\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. Suppose \(g_1\) is a Riemannian metric on \(M^4\) with scalar curvature \(\mathrm{Sc}_{g_1} \ge -12\). Assume in addition that at least one of the following is true: (i) \(g_1\) is self-dual or anti-self-dual; or (ii) the Yamabe metric \(g\in[g_1]\), meaning a metric conformal to \(g_1\) with constant scalar curvature \(\mathrm{Sc}_g=-12\), satisfies 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 \(r\)-balls in the two metrics satisfy
\[
\mathrm{Vol}_g\bigl(B_r^g(x)\bigr)\ge \mathrm{Vol}_{g_H}\bigl(B_r^{g_H}(x)\bigr),
\]
where \(B_r^g(x)\) and \(B_r^{g_H}(x)\) denote the radius-\(r\) geodesic balls centered at \(x\) with respect to \(g\) and \(g_H\), respectively. Under these assumptions, which statement about the total volume of \(M^4\) is valid?

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

Choices:
B. \(\mathrm{Vol}_{g_1}(M^4)\ge \mathrm{Vol}_{g_H}(M^4)\). Moreover, if \(\mathrm{Vol}_{g_1}(M^4)=\mathrm{Vol}_{g_H}(M^4)\), then the Yamabe metric \(g\in[g_1]\) 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 \(x\in M^4\), the geodesic \(r\)-balls satisfy \(\mathrm{Vol}_g(B_r^g(x))\ge \mathrm{Vol}_{g_H}(B_r^{g_H}(x))\) for all \(0<r\le r_0\); moreover, if equality holds, then \(g_1\) is isometric to \(g_H\).
E. \(\mathrm{Vol}_{g_1}(M^4)\ge \mathrm{Vol}_{g_H}(M^4)\) still follows if in \((ii)\) one only assumes that the Yamabe metric \(g\in[g_1]\) has constant scalar curvature satisfying \(\mathrm{Sc}_g\ge -12\) and either \(g\) is Einstein or, for every point \(x\in M^4\), there exists \(r(x)>0\) such that for all \(0<r\le r(x)\), \(\mathrm{Vol}_g(B_r^g(x))\ge \mathrm{Vol}_{g_H}(B_r^{g_H}(x))\); moreover, equality of total volumes forces \(g_1\) to be isometric to \(g_H\).

rigidity conclusion transferred from original metric to Yamabe metric only

dropped the rigidity/equality conclusion

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

exact normalization to the Yamabe metric with \(\mathrm{Sc}_g=-12\) weakened to \(\mathrm{Sc}_g\ge -12\)


[{\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 relationship between the geometry of manifolds and the analytic properties of functions on manifolds is a defining theme of geometric analysis. Our direction starts with the Liouville theorems for harmonic functions on \(\mathbb{R}^n\) and Yau's generalization.

Yau proved in \cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \(M\) with nonnegative Ricci curvature, the space \(\mathcal{H}_d(M)\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \cite{colding1997harmonic}.

A natural generalization is to try to show this result for solutions of the heat equation. However, the heat equation is very flexible compared to the Laplace equation, and since there are bounded solutions to the heat equation, no Liouville theorem is possible in general.

Despite this, if we restrict attention to specifically \textit{ancient} solutions of the heat equation, that is, solutions which are defined for all time going back to \(-\infty\), then Liouville theorems actually do become possible. Indeed, in \cite{colding2021optimal}, Colding and Minicozzi generalize \cite{colding1997harmonic} to show that the space \(\mathcal{P}_d(M)\) of ancient solutions of the heat equation with polynomially bounded growth also has finite dimension. In (\cite{colding2020complexity}, \cite{colding2019search}, \cite{colding2019liouville}), Colding and Minicozzi show how the spaces \(\mathcal{P}_d(M)\) are relevant to geometric flows.

Continuing to more types of equations, Wang and Zhu recently generalized the result of \cite{colding1997harmonic} to biharmonic functions \cite{wang2025qualitativebehaviorbiharmonicfunctions}, i.e., functions \(u: M \rightarrow \mathbb{R}\) solving
\begin{equation*}
    \Delta\Delta u = 0.
\end{equation*}

This equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.

