duality, that the topological dimension satisfies $\dim(\Lambda(\Gamma)) \geq 2$.
Finally, by Kapovich's theorem \cite{MR2491697}, $\Lambda(\Gamma)$ is embedded as a round $2$-sphere, which completes the proof. The proof also holds for manifolds $M^{2n}$ ($n \geq 2$) if one replaces the condition $\pi_2(M^4) \neq 0$ by $\widetilde{H}_n(\widetilde{M}; \mathbb{Z}) \neq 0$.

\paragraph*{Organization of the Paper.}
In Section~\ref{2}, we partially verify Schoen's conjecture in dimension~4. In Section~\ref{3}, we  characterize Riemannian $4$–manifolds with positive scalar curvature that arise as the total space of a fiber bundle. In Section~\ref{4}, results and questions related to the rigidity of negative scalar curvature are  discussed. In Section~\ref{5},  we  characterize the universal cover of locally conformally flat scalar-flat $2n$–manifolds under an additional condition.

\paragraph*{Acknowledgment.}
The author acknowledges support from the Oberwolfach Leibniz Fellows programme (MFO), the YMSC Overseas Shuimu Scholarship, the Simons Center for Geometry and Physics, and ICMS Edinburgh (workshops on Geometric Measure Theory on Metric Spaces with Applications to Physics and Geometry and Geometric Moduli Spaces, respectively). I  thank Gerhard Huisken for discussions on the Ricci flow. This work originates from a broader project initiated during my postdoctoral stay at the Yau Center. During that time, this work was also supported by NSFC  12401063 and partially by  NSFC  12271284. I am deeply grateful to Shing-Tung Yau and Akito Futaki for their trust and support, which allowed me to pursue independent research.

To prove Theorem~\ref{A}, the proof strategy is to \emph{reduce the problem to the Yamabe metric}. First, the authors \emph{establish a Schoen--conjecture--type volume inequality} for the Yamabe metric of negative scalar curvature in the given conformal class. Second, they \emph{apply the Gauss--Bonnet--Chern formula} to the Yamabe metric to complete the argument.

To prove the main theorem, the proof strategy is to \emph{reduce the problem to the Yamabe metric}. First, the authors \emph{establish a Schoen--conjecture--type volume inequality} for the Yamabe metric of negative scalar curvature in the given conformal class. Second, they \emph{apply the Gauss--Bonnet--Chern formula} to the Yamabe metric to complete the argument.

\label{A}
Let \( (M^4, g_H) \) be a closed hyperbolic \(4\)-manifold.
Suppose \( g_1 \) is a Riemannian metric on \( M^4 \) with
\( \mathrm{Sc}_{g_1} \ge -12 \).
Assume in addition that one of the following holds:
\begin{enumerate}
    \item[(i)] \( g_1 \) is self–dual or anti–self–dual;
    \item[(ii)] the Yamabe metric \( g \in [g_1] \) with \( \mathrm{Sc}_g = -12 \)
    satisfies one of the conditions:
    \begin{enumerate}
        \item[(a)] \( g \) is Einstein;
        \item[(b)] for every point \( x \in (M^4, g) \), there exists
        \( r(x) > 0 \) such that for all \( 0 < r \le r(x) \), the geodesic \( r \)-balls satisfy $  \mathrm{Vol}_g(B^g_r(x)) \ge  \mathrm{Vol}_{g_H}(B^{g_H}_r(x)). $
    \end{enumerate}
\end{enumerate}

Then $\mathrm{Vol}_{g_1}(M^4) \geq \mathrm{Vol}_{g_H}(M^4)$. Moreover, if equality holds, i.e., \( \mathrm{Vol}_{g_1}(M^4) = \mathrm{Vol}_{g_H}(M^4) \), then  \( g_1 \) is isometric to \( g_H \).,

