\begin{theorem}\label{thm_specialcase}
    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.
\end{theorem}

\subsection{Background}

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
 and has field of moduli a number field $L$ with at least one real embedding.
 Write $j_0:=j(C)\in L$ and $\Nm_{L/\bQ}(j_0) = \frac{n}{d}$ with $n,d\in \bZ, (n,d,6) = 1, d>0$.
 Assume at least one of the following conditions:
 \begin{enumerate}
 \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\bQ]+v_3(d\Nm_{L/\bQ}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
 \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
 \item $[L:\bQ]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
 \end{enumerate}
 Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

In each case, we are going to find some discriminant $D$ with a rational prime $p\notin \Nm_{L/\bQ}(S)$ such that $v_p(P_{D}(j_0)) > 0$ and $\left(\frac{D}{p}\right) \neq 1$, then as in \cite{MR2414789}, there is a prime $\fp \notin S$ above $p$, such that the Jacobian of $C$ has supersingular, and hence superspecial reduction.
 \begin{enumerate}
 \item Choose a prime $l$ satisfying the conditions:
 \begin{itemize}
 \item $l\equiv 13 \pmod{24}$,
 \item $\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-4l}(x)$ has a real root $j(a_i)$ in a subinterval of $(-\frac{16}{27},0)$ to be specified later.
 \end{itemize}
 Let $D = -4l$. By \cite{MR2414789}*{Lemma 6} and \Cref{lem:red23}, \begin{align*}
 P_{-4l}(x) &\equiv x^{h'} \pmod{4}, \\
 P_{-4l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-4l}(x) &\equiv (27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \leq 0$ for all $\mathfrak{p}$ above $2,3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)$, then
 $N, dQ\in \bN$ and $(N, 6) = 1,\, (dQ,2) = 1$.
 Suppose $l\nmid N$, then \begin{align*}
 \left(\frac{-4l}{N}\right) &= \left(\frac{-1}{N}\right)\left(\frac{l}{N}\right)= \left(\frac{-1}{N}\right)\left(\frac{N}{l}\right) \\
 &= \left(\frac{-1}{N}\right)\left(\frac{dQ}{l}\right) = \left(\frac{-1}{N}\right)\left(\frac{l}{dQ}\right)\\
 &= \left(\frac{-1}{N}\right)\left(\frac{-1}{dQ}\right) \left(\frac{-l}{dQ}\right) \\ &= \left(\frac{-1}{s(d\Nm_{L/\bQ} j_0)^{h'}}\right)\left(\frac{-1}{s'3^{[L:\bQ]}d\Nm_{L/\bQ}(j_0)}\right)\left(\frac{-l}{3^{v_3(dQ)}}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)}.\end{align*}
 If $[L:\bQ]+v_3(dQ)$ is odd, let \[j(a_1)\in\left(-\frac{16}{27}, -\frac{16}{27} + \min_{1\leq i \leq r} \left|j_i + \frac{16}{27}\right|\right)\]
 so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r$.
 If $[L:\bQ]+v_3(dQ)$ is even, by assumption let $j_t$ be the minimal real conjugate of $j_0$ in $( -\frac{16}{27} , 0)$, and \[j(a_1)\in\left(j_t, j_{t+1}\right)\] so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r, i\neq t$ and $(27j_t + 16) P_{-4l}(j_t) < 0$. In either case we have \[\left(\frac{-4l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-4l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.
 \item Choose a prime $l$ satifying the conditions:
 \begin{itemize}
 \item $l\equiv 19\pmod{24}$,
 \item $\left(\frac{q}{l}\right)=\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-l}(x)$ has a real root $j(a_1)$ in a subinterval of $(0, \infty)$ to be specified later.
 \end{itemize} Let $D = -l$. By \Cref{lem:red23}, \Cref{lem:pair}, and \cite{MR2414789}*{Lemma 4},
 \begin{align*}
 P_{-l}(x) &\equiv 1 \pmod{2}, \\
 P_{-l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-l}(x) &\equiv 3(27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$ and $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)\backslash\{2\}$, then
 $N, dQ\in \bN$ and $(N, 6) = 1$ .
 Suppose $l\nmid N$, then
 \begin{align*}
 \left(\frac{-l}{N}\right) &= \left(\frac{N}{l}\right) \\
 &= \left(\frac{s(d\Nm_{L/\bQ}(3(27j_0+16)))}{l}\right)\\
 &= \left(\frac{ss'3^{[L:\bQ]}dQ}{l}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)+v_2(Q)}
 \end{align*}
 Let $j_t$ be the minimal real conjugate of $j_0$ in $(0, \infty)$ and $n$ ($0\leq n < r$) be the number of real conjugates of $j_0$ in $(-\frac{16}{27}, 0)$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is odd, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is odd}{(0, j_t)}{$n$ is even}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)>0$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is even, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is even}{(0, j_t)}{$n$ is odd}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)<0$. In either case we have \[\left(\frac{-l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.