Our goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \cite{colding2021optimal}. Our main result is
\begin{theorem*}
    Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(u: M \times (-\infty, 0] \rightarrow \mathbb{R}\) be an ancient solution of
    \begin{equation*}
        \partial_t u(x, t) + \Delta\Delta u(x, t) = 0
    \end{equation*}
    such that \(|u(x, t)|\) and \(|\nabla u(x, t)|\) have polynomially bounded growth in the heat balls \(B_R(x) \times [-R^4, 0]\). The space of all such solutions \(u(x, t)\) is finite dimensional.
\end{theorem*}

\subsection{Definitions and Notation}
We now give more precise definitions and statements. Given a manifold \(M\) and an interval \(I \subset \mathbb{R}\), a function \(u: M \times I \rightarrow \mathbb{R}\) satisfies the biharmonic heat equation if
\begin{equation}\label{first instance of biharmonic heat equation}
    \partial_t u(x, t) + \Delta\Delta u(x, t) = 0.
\end{equation}
We will call such a function ``bicaloric'' for brevity. A bicaloric function \(u\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \(t \in (-\infty, 0]\). We say that \(u \in \mathcal{P}_{d, d'}(M)\) for \(d, d' > 0\) if \(\partial_t u + \Delta\Delta u = 0\), \(u\) is ancient, and for some constants \(C, C' > 0\),
\begin{equation}\label{definition of polynomially bounded growth in introduction}
    \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}
for any \(p \in M\) and \(R > 0\). We similarly say that \(u \in \mathcal{H}_{d, d'}(M)\) if \(\Delta\Delta u = 0\) and the same bounds in (\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \(B_R(p)\).

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
    \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

With these definitions, our main results are more precisely stated as

Yau proved in \cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \(M\) with nonnegative Ricci curvature, the space \(\mathcal{H}_d(M)\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \cite{colding1997harmonic}.

This equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.

Our goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \cite{colding2021optimal}. Our main result is
\begin{theorem*}
 Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(u: M \times (-\infty, 0] \rightarrow \mathbb{R}\) be an ancient solution of
 \begin{equation*}
 \partial_t u(x, t) + \Delta\Delta u(x, t) = 0
 \end{equation*}
 such that \(|u(x, t)|\) and \(|\nabla u(x, t)|\) have polynomially bounded growth in the heat balls \(B_R(x) \times [-R^4, 0]\). The space of all such solutions \(u(x, t)\) is finite dimensional.
\end{theorem*}

\subsection{Definitions and Notation}
We now give more precise definitions and statements. Given a manifold \(M\) and an interval \(I \subset \mathbb{R}\), a function \(u: M \times I \rightarrow \mathbb{R}\) satisfies the biharmonic heat equation if
\begin{equation}\label{first instance of biharmonic heat equation}
 \partial_t u(x, t) + \Delta\Delta u(x, t) = 0.
\end{equation}
We will call such a function ``bicaloric'' for brevity. A bicaloric function \(u\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \(t \in (-\infty, 0]\). We say that \(u \in \mathcal{P}_{d, d'}(M)\) for \(d, d' > 0\) if \(\partial_t u + \Delta\Delta u = 0\), \(u\) is ancient, and for some constants \(C, C' > 0\),
\begin{equation}\label{definition of polynomially bounded growth in introduction}
 \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}
for any \(p \in M\) and \(R > 0\). We similarly say that \(u \in \mathcal{H}_{d, d'}(M)\) if \(\Delta\Delta u = 0\) and the same bounds in (\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \(B_R(p)\).

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
 \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

With these definitions, our main results are more precisely stated as

\begin{equation}\label{definition of polynomially bounded growth in introduction}
    \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
 \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

Combining this with Wang and Zhu's result \cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:

\begin{corollary}
 Let \(M\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \(k, \ell \ge 0\) the spaces \(\mathcal{P}_{4k, 4\ell}(M)\) are finite dimensional.
\end{corollary}