Question:
Let \((M^4,g_H)\) be a closed hyperbolic 4-manifold, where \(g_H\) is the hyperbolic metric. Let \(g_1\) be a Riemannian metric on \(M^4\) with scalar curvature \(\mathrm{Sc}_{g_1}\ge -12\). Assume in addition that either
(i) \(g_1\) is self-dual or anti-self-dual, or
(ii) if \(g\in[g_1]\) denotes the Yamabe metric in the conformal class of \(g_1\) with constant scalar curvature \(\mathrm{Sc}_g=-12\), then either
(a) \(g\) is Einstein, or
(b) for every point \(x\in M^4\), there exists \(r(x)>0\) such that for every \(0<r\le r(x)\), the geodesic balls satisfy
\[
\mathrm{Vol}_g(B_r^g(x))\ge \mathrm{Vol}_{g_H}(B_r^{g_H}(x)),
\]
where \(B_r^g(x)\) and \(B_r^{g_H}(x)\) are the radius-\(r\) geodesic balls centered at \(x\) in the metrics \(g\) and \(g_H\), respectively.

Which quantitative global volume statement holds under these assumptions?

\(\mathrm{Vol}_{g_1}(M^4)\ge \mathrm{Vol}_{g_H}(M^4)\). Moreover, if \(\mathrm{Vol}_{g_1}(M^4)=\mathrm{Vol}_{g_H}(M^4)\), then \(g_1\) is isometric to \(g_H\).

Choices:
B. \(\mathrm{Vol}_{g_1}(M^4)\le \mathrm{Vol}_{g_H}(M^4)\). Moreover, if \(\mathrm{Vol}_{g_1}(M^4)=\mathrm{Vol}_{g_H}(M^4)\), then \(g_1\) is isometric to \(g_H\).
C. \(\mathrm{Vol}_{g_1}(M^4)\ge \mathrm{Vol}_{g_H}(M^4)\).
D. \(\mathrm{Vol}_{g_1}(M^4)\ge \mathrm{Vol}_{g_H}(M^4)\) provided that in case \((ii)(b)\) there exists a single radius \(r_0>0\) such that for every point \(x\in M^4\) and every \(0<r\le r_0\), the geodesic balls satisfy \(\mathrm{Vol}_g(B_r^g(x))\ge \mathrm{Vol}_{g_H}(B_r^{g_H}(x))\). Moreover, if equality holds, then \(g_1\) is isometric to \(g_H\).
E. \(\mathrm{Vol}_{g}(M^4)\ge \mathrm{Vol}_{g_H}(M^4)\). Moreover, if \(\mathrm{Vol}_{g}(M^4)=\mathrm{Vol}_{g_H}(M^4)\), then the Yamabe metric \(g\) is isometric to \(g_H\).

direction of global volume inequality after Yamabe reduction

pointwise-dependent small-radius condition replaced by a uniform radius

global conclusion for \(g_1\) replaced by conclusion only for the Yamabe metric \(g\)

The stem does not explicitly reveal the correct option. It sets up a theorem-like hypothesis package, but the exact direction, sharpness, and rigidity conclusion still have to be chosen from the options.

This is essentially a theorem-completion item: the hypotheses are stated in full and the student is asked to identify the conclusion. That makes it very close to a direct restatement rather than an independently posed problem.

There is some reasoning pressure because the student must distinguish the exact sharp conclusion from nearby variants (wrong inequality direction, weaker true statement, altered quantifier, wrong metric in the conclusion). However, it mostly tests precise theorem recall rather than generative mathematical reasoning.

The distractors are strong and mathematically meaningful: B reverses the inequality, C gives a weaker true statement, D subtly changes the quantifier structure in the local ball condition, and E shifts the conclusion from g1 to the Yamabe metric g. These reflect plausible failure modes.

A solid theorem-recall MCQ with high-quality distractors and little direct answer leakage, but it is strongly tautological and only moderately tests genuine reasoning.


[{\cite[Conjecture 5]{motivatedintro}}]\label{conj}
    Let $G$ be a unipotent group over $\overline{\F}_q$. $G$ is easy if and only if it has trivial $\mathbb{L}$-packets.