\begin{rem} \label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$.
For primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\cP_{D, \cE}$ avoids intersection with one of $\cP_{-3, \cE}$ and $\cP_{-4, \cE}$ by \Cref{lem:avoidintersection}.
The local conditions on $j_0$ are determined so that it avoids intersection with the these $\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of $P_D$ lie in the subinterval corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result.
\end{rem}
\bibliographystyle{amsalpha}
\bibliography{references}

More generally, the coarse moduli variety of principally polarized abelian surfaces with potential multiplication by the maximal quaternion order of discriminant $6$ is isomorphic to $\mathbb{P}^1$, and an arithmetic $j$-function (see \eqref{eq:j} in \cref{sec:coord}) is defined in \cite{MR2423455}. In terms of this coordinate, the assumption at primes $\mathfrak{p}$ above $2$ and $3$ in \Cref{thm_specialcase} corresponds to the case $v_{\mathfrak{p}}(j(C)) = 0$ by \cite{MR2414789}*{Proposition 2}, and our main theorem can be stated as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Write $j_0:=j(C)\in L$ and $\Nm_{L/\mathbb{Q}}(j_0) = \frac{n}{d}$ with $n,d\in \mathbb{Z}, (n,d,6) = 1, d>0$.
    Assume at least one of the following conditions:
    \begin{enumerate}
        \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\mathbb{Q}]+v_3(d\Nm_{L/\mathbb{Q}}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
        \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
        \item $[L:\mathbb{Q}]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
    \end{enumerate}
    Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

As in \cite{MR2414789}, we need to consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$. We use \cite{MR2704678} to generalize \cite{MR2414789} through a more detailed, case-by-case study, where in each case the discriminant $D$ is chosen in a form adapted to the conditions.
In characteristic $2$ and $3$, \cite{MR2414789} shows that the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$, and the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle. The local conditions on $j_0$ at primes above  $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes.
For more on the choice of local conditions and Heegner cycles, and for potential further weakenings of these conditions, see \Cref{rem:conclude}.

\subsection{Notation and conventions}

Assume the following unless specified otherwise.

Let $B = B_{\Delta}$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $\Delta$,  $\Lambda = \Lambda_{\Delta}$ be a maximal order of $B$, and $\Lambda^1 = \Lambda_{\Delta}^1$ be the group of units in $\Lambda$ of norm $1$. Fix an element $\mu \in \Lambda_{\Delta}$ such that $\mu^2 = -\Delta$,\footnote{The element $\mu$ exists because the field $\mathbb{Q}(\sqrt{-\Delta})$ embeds in $B$ by the local-global principle, and any two maximal orders in $B$ are conjugate to each other by strong approximation.} then the involution $\alpha \mapsto \alpha' = \mu^{-1} \bar{\alpha} \mu$ is a positive anti-involution on $B$.

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}), so “it therefore suffices to construct primes of supersingular reduction,” following “Elkies' general strategy.” Given $C$, one finds supersingular reduction of its Jacobian “from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$.” The intersection at a prime $p$ is “captured by $v_p(P_D(j_0))>0$,” where $P_D(x)$ is “the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$,” and when this occurs, supersingular reduction at $p$ is “detected by $\left(\frac{D}{p}\right) \neq 1$.”

As in \cite{MR2414789}, one must “consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$.” Using \cite{MR2704678}, the argument proceeds via “a more detailed, case-by-case study,” where “in each case the discriminant $D$ is chosen in a form adapted to the conditions.” In characteristic $2$ and $3$, \cite{MR2414789} shows “the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$,” and “the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle.” Finally, “the local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes,” yielding infinitely many primes of superspecial reduction.

Question: Let $C$ be a genus $2$ curve whose Jacobian $J(C)$ admits an action of the maximal order in the rational quaternion algebra of discriminant $6$, and suppose the field of moduli of $C$ is a number field $L$ having at least one real embedding. Assume moreover that for every prime of $L$ above $2$ or $3$, the curve $C$ has potentially smooth stable reduction there (that is, after a finite extension, its stable reduction becomes smooth). Under these hypotheses, which statement about the reduction of $J(C)$ holds? Here, saying that $J(C)$ has superspecial reduction at a prime means that its reduction modulo that prime is a superspecial abelian surface.