Since there are \(d + 1\) vectors \((1, t_i, \ldots, t_i^{d + 1})\), they span \(\mathbb{R}^{d + 1}\), and so there are constants \(b_i^j\) such that
 \begin{equation}\label{bdefinitions}
 e_j = b_i^j(1, t_i, \ldots, t_i^{d}).
 \end{equation}
 It now follows that
 \begin{equation}\label{expression of p in b}
 p_j(x) = b_i^ju(x, t_i), \quad \nabla p_j(x) = b_i^j\nabla u(x, t_i),
 \end{equation}
 and we conclude that \(p_j\) can grow at most polynomially of degree \(4k\) and \(\nabla p_j\) can grow at most polynomially of degree \(4\ell\). Because \(p_j\) vanishes when \(j > k\) and \(\nabla p_j\) vanishes when \(j > \ell + 1\), we have
 \begin{equation}
 u = p_0 + t p_1 + \ldots + t^{k}p_{k} \quad \text{and} \quad \nabla u = \nabla p_0 + t\nabla p_1 + \ldots + t^{\ell}\nabla p_{\ell},
 \end{equation}
 it follows that
 \begin{equation}
 \begin{split}
 |u(x, t)| \le C(1 + |t|^k + |x|^{4k}) \quad \text{and} \quad |\nabla u(x, t)| \le C(1 + |t|^{\ell} + |x|^{4\ell})
 \end{split}
 \end{equation}
 From equation (\ref{bdefinitions}) we have
 \begin{equation}
 \begin{split}
 \sum_i b_i^ju(x, R^4t_i) & = \sum_i\sum_{m}b_i^jp_{m}(x)R^{4j}t_i^{m} = R^{4j}\sum_mp_m(x)\left(\sum_ib_{i}^jt_i^m\right) \\ & = \sum R^{4j}\sum_m p_m(x)\delta_{mj} \\ & = R^{4j}p_j(x).
 \end{split}
 \end{equation}
 Similarly,
 \begin{equation}
 \sum_i b_i^j\nabla u(x, R^4t_i) = R^{4j}\nabla p_j(x)
 \end{equation}
 Thus
 \begin{equation}
 \begin{split}
 |R^{4j}p_j(x)| & = \left|\sum_i b_i^ju(x, R^4t_i)\right| \le A\sum_i\left|u(x, R^4t_i)\right| \\ & \le A(1 + |x|^{4k} + \sum_i|Rt_i|^{4d}) \le AR^{4k}
 \end{split}
 \end{equation}
 and similarly
 \begin{equation}
 \begin{split}
 |R^{4j}\nabla p_j(x)| & \le A'(1 + |x|^{4\ell} + \sum_i|Rt_i|^{4\ell}) \le A'R^{4\ell},
 \end{split}
 \end{equation}
 so that \(|p_j(x)| \le A_jR^{4(k - j)}\) and \(|p_j(x)| \le A_j'R^{4(\ell - j)}\).
\end{proof}
\begin{proof}[Proof of Theorem \ref{dimension bound theorem}]
 Choose some \(u \in {P}_{4k, 4\ell}(M)\) and suppose \(u = p_0(x) + tp_1(x) + \cdots + t^dp_d(x)\), as in lemma \ref{coefficientbound}. Then \(\Delta\Delta p_{d} = 0\) and for \(j < d\), \(\Delta\Delta p_j = -(j+1)p_{j + 1}\). Thus, there a linear map \(\Psi_0: \mathcal{P}_{4k, 4\ell} \rightarrow \mathcal{H}_{4k, 4\ell}\) defined by \(\Psi_0u = p_{d}\) (here we use the coefficient estimate in (\ref{coefficients in polynomial expression bounds})). If we let \(\mathcal{K}_0 = \ker \Psi_0\) we find
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell} \le \dim \mathcal{K}_0 + \dim \mathcal{H}_{4k, 4\ell}
 \end{equation}
 If \(u \in \mathcal{K}_0\), then \(p_{d} = 0\) and \(\Delta\Delta p_{d - 1} = -dp_{d} = 0\), and so we have a map \(\Psi_1: \mathcal{K}_0 \rightarrow \mathcal{H}_{4(k - 1), 4(\ell - 1)}\) defined by setting \(\Psi_1u = p_{d - 1}\). Letting \(\ker \Psi_1 = \mathcal{K}_1\) then
 \begin{equation}
 \dim \mathcal{K}_0 \le \dim \mathcal{K}_1 + \dim \mathcal{H}_{4k, 4\ell}
 \end{equation}
 When \(k \le \ell + 1\), we can repeat this \(k\) times to get
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell}(M) \le \sum_{i = 0}^{k} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M)
 \end{equation}
 When \(k > \ell + 1\), we have from lemma \ref{coefficientbound} that \(\nabla p_{\ell + 1} = 0\), and so \(p_{\ell + 1} = p_d\) is a constant. Thus in this case \(p_d\) lies in a one dimensional subspace of \(H_{4k, 4\ell}(M)\), so that
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell} = \dim \mathcal{K}_0 + 1.
 \end{equation}
 We then iterate the same argument as before, \(\ell\) times, to get the second inequality in (\ref{equation in the first statement of the main theorem}).