The theory of character sheaves for unipotent groups was first conjectured by George Lusztig in \cite{lusztig2006}. This theory was later developed by Mitya Boyarchenko and Vladimir Drinfeld in \cite{foundations}, \cite{characters}, and \cite{chsheaves}. In 2006, Boyarchenko and Drinfeld proposed six initial conjectures in the development of this theory (\cite{motivatedintro}). Five of the six were proved in \cite{foundations} and \cite{chsheaves}. The goal of this paper is to complete the proof of the last remaining conjecture:

\begin{abstract}
 In 2006, Boyarchenko and Drinfeld conjectured that for a unipotent algebraic group over $\fqb$, every geometric point is contained in the neutral connected component of its centralizer if and only if its $\L$-packets of character sheaves are singletons. In 2013, Boyarchenko proved the ``only if" direction of this conjecture. In this paper, we complete the proof. Along the way, we explore the relationship between general algebraic groups satisfying this property and their Asai twisting operator.
\end{abstract}
\section*{Introduction}
The theory of character sheaves for unipotent groups was first conjectured by George Lusztig in \cite{lusztig2006}. This theory was later developed by Mitya Boyarchenko and Vladimir Drinfeld in \cite{foundations}, \cite{characters}, and \cite{chsheaves}. In 2006, Boyarchenko and Drinfeld proposed six initial conjectures in the development of this theory (\cite{motivatedintro}). Five of the six were proved in \cite{foundations} and \cite{chsheaves}. The goal of this paper is to complete the proof of the last remaining conjecture:

In \cite{chsheaves}, Boyarchenko proved the ``only if" direction. The following theorem completes the proof of the conjecture:

\begin{theorem}[{\cite[Theorem 1.15]{foundations}}]\label{edgg cat thm}
 Let $G$ be a unipotent algebraic group over $k$, and let $e\in \dgg$ be a minimal closed idempotent.
 \begin{itemize}
 \item[(a)]$\mathscr{M}_e^{perv}$ is a semisimple abelian category with finitely many simple objects. In particular, $\L$-packets of character sheaves on $G$ are finite.
 \item[(b)]There exists a (necessarily unique) integer $n_e$ such that $e[-n_e] \in \mathscr{M}_e^{perv}$. One has $0\le n_e \le \dim G$. The subcategory $\mathscr{M}_e := \mathscr{M}_e^{perv}[n_e]$ of the monoidal category $e\dgg$ is monoidal.
 \item[(c)]The perverse t-structure on $\dg$ induces a t-structure on $e\dgg$, and the canonical functor $D^b(\mathscr{M}_e^{perv}) \rightarrow e\dgg$ is an equivalence.
 \end{itemize}
\end{theorem}

\begin{prop}[{\cite[Proposition 4.4]{foundations}}]\label{ad pair prop}
 Let $G$ be a unipotent group over $k$ and $M \in \dg$, $M \ne 0$. Suppose that $\hl \in \mathscr{P}_{norm}(G)$ is maximal among all pairs $(A,\NN) \in \mathscr{P}_{norm}(G)$ that are compatible with $M$. If $\LL$ is invariant under the conjugation action of $G$, then the pair $(H,\LL)$ is admissible for $G$.
\end{prop}

\begin{lemma}[{\cite[Lemma 4.16]{characters}}]\label{easy twists}
 Let $G$ be an easy unipotent group over a field $k$ of characteristic $p>0$. Then, for every object $M \in \dgg$, its twist automorphism $\theta_M$ is trivial.
\end{lemma}

\begin{theorem}[{\cite[Theorem 1.8(b)]{chsheaves}}]\label{basis}
 Let $G$ be a connected unipotent group. The functions
 $$\{t_M : G^F \to \qlb \;| \;M\in CS(G)^F\} $$
 form a basis for the space $C(G^F/ \sim)$ which is orthonormal with respect to the inner product
 $$\langle f_1 \; | \;f_2\rangle = \sum_{g \in G^F}f_1(g) \overline{f_2(g)}.$$
\end{theorem}

\begin{theorem}\label{asai prop}
 Let $G$ be a connected algebraic group. The Asai twisting operator is trivial on $C(G^{F^m}/\sim)$ for all $m \in \Z_{>0}$ if and only if $G$ is easy.
\end{theorem}

From now on, let $G$ be a unipotent group over $\fqb$. The proof of \Cref{thm} relies on the structure of twists in the category $\dgg$ and their relationship to the Asai twisting operator.
\begin{lemma}\label{triv twists}
 If the $\L$-packet of character sheaves corresponding to $e$ is a singleton, then the twists in the Hecke subcategory $e\dgg$ are trivial.
\end{lemma}

\begin{theorem}\label{thm}
    If $G$ has trivial $\L$-packets, then $G$ is easy.
\end{theorem}