The Jacobian $J(C)$ has superspecial reduction at infinitely many primes.

Choices:
B. The Jacobian $J(C)$ has superspecial reduction at all but finitely many primes.
C. The Jacobian $J(C)$ has superspecial reduction at at least one prime.
D. There exists a finite extension $L'/L$ such that the base change $J(C)_{L'}$ has superspecial reduction at infinitely many primes of $L'$.
E. The Jacobian $J(C)$ has infinitely many primes of supersingular reduction.


\label{examples}
In this section, we provide a concrete example to show how specific  high  rank  elliptic curve of the form
$y^2=x(x^2+B)$  with a known set of rational points correspond to a single

\begin{thm} \label{main_theorem}
		Let $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \eqref{curve_main} with genus $g_{r,s}(C) \geq 1$.
	\end{thm}

A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \cite{Silverman1986}. Thus, $A(k) \cong A(k)_{\text{tors}} \oplus \mathbb{Z}^r$, where $A(k)_{\text{tors}}$ is the finite torsion subgroup and $r = \rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \geq 1$ defined over $k$.

Even for elliptic curves ($g=1$) over $k=\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.

An alternative approach, pioneered by Yamagishi \cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves
 $ y^s = ax^r+b$.

This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation
 \begin{equation} \label{curve_main}
 C_{a,b}: y^s=x(ax^r+b),
 \end{equation}
 where $r \geq 1, s \geq 2$ are fixed integers, and $k$ is a number field containing $\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.

\begin{conj}[Lang, \cite{Lang1986, Lang1991}]
		\label{conj1}
		Let $k$ be a number field and $X$ a smooth variety of general type over $k$.
		\begin{enumerate}
			\item[(a)] \textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.
			\item[(b)] \textbf{(Strong)} There exists a proper Zariski-closed subset $Z \subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \setminus Z$ is finite.
		\end{enumerate}
	\end{conj}

\begin{equation} \label{curve_main}
		C_{a,b}: y^s=x(ax^r+b),
	\end{equation}

\begin{abstract}
 Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$), and $r \geq 1, s \geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family.
 Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\Wc_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\Xc_n$, a complete intersection variety. The geometric properties of the fibers of $\Xc_n \to \Sym^{n+1}(\Pp^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it
 states that the rank of elliptic curves $y^2=x (x^2+B)$
 is uniformly bounded subject to the strong version of Lang's conjecture.
\end{abstract}

A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \cite{Silverman1986}. Thus, $A(k) \cong A(k)_{\text{tors}} \oplus \mathbb{Z}^r$, where $A(k)_{\text{tors}}$ is the finite torsion subgroup and $r = \rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \geq 1$ defined over $k$.

This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation
 \begin{equation} \label{curve_main}
 C_{a,b}: y^s=x(ax^r+b),
 \end{equation}
 where $r \geq 1, s \geq 2$ are fixed integers, and $k$ is a number field containing $\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.

The main result and its consequences provide new geometric evidence for rank boundedness conjectures for elliptic curves similar to that given in author's previous work \cite{Salami2025toappear}
different from that of given in \cite{Pasten2019}.
To be explicit, let us consider
the case $g_{r,s}(C) = 1$ and $k=\Q$, where the curve $C $ is an elliptic curves of the form $y^2= x(x^2+ b)$ with $\Z$.
By Theorem\ref{main_theorem}, one can conclude that the rank of elliptic curves
 is uniformly bounded subject to the strong version of Lang's conjecture, see Section \ref{sec:applications}.
 We note that the record of rank in this family is 14 due to M. Watkins obtained in 2002, according to \cite{DujellaRankRecord}.
 As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic curve of rank 8, see \cite{Watkins2014}.
Theorem\ref{main_theorem} implies the uniformity of rank in family of congruent number elliptic curves, subject to the strong version of Lang's conjecture, see Section \ref{sec:applications}.

\begin{thm} \label{uniformity_theorem}
 Assume the weak version of Lang's conjecture (Conjecture \ref{conj1}) holds over $k$. Let $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, and let $n_0$ be as in Theorem \ref{finiteness_theorem}. Then for $n \geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\#\Cc_{\ba_n}(k) < M_0$ for all valid choices of $\ba_n$. Furthermore, if $g_{r,s}(C) \geq 2$, there exists an integer $m_0 > n_0$ such that $\Cc_{\ba_m}(k) = \emptyset$ for all $m \geq m_0$ and all valid $\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)
 \end{thm}