\end{proof}
\subsection{Bicaloric Polynomials in \(\mathbb{R}^n\)}
Now we consider polynomially bounded solutions to the biharmonic heat equation in \(\mathbb{R}^n\).
\begin{proposition}\label{bicaloric functions in Rn are polynomials}
 Let \(u \in \mathcal{P}_{k, \ell}(\mathbb{R}^n)\). Then \(u\) is a polynomial in \(x_i\) and \(t\).
\end{proposition}
\begin{proof}
 As before, this follows from the reverse Poincaré estimate and the fact that the operators \(\partial_{x_i}\), \(\partial_t\), and \(\Delta\) commute in \(\mathbb{R}^n\).
\end{proof}
\begin{corollary}
 If \(u \in \mathcal{P}_{k, \ell}(\mathbb{R}^n)\), then there is some \(d\) such that \(u \in \mathcal{P}_{d, d-1}(\mathbb{R}^n)\).
\end{corollary}\label{fewer caloric spaces in R^n}
\begin{proof}
 This follows the same way as before.
\end{proof}
Given a monomial in \(x_i\) and \(t\), we define its biparabolic degree as follows: for \(t^{n_0}\prod x_i^{n_i}\), the biparabolic degree is \(4n_0 + \sum n_i\). The degree of a polynomial is then the maximal degree of the monomials summing to it. Let \(\mathcal{A}_j^n\) be the set of homogeneous polynomials in \(\mathbb{R}^n\) of biparabolic degree \(j\). We have
\begin{equation}
 \mathcal{A}_d^n = A_d^n \oplus t A_{d - 4}^n \oplus t^2A_{d - 8}^n \oplus \cdots
\end{equation}
\begin{lemma}\label{dim for bicaloric polynomials}
 For \(d > 0\) we have \(\dim(\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n) = \dim A_d^n\) and
 \begin{equation}
 \dim \mathcal{P}_{d, d-1}(\mathbb{R}^n) = \sum_{j = 0}^d\dim A_j^n.
 \end{equation}
\end{lemma}
\begin{proof}
 Both \(\partial_t\) and \(\Delta\Delta\) map \(\mathcal{A}_d^n\) to \(\mathcal{A}_{d-4}^n\). We note that for \(u \in \mathcal{A}_{d-4}^n\) we have
 \begin{equation}
 (\partial_t + \Delta\Delta)\left(tu - \frac{1}{2}t^2(\partial_t + \Delta\Delta)u + \frac{1}{6}t^3(\partial_t + \Delta\Delta)^2 u - \cdots\right) = u,
 \end{equation}
 so that \(\partial_t + \Delta\Delta\) is surjective. Thus,
 \begin{equation}
 \dim (\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n) = \dim \mathcal{A}_d^n - \dim\mathcal{A}_{d-4}^n = \dim A_d^n
 \end{equation}
 since \(\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n\) is the kernel of \(\partial_t + \Delta\Delta\) restricted to \(\mathcal{A}_d^n\). Summing gives the second claim.