In \cite{chsheaves}, Boyarchenko proved the ``only if" direction. The following theorem completes the proof of the conjecture:

\begin{mainthm}[\ref{thm}]
    If $G$ is a unipotent group with trivial $\mathbb{L}$-packets, then $G$ is easy.
\end{mainthm}

The proof relies on the relationship between character sheaves and Shintani descent described by Deshpande in \cite{unipshint}.  In the process, we prove a statement about general algebraic groups:

\begin{mainthm} [\ref{asai prop}]  Let $G$ be a connected algebraic group over $\overline{\F}_q$. The Asai twisting operator is trivial on $C(G^{F^m}/\sim)$ for all $m \in \Z_{>0}$ if and only if $G$ is easy.
\end{mainthm}

\subsection*{Acknowledgments} I thank Tanmay Deshpande for generously sharing his expertise and ideas. I am grateful to my advisor, Charlotte Chan, for introducing me to the subject area, many helpful meetings, and comments on a previous draft. I also thank Sydney Mathematical Research Institute for excellent working conditions. This work was partially supported by NSF grants DMS-2507946 and DMS-1840234 (RTG) as well as a Simons Dissertation Fellowship (MPS-SDF-00014744).

The proof is said to rely on “the relationship between character sheaves and Shintani descent described by Deshpande in \cite{unipshint}.” In the process, the authors “prove a statement about general algebraic groups,” namely: for a connected algebraic group $G$ over $\overline{\F}_q$, “the Asai twisting operator is trivial on $C(G^{F^m}/\sim)$ for all $m\in\Z_{>0}$ if and only if $G$ is easy.”,

[{{\cite[Conjecture 5]{motivatedintro}}}]\label{conj}
    Let $G$ be a unipotent group over $\overline{\F}_q$. $G$ is easy if and only if it has trivial $\mathbb{L}$-packets.

Question: Let $G$ be a unipotent algebraic group over $\overline{\mathbb F}_q$. Call $G$ easy if for every geometric point $g$ of $G$, one has $g \in Z_G(g)^\circ$, where $Z_G(g)^\circ$ is the neutral connected component of the centralizer of $g$. Say that $G$ has trivial $\mathbb L$-packets if every $\mathbb L$-packet of character sheaves on $G$ is a singleton. Which statement holds for such a group $G$?

$G$ is easy if and only if it has trivial $\mathbb L$-packets.

Choices:
B. $G$ is easy if and only if at least one $\mathbb L$-packet of character sheaves on $G$ is a singleton.
C. If $G$ is easy, then it has trivial $\mathbb L$-packets.
D. If $G$ has trivial $\mathbb L$-packets, then the Asai twisting operator on $C(G^{F^m}/\sim)$ is trivial for some $m\in \mathbb Z_{>0}$, and hence $G$ is easy.
E. $G$ is easy if and only if, for every geometric point $g$ of $G$, the full centralizer $Z_G(g)$ is connected.

global singleton condition replaced by existence of one singleton packet

uniformity in the Asai criterion over all positive integers m

membership in the neutral connected component of the centralizer replaced by connectedness of the entire centralizer

The stem defines the two properties but does not explicitly reveal the equivalence. There is no direct verbal cue that singles out choice A.

The item is essentially asking for the exact theorem statement: whether 'easy' is equivalent to having trivial ℒ-packets. This is close to a direct restatement rather than an application or consequence.

There is some logical comparison among choices (equivalence vs one-way implication vs altered conditions), but the problem mainly tests recall/recognition of a theorem rather than generating a conclusion from mathematical reasoning.

The distractors are plausible and mathematically distinct: a weakened implication, a quantifier error, a technical but incorrect criterion involving the Asai operator, and a nearby but stronger connectedness condition. These reflect realistic failure modes.

A solid recall-style MCQ with strong distractors and no answer leakage, but it is largely theorem restatement and therefore only moderately effective at testing genuine generative reasoning.