\begin{proof}[Proof of Theorem \ref{main_theorem}]
 Let $k$ be a number field, $[k:\Q]=d$. Let $\mathcal{F}$ be the family of smooth curves $C: y^s=x(ax^r+b)$ over $k$ with genus $g=g_{r,s}(C) \geq 1$. Assume, for contradiction, that the rank $r(J_C(k))$ is unbounded for $C \in \mathcal{F}$.
 Then there exists a sequence of non-isomorphic curves $C_j \in \mathcal{F}$ such that $r_j = r(J_{C_j}(k)) \to \infty$ as $j \to \infty$.
 By the result of Dimitrov, Gao, and Habegger (Theorem \ref{dgh}), there is a constant $c=c(g,d)$ such that $\#C_j(k) \leq c^{1+r_j}$. Since $r_j \to \infty$, it follows that $\#C_j(k) \to \infty$.

Now, assume the Strong Lang Conjecture (Conjecture \ref{conj1}). This implies the strong version of Theorem \ref{uniformity_theorem}: there exists an integer $m_0$ (depending only on $r, s$, possibly $g$, but not $k$) such that for any $m \geq m_0$, $\Cc_{\ba_m}(K) = \emptyset$ for any number field $K$ and any valid $\ba_m \in \Uc_{r,m}(K)$. This means no smooth curve in the family \eqref{curve_main} over any number field $K$ can possess $m+1$ points $(\alpha_i, \beta_i)$ where the $\alpha_i$ are distinct, non-zero, and satisfy $\alpha_i^r \neq \alpha_j^r$, provided $m \ge m_0$.

\begin{thm}
 \label{CN-rank1}
 Assume the strong version of Lang's conjecture (Conjecture \ref{conj1}) holds. Then the Mordell-Weil ranks
$r(E_B(\Q))$ of elliptic curves $E_B: y^2=x (x^2+B)$ are uniformly bounded as $B$ varies over all integers.
\end{thm}

The main  result and its consequences provide new geometric evidence for rank boundedness  conjectures  for elliptic curves  similar to  that given in author's previous work \cite{Salami2025toappear}
different from that of given in \cite{Pasten2019}.
To be explicit, let us consider
the case $g_{r,s}(C) = 1$ and $k=\mathbb{Q}$, where the curve $C $ is an elliptic curves of the form  $y^2= x(x^2+ b)$ with  $\mathbb{Z}$.
By  Theorem\ref{main_theorem}, one can conclude that the  rank of elliptic curves
 is  uniformly bounded subject to the strong version of Lang's conjecture,   see Section \ref{sec:applications}.
    We note that  the record of rank in this family is 14 due to  M. Watkins obtained in 2002, according to \cite{DujellaRankRecord}.
 As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic  curve of rank 8, see   \cite{Watkins2014}.
Theorem\ref{main_theorem}  implies the uniformity of rank  in family of congruent number elliptic  curves, subject to the strong version of Lang's conjecture,   see Section \ref{sec:applications}.

	The proof of Theorem\ref{main_theorem} relies crucially on studying the set $\Cc_{\ba_n}(k)$ consisting of smooth curves $C$ in the family \eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\alpha_i \in k$ ($i=0,\dots,n$) where $\ba_n$ represents the collection $\{\alpha_0, \dots, \alpha_n\}$ of distinct, non-zero elements of $k$ satisfying $\alpha_i^r \neq \alpha_j^r$ for $i \neq j$. We establish the following finiteness and uniformity results for these sets.

	\begin{thm} \label{finiteness_theorem}
		Let $k$ be a number field containing $\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\ba_n = \{\alpha_0, \ldots, \alpha_n\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\alpha_i^r \neq \alpha_j^r$ for $i \neq j$, and for integers $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, the set $\Cc_{\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \geq n_0$. Consequently, for a fixed $\ba_n$ with $n \geq n_0$, the rank $r(J_C(k))$ is bounded for $C \in \Cc_{\ba_n}(k)$.
	\end{thm}

We use the uniformity results given this non-conditional result to discuss  a solution for the problem of arithmetic progression  on the points on the elliptic curves $y^2=x (x^2+B)$ as well as the congruent number elliptic curves, in Section~\ref{sec:applications}.