\end{proof}
Now finally we show that the estimate in Theorem \ref{dimension bound theorem} is sharp in \(\mathbb{R}^n\).
\begin{corollary}\label{sharp in Rn}
 For positive integers \(d > 0\)
 \begin{equation}
 \dim \mathcal{P}_{4d, 4d - 1}(\mathbb{R}^n) = \sum_{i = 0}^{d} \dim \mathcal{H}_{4(d - i), 4(d - i) - 1}(\mathbb{R}^n).
 \end{equation}
\end{corollary}
\begin{proof}
 From lemmas \ref{dim for bicaloric polynomials} and \ref{dim for biharmonic polynomials} we have
 \begin{equation}
 \begin{split}
 \dim \mathcal{P}_{4d, 4d-1}(\mathbb{R}^n) & = \sum_{j = 0}^{4d}\dim A_j^n = \sum_{j = 0}^d(\dim A_{4j}^n + \dim A_{4j - 1}^n + \dim A_{4j - 2}^n + \dim A_{4j - 3}^n) \\ & = \sum_{j = 0}^d\dim \mathcal{H}_{4d, 4d - 1}(\mathbb{R}^n).
 \end{split}
 \end{equation}
 Setting \(k = \ell = d\) in Theorem \ref{dimension bound theorem} and noting \(\mathcal{P}_{4d, 4d}(\mathbb{R}^n) = \mathcal{P}_{4d, 4d-1}(\mathbb{R}^n)\) and \(\mathcal{H}_{4d, 4d}(\mathbb{R}^n) = \mathcal{H}_{4d, 4d-1}(\mathbb{R}^n)\) shows that the estimate in Theorem \ref{dimension bound theorem} is sharp.

Combining this with Wang and Zhu's result \cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:

\begin{corollary}
    Let \(M\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \(k, \ell \ge 0\) the spaces \(\mathcal{P}_{4k, 4\ell}(M)\) are finite dimensional.
\end{corollary}

\subsection{Harmonic and biharmonic functions}

Biharmonic functions arise in several variational problems. Just as minimizing \(\int|\nabla u|^2\) leads one to the Laplace and heat equations, minimizing \(\int |\Delta u|^2\) leads to the biharmonic and biharmonic heat equations.

In general, fourth order elliptic operators arise naturally when taking variations involving second order objects, one major example being variations of metrics in conformal geometry (see \cite{chang1995extremal}, \cite{Lin1998classification}). They also arise in the study of the Willmore energy. For an immersed surface \(\phi: M^2 \rightarrow \mathbb{R}^3\), the Willmore energy is defined as
\begin{equation}\label{Willmore functional}
    \mathcal{W}(\phi) = \int_{M}H^2\,dA
\end{equation}
where \(dA\) is the induced volume element and \(H\) is the mean curvature \cite{willmore2000surfaces}. In studying critical points of this functional one arrives at the Euler-Lagrange equation
\begin{equation}\label{euler-lagrange equation}
    \Delta H + 2H(H^2 - K) = 0,
\end{equation}
a fourth order elliptic operator. The biharmonic heat equation similarly arises when studying the gradient flow of the Willmore energy (\cite{kuwert2002gradient}, \cite{lamm2005biharmonic}). Ancient solutions to heat equations often appear when doing blowup analysis of general solutions to a variational problem. See \cite{kuwert2004removable} for blowup analysis of singularities of Willmore flows. We also again reference (\cite{colding2020complexity}, \cite{colding2019search}, \cite{colding2019liouville}) for more on how ancient solutions to heat equations with polynomially bounded growth are relevant to geometric flows.

Although both arise from variational problems, biharmonic functions in general differ significantly from harmonic functions, because no maximum principle holds for biharmonic functions. This limits the kinds of estimates we can find for biharmonic functions. In particular, the usual pointwise derivative estimates one can find for harmonic functions on a ball cannot be found for a biharmonic function.

On the bright side, energy methods for harmonic and caloric functions seem to have analogs for biharmonic and bicaloric functions, which we will see as we prove Theorem \ref{dimension bound theorem}. We are still limited to some extent, however, because when performing integrations by parts we are forced to use the Bochner formula
\begin{equation*}
    \frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\end{equation*}
to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\). It is the appearance of the Ricci term here that makes the decay on Ricci curvature crucial for our result.

Our methodology is inspired by Colding and Minicozzi's in \cite{colding2021optimal}. We will show a reverse Poincaré inequality for bicaloric functions on ``heat balls'' \(B_R(p) \times [-R^4, 0]\). Because we are considering ancient bicaloric functions, we will be able to apply the inequality as \(R \rightarrow \infty\) to get strong, global control of their behavior. In particular we will see that high order time derivatives \(\partial_t^ku\) must vanish identically, allowing us to write for some finite \(d\):
\[u(x, t) = p_d(x)t^d + \cdots + p_1(x)t + p_0(x)\]
with \(\Delta \Delta p_d = 0\) and \(\Delta \Delta p_j = -(j + 1)p_{j + 1}\). This will allow us to directly compare the spaces \(\mathcal{H}_{4k, 4\ell}(M)\) with \(\mathcal{P}_{4k, 4\ell}(M)\).

To show the dimension estimates are sharp in \(\mathbb{R}^n\), we will consider biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions of the spaces \(\mathcal{H}_{4k, 4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k, 4\ell}(\mathbb{R}^n)\).

The introduction indicates the proof strategy for Theorem~\ref{dimension bound theorem} as follows.

1. **Use energy methods and Bochner to handle fourth order terms.** It says that while working "as we prove Theorem \ref{dimension bound theorem}" one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is "forced to use the Bochner formula"
\[
\frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\]
"to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\)," and "the appearance of the Ricci term here" is why the quadratic lower bound/decay assumption on Ricci is crucial.

2. **Establish a reverse Poincar\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will "show a reverse Poincar\'e inequality for bicaloric functions on `heat balls' \(B_R(p)\times[-R^4,0]\)."

3. **Send \(R\to\infty\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, "we will be able to apply the inequality as \(R\to\infty\) to get strong, global control of their behavior."

4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, "high order time derivatives \(\partial_t^k u\) must vanish identically," so for some finite \(d\)
\[
u(x,t)=p_d(x)t^d+\cdots+p_1(x)t+p_0(x),
\]
with relations "\(\Delta\Delta p_d=0\)" and "\(\Delta\Delta p_j=-(j+1)p_{j+1}\)."

5. **Compare \(\mathcal{H}_{4k,4\ell}(M)\) and \(\mathcal{P}_{4k,4\ell}(M)\) via the coefficient relations.** The stated expansion and recursions "will allow us to directly compare the spaces \(\mathcal{H}_{4k,4\ell}(M)\) with \(\mathcal{P}_{4k,4\ell}(M)\)," yielding the dimension bound.

6. **Sharpness in \(\mathbb{R}^n\) by explicit polynomial models.** To show sharpness, it says they will consider "biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions" of \(\mathcal{H}_{4k,4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k,4\ell}(\mathbb{R}^n)\).

The introduction indicates the proof strategy for the following theorem.

\begin{theorem}\label{dimension bound theorem}
    Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(k\), \(\ell\) be nonnegative integers. Then
    \begin{equation}\label{equation in the first statement of the main theorem}
        \dim \mathcal{P}_{4k, 4\ell}(M) \le \begin{cases}
             \displaystyle \sum_{i = 0}^{k} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M) & \quad k \le \ell + 1, \\ \displaystyle
            1 + \sum_{i = 0}^{\ell} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M) & \quad k > \ell + 1
        \end{cases}
    \end{equation}
    Moreover, these inequalities are sharp in \(\mathbb{R}^n\).
\end{theorem}

1. **Use energy methods and Bochner to handle fourth order terms.** In establishing the main theorem, one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is forced to use the Bochner formula
\[
\frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\]
so as to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\); the appearance of the Ricci term here is why the quadratic lower bound/decay assumption on Ricci is crucial.

2. **Establish a reverse Poincar\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will show a reverse Poincar\'e inequality for bicaloric functions on “heat balls” \(B_R(p)\times[-R^4,0]\).

3. **Send \(R\to\infty\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, one will be able to apply the inequality as \(R\to\infty\) to get strong, global control of their behavior.

4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, high order time derivatives \(\partial_t^k u\) must vanish identically, so for some finite \(d\)
\[
u(x,t)=p_d(x)t^d+\cdots+p_1(x)t+p_0(x),
\]
with relations “\(\Delta\Delta p_d=0\)” and “\(\Delta\Delta p_j=-(j+1)p_{j+1}\).”