	Using the uniformity theorems on the number of $k$-rational points on curve of genus $\geq 2$ gibem in \cite{Caporaso1997}, as
	consequences of weak version of Lang's conjecture, we showed the cardinal number of $\Cc_{\ba_n}(k)$ does not depends on the choice of ${\ba_n}$ and  is   uniformly bounded and eventually empty for large  n.

	\begin{thm} \label{uniformity_theorem}
		Assume the weak version of Lang's conjecture (Conjecture \ref{conj1}) holds over $k$. Let $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, and let $n_0$ be as in Theorem \ref{finiteness_theorem}. Then for $n \geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\#\Cc_{\ba_n}(k) < M_0$ for all valid choices of $\ba_n$. Furthermore, if $g_{r,s}(C) \geq 2$, there exists an integer $m_0 > n_0$ such that $\Cc_{\ba_m}(k) = \emptyset$ for all $m \geq m_0$ and all valid $\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)
	\end{thm}

	The structure of the paper mirrors the logical progression of the proof. Section \ref{sec_geom_constr} details the construction of the parameter space $\Wc_n$ using twists and its birational model $\Xc_n$, a complete intersection variety, establishing their relationship. Section \ref{sec_fiber_analysis} focuses on the geometry of the fibers $\Xc_{\ba_n}$ of $\Xc_n \to \Sym^{n+1}(\mathbb{P}^1)$, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case, integrating necessary background results. Section \ref{sec_finiteness_uniformity_proofs} provides proofs for Theorems \ref{finiteness_theorem} and \ref{uniformity_theorem}, incorporating Faltings' theorem and Lang's conjectures. In Section \ref{sec_main_theorem_proof}, we prove  the main result, Theorem \ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.
	 Throughout, we cite analogous arguments from \cite{Salami2025toappear} where applicable.
	In Section~\ref{sec:applications}, we  have provided consequences of our results for   elliptic curves of the form $y^2=x(x^2+B)$ and  in particular the congruent number elliptic curves.
	 Finally, in   Section \ref{examples}, we have provided a computational example of the correspondence between
	the set of curves  $\Cc_{\ba_n}$ and rational points on the fiber $\Xc_{\ba_n}$.

The post-theorem introduction gives the following proof outline for Theorem\ref{main_theorem}.

- The proof \emph{“relies crucially on studying the set $\Cc_{\ba_n}(k)$ consisting of smooth curves $C$ in the family \eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\alpha_i\in k$”}, where $\ba_n=\{\alpha_0,\dots,\alpha_n\}$ are distinct nonzero elements with $\alpha_i^r\neq \alpha_j^r$ for $i\neq j$.

- One first \emph{“establish[es]… finiteness and uniformity results for these sets”} via Theorems \ref{finiteness_theorem} and \ref{uniformity_theorem} (finiteness for $n\ge n_0$; uniform bounds $\#\Cc_{\ba_n}(k)<M_0$ for all valid $\ba_n$, and eventual emptiness for large $n$ when $g_{r,s}(C)\ge 2$). The proofs of these results are said to be in Section~\ref{sec_finiteness_uniformity_proofs}, \emph{“incorporating Faltings' theorem and Lang's conjectures.”}

- Geometric input is developed beforehand: Section~\ref{sec_geom_constr} \emph{“details the construction of the parameter space $\Wc_n$ using twists and its birational model $\Xc_n$, a complete intersection variety”}; Section~\ref{sec_fiber_analysis} \emph{“focuses on the geometry of the fibers $\Xc_{\ba_n}$…, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case.”}

- Finally, \emph{“In Section \ref{sec_main_theorem_proof}, we prove the main result, Theorem \ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.”}

\label{main_theorem}
		Let $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture holds for varieties defined over $k$, namely:
		\begin{conj}[Lang, \cite{Lang1986, Lang1991}]
		\label{conj1}
		Let $k$ be a number field and $X$ a smooth variety of general type over $k$.
		\begin{enumerate}
			\item[(a)] \textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.
			\item[(b)] \textbf{(Strong)} There exists a proper Zariski-closed subset $Z \subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \setminus Z$ is finite.
		\end{enumerate}
	\end{conj}
		Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by the equation
		\begin{equation} \label{curve_main}
		C_{a,b}: y^s=x(ax^r+b),
	\end{equation}
		with genus $g_{r,s}(C) \geq 1%.

Question:
Let $k$ be a number field containing a primitive $s$-th root of unity, and fix integers $r\ge 1$ and $s\ge 2$. Assume the strong version of Lang's conjecture over $k$: for every smooth variety $X$ of general type over $k$, there exists a proper Zariski-closed subset $Z\subset X$ such that for every finite extension $K/k$, the set $(X\setminus Z)(K)$ is finite. For each smooth curve $C=C_{a,b}$ over $k$ given by
\[
C_{a,b}: y^s=x(ax^r+b),
\]
with genus $g_{r,s}(C)\ge 1$, let $J_C$ be its Jacobian and let $r(J_C(k))$ denote the Mordell-Weil rank of the finitely generated abelian group $J_C(k)$. Which conclusion about these ranks is valid under these hypotheses?