5. **Compare \(\mathcal{H}_{4k,4\ell}(M)\) and \(\mathcal{P}_{4k,4\ell}(M)\) via the coefficient relations.** The stated expansion and recursions allow one to directly compare the spaces \(\mathcal{H}_{4k,4\ell}(M)\) with \(\mathcal{P}_{4k,4\ell}(M)\), yielding the dimension bound \eqref{equation in the first statement of the main theorem}.

6. **Sharpness in \(\mathbb{R}^n\) by explicit polynomial models.** To show sharpness, one considers biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling one to explicitly compute the dimensions of \(\mathcal{H}_{4k,4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k,4\ell}(\mathbb{R}^n)\).

Question:
Let \(M\) be a complete Riemannian manifold with polynomial volume growth, meaning that there exist constants \(C,d_V>0\) and a point \(p\in M\) such that \(\operatorname{Vol}(B_R(p))\le C(1+R)^{d_V}\) for all \(R>0\), and assume its Ricci curvature is bounded below quadratically, i.e. there exist \(K>0\) and \(p\in M\) such that for all \(R>0\),
\[
\sup_{v\in T B_R(p)} \frac{\operatorname{Ric}(v,v)}{|v|^2} \ge -\frac{K}{R^2}.
\]
For exponents \(d,d'\), let \(\mathcal P_{d,d'}(M)\) denote the space of ancient solutions \(u:M\times(-\infty,0]\to\mathbb R\) of the biharmonic heat equation
\[
\partial_t u+\Delta\Delta u=0
\]
such that for some constants \(C,C'>0\), for every point \(q\in M\) and every \(R>0\),
\[
\sup_{B_R(q)\times[-R^4,0]} |u(x,t)|\le C(1+R)^d,
\qquad
\sup_{B_R(q)\times[-R^4,0]} |\nabla u(x,t)|\le C'(1+R)^{d'}.
\]
Let \(\mathcal H_{d,d'}(M)\) denote the space of biharmonic functions \(v:M\to\mathbb R\) with \(\Delta\Delta v=0\) satisfying the analogous bounds over \(B_R(q)\). For nonnegative integers \(k\) and \(\ell\), which dimension estimate (including the Euclidean sharpness statement) holds?

For every nonnegative integers \(k,\ell\),
\[
\dim \mathcal P_{4k,4\ell}(M)\le
\begin{cases}
\displaystyle \sum_{i=0}^{k} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M), & \text{if } k\le \ell+1,\\[1ex]
\displaystyle 1+\sum_{i=0}^{\ell} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M), & \text{if } k>\ell+1.
\end{cases}
\]
Moreover, these inequalities are sharp when \(M=\mathbb R^n\).

Choices:
B.
For every nonnegative integers \(k,\ell\),
\[
\dim \mathcal P_{4k,4\ell}(M)\le
\sum_{i=0}^{\min\{k,\ell\}} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M).
\]
Moreover, this inequality is sharp when \(M=\mathbb R^n\).
C.
For every nonnegative integers \(k,\ell\),
\[
\dim \mathcal P_{4k,4\ell}(M)<\infty.
\]
Moreover, this finiteness statement holds when \(M=\mathbb R^n\).
D.
For every nonnegative integers \(k,\ell\),
\[
\dim \mathcal P_{4k,4\ell}(M)\le
\begin{cases}
\displaystyle \sum_{i=0}^{k} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M), & \text{if } k\le \ell,\\[1ex]
\displaystyle 1+\sum_{i=0}^{\ell} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M), & \text{if } k>\ell.
\end{cases}
\]
Moreover, these inequalities are sharp when \(M=\mathbb R^n\).
E.
For every nonnegative integers \(k,\ell\),
\[
\dim \mathcal P_{4k,4\ell}(M)=
\begin{cases}
\displaystyle \sum_{i=0}^{k} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M), & \text{if } k\le \ell+1,\\[1ex]
\displaystyle 1+\sum_{i=0}^{\ell} \dim \mathcal H_{4(k-i),\,4(\ell-i)}(M), & \text{if } k>\ell+1.
\end{cases}
\]
In particular, the same equalities hold for every such manifold \(M\), not only for \(M=\mathbb R^n\).