The Mordell-Weil ranks are uniformly bounded in this family: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, one has
\[
r(J_C(k))\le M.
\]
Equivalently, as $J_C$ varies over the Jacobians of all such curves, the ranks $r(J_C(k))$ remain bounded by a single constant depending only on the fixed data of the family.

Choices:
B.
The Mordell-Weil ranks are uniformly bounded in this family after allowing finite extensions: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, and for every finite extension $K/k$, one has
\[
r(J_C(K))\le M.
\]
Equivalently, as $J_C$ varies over the Jacobians of all such curves and $K$ varies over finite extensions of $k$, the ranks remain bounded by a single constant depending only on the fixed data of the family.
C. For each smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, the Mordell-Weil group $J_C(k)$ is finitely generated, so in particular the rank $r(J_C(k))$ is finite for every individual curve.
D.
The Mordell-Weil ranks are uniformly bounded in this family only for the subfamily of curves with genus $g_{r,s}(C)\ge 2$: there exists a constant $M$ such that every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 2$ satisfies
\[
r(J_C(k))\le M,
\]
while no conclusion is asserted for the genus-one members of the family.
E.
There exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, one has a uniform bound on rational points
\[
\#C(k)\le M.
\]
Equivalently, the family has uniformly bounded numbers of $k$-rational points, and hence the Jacobian ranks are bounded as well.


A classical conjecture of Geroch asserts that the $n$-torus $\mathbb{T}^n$ admits no Riemannian metric of positive scalar curvature. The conjecture (together with its rigidity case) was settled by Schoen--Yau for $3\le n\le 7$ via minimal hypersurfaces \cite{SY79}, and in all dimensions by Gromov--Lawson using the Dirac operator \cite{GL}. At the opposite end, Gromov asked whether a Riemannian metric $g$ on $S^n$ that strictly dominates the round metric $g_{S^n}$ must have scalar curvature strictly less than $n(n-1)$ somewhere. Llarull answered this by proving that if a closed spin $n$-manifold $(M^n,g)$ satisfies $R_M\ge n(n-1)$ and admits an area non-increasing map of nonzero degree to the round sphere, then rigidity holds in the strongest sense: the map is an isometry and hence $g$ is the pullback of the round metric \cite{Llarull}. Llarull's argument proceeds via spectral estimates for a twisted Dirac operator and index theoretic input tied to the nonzero degree. Subsequent work extended Llarull's theorem by lowering the regularity of the map \cite{Bar24, CHS24, LT22}, allowing targets beyond the sphere \cite{GS02, HSS24}, and treating manifolds with boundary under mean curvature assumptions \cite{BBHW24, CZ24, HLS, Lott21}.\\

These two results-—nonexistence and rigidity on tori, and Llarull-type rigidity on spheres--mark the extremes for scalar curvature on basic topological models. Between them lies the mixed geometry of products $S^{n-m}\times\mathbb{T}^m$. The product metric $g_{S^{n-m}}+ g_{\mathbb{T}^m}$ has scalar curvature exactly $(n-m)(n-m-1)$, the Llarull threshold for the spherical factor, yet it carries $m$ macroscopic flat directions encoding torus topology. Studying rigidity at this borderline illuminates how positive scalar curvature interacts with large-scale topology: it asks how much of the sphere’s rigidity persists once one permits $m$ flat directions, and conversely how enlargeability phenomena behind the torus obstruction constrain geometry when a positively curved factor is present. In this sense, $S^{n-m}\times\mathbb{T}^m$ is a natural ``midpoint" between the sphere and the torus, and scalar-curvature rigidity for degree–nonzero maps to this product probes the precise balance between curvature and topology.\\

In this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\times\mathbb{T}^m$ for $3\leq n\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:

\begin{abstract}
We prove Llarull-type rigidity for $S^{n-m}\times\T^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\T^m$ whose spherical projection is area non-increasing, and there exists $\psi\in C^\infty(M)$ with $-\Delta_M\psi-\tfrac12|D_M\psi|^2+\tfrac12\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\textup{dist}(\partial_-M,\partial_+M)\le 2\pi\sqrt{n/((n+1)\sigma)}$ when $R_M\ge (n-m)(n-m-1)+\sigma$. The method combines stable weighted slicing with a spectral Dirac operator argument.
\end{abstract}

In this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\times\T^m$ for $3\leq n\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:

Our approach combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$
with positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\ge 6$ was proved by different methods in \cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \cite{Tony}, which shows that when $\pr_{S^{n-m}}\circ\Phi:M^n\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\

\begin{maintheorem}\label{main;thm;2}
 Let $3\leq n+1\leq 7$ and $1\leq m\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\partial M = \partial_-M\sqcup\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\sigma > 0$ be a positive real number. Let $\Phi:(M,g)\to (S^{n-m}\times \T^m \times [-1,1], g_{S^{n-m}} + g_{\T^m} + dt^2)$ be a smooth map with the following properties:
 \begin{itemize}
 \item $\Phi$ has non-zero degree.
 \item $\Phi(\partial_{\pm}M)\subset S^{n-m}\times \T^m \times\{\pm1\}.$
 \item $\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\geq 3$ and 1-Lipschitz when $n-m=2$.
 \end{itemize}
The followings hold for the bandwidth of $M$:
\begin{enumerate}[(i)]
 \item Suppose that $R_M \geq (n-m)(n-m-1) + \sigma$. Then
 \[ \textup{dist}_g(\partial_-M, \partial_+M) \leq 2\pi\sqrt{\frac{n}{(n+1)\sigma}}.\]
 \item Suppose that $\psi$ is a smooth function on $M$ such that
 \[ -\Delta_M\psi -\frac{1}{2}|D_M\psi|^2 + \frac{1}{2}\Big(R_M - (n-m)(n-m-1) - \sigma \Big) \geq 0.\]
 Then
 \[ d(\partial_-M, \partial_+M) \leq \frac{2\pi}{\sqrt{\sigma}}.\]
\end{enumerate}
\end{maintheorem}

\begin{proposition}\label{slicing;exist;prop}
 Let $2\leq n\leq 7$ and $1\leq m\leq n-2$. Let $(M^n, g)$ be a closed, connected Riemannian manifold of dimension $n$ with a smooth map $\Phi:(M^n,g)\to (S^{n-m}\times \T^m, g_{S^{n-m}} + g_{\mathbb{T}^{m}})$ of non-zero degree such that $\pr_{S^{n-m}}\circ\Phi :(M^n,g)\to (S^{n-m}, g_{S^{n-m}})$ is 1-Lipschitz. Then there exists a stable weighted slicing
 \[ \Sigma_m\subset\Sigma_{m-1}\subset\cdots\subset\Sigma_1\subset\Sigma_0 = M^n\]
 of order $m$.

\begin{theorem}\label{spectral;Llarull;thm}
 Let $N^n$ be a closed, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $N$. Suppose that there exists a smooth function $\rho >0$ on $N$ such that
\begin{align}\label{spectral;Llarull;assumption}
 0 \leq& \int_{N}\rho|D_{N}f|^2 - \frac{1}{2}\int_{N}(n(n-1) - R_{N})\rho\, f^2 - \int_{N} \left(\Delta_{N}\log\rho + \frac{1}{2} |D_{N}\log\rho|^2
 \right) \rho\, f^2
\end{align}
for all $f\in C^{\infty}(N)$.
Suppose that $\Phi:(N,g)\to (S^n, g_{S^n})$ is a smooth map with the following properties:
 \begin{itemize}
 \item $\Phi$ has non-zero degree,
 \item $\Phi$ is 1-Lipschitz.
 \end{itemize}
 Then $\rho$ is a constant on $N$ and $\Phi$ is a Riemannian isometry.
\end{theorem}

\subsection{Proof of Theorem \ref{main;thm;2}}
In the first step, we construct a $\mu$-bubble that allows us to apply Theorem \ref{main;thm;1}. Denote the projection of $\Phi$ onto the factors by $\phi:M\to S^{n-m}\times \T^m$ and $\varphi:M\to [-1,1]$. By assumption, the map $\pr_{S^{n-m}}\circ\phi = \pr_{S^{n-m}}\circ\Phi$ is 1-Lipschitz. Let $\Theta$ be a top-form of $S^{n-m}\times\T^m$ such that $\int_{S^{n-m}\times\T^m}\Theta = 1$. Define the pull-back form $\omega = \phi^*\Theta$. By Sard's theorem, we can find a regular value $t_0\in (-1,1)$ of the map $\varphi$. Let $\hat{\Sigma} = \varphi^{-1}(t_0)$. Because $M$ is connected and $\varphi$ is continuous, and by assumption $\Phi(\partial_{-}M)\subset S^{n-m}\times \T^m \times\{-1\}$ and $\Phi(\partial_{+}M)\subset S^{n-m}\times \T^m \times\{1\}$, we see that $\varphi(M)$ is a connected subset of $[-1,1]$ containing both $-1$ and $1$. It follows that $\varphi(M) = [-1,1]$ and $ \varphi^{-1}(t)$ is non-empty for all regular values $t$. So $\hat{\Sigma}$ is a smooth, orientable and embedded hypersurface in $M$. By the coarea formula,
\begin{align}\label{bandwidth;eqn1}
 \deg(\Phi) = \int_M d\varphi\wedge\omega = \int_{-1}^1\left(\int_{\varphi^{-1}(t)}\omega\right) dt
\end{align}
On the other hand, if $t_1 < t_2$ are two regular values, the Stokes theorem gives
\begin{align}\label{bandwidth;eqn2}
 0 = \int_{\varphi^{-1}([t_1, t_2])} d\omega = \int_{\varphi^{-1}(t_2)}\omega - \int_{\varphi^{-1}(t_1)}\omega.
\end{align}
Putting (\ref{bandwidth;eqn1}) and (\ref{bandwidth;eqn2}) together, we obtain
\begin{align}\label{bandwidth;eqn3}
 \int_{\hat{\Sigma}}\omega = \deg(\Phi).
\end{align}

In conclusion, we have obtained a smooth map $\Psi:\Sigma^n\to S^{n-m}\times\T^m$ of non-zero degree such that $\pr_{S^{n-m}}\circ\Psi$ is 1-Lipschitz, and a smooth function $\log\rho$ on $\Sigma$ satisfying
\begin{align}\label{bandwidth;eqn6}
 -\Delta_{\Sigma}\log\rho -\frac{1}{2}|D_{\Sigma}\log\rho|^2 + \frac{1}{2}(R_{\Sigma} - (n-m)(n-m-1) ) > 0.
\end{align}
 Theorem \ref{main;thm;1} then implies $R_{\Sigma} = (n-m)(n-m-1)$ and $\rho$ is a constant function, contradicting to (\ref{bandwidth;eqn6}). This finishes the proof of Theorem \ref{main;thm;2}.

Our approach combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$
with positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\ge 6$ was proved by different methods in \cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \cite{Tony}, which shows that when $\pr_{S^{n-m}}\circ\Phi:M^n\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\

In the incomplete setting, we extend Gromov's torical band inequality \cite{Gro18,Gro19} to bands over sphere--torus products. Bands are among the most flexible test-objects for scalar curvature: they detect how lower bounds on scalar curvature constrain the macroscopic separation of boundary components, yielding distance-type obstructions to fill-ins, doubling, and collaring. Recent progress has deepened the connection between scalar curvature and band width; see for instance \cite{CZ24, HKKZ23, HKKZ25}. For targets of the form $S^{n-m}\times \mathbb{T}^m$, the interplay between a positively curved spherical factor and $m$ flat directions raises a natural quantitative question: how far apart can two boundary components be kept under a scalar curvature lower bound that matches the spherical threshold up to a gap $\sigma>0$? Our results below give sharp-in-scale ($\sim\sigma^{-1/2}$) upper bounds, reflecting the model behavior of constant-curvature metrics and extending the torical width control to the mixed sphere-torus regime. In particular, requiring the spherical projection to be area non-increasing  prevents macroscopic stretching in the directions where scalar curvature is most constrained, and the band-width estimates quantify this restriction. To that end, we prove:\\

\begin{maintheorem}\label{main;thm;2}
	Let $3\leq n+1\leq 7$ and $1\leq m\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\partial M = \partial_-M\sqcup\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\sigma > 0$ be a positive real number. Let  $\Phi:(M,g)\to (S^{n-m}\times \mathbb{T}^m \times [-1,1], g_{S^{n-m}} + g_{\mathbb{T}^m} + dt^2)$ be a smooth map with the following properties:
	\begin{itemize}
		\item $\Phi$ has non-zero degree.
		\item $\Phi(\partial_{\pm}M)\subset S^{n-m}\times \mathbb{T}^m \times\{\pm1\}.$
		\item $\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\geq 3$ and 1-Lipschitz when $n-m=2$.
	\end{itemize}
The followings hold for the bandwidth of $M$:
\begin{enumerate}[(i)]
    \item Suppose that $R_M \geq  (n-m)(n-m-1) + \sigma$. Then
    \[	\textup{dist}_g(\partial_-M, \partial_+M) \leq  2\pi\sqrt{\frac{n}{(n+1)\sigma}}.\]
    \item Suppose that $\psi$ is a smooth function on $M$ such that
	\[	-\Delta_M\psi -\frac{1}{2}|D_M\psi|^2 + \frac{1}{2}\Big(R_M - (n-m)(n-m-1) - \sigma \Big) \geq 0.\]
    Then
    \[	d(\partial_-M, \partial_+M) \leq  \frac{2\pi}{\sqrt{\sigma}}.\]
\end{enumerate}
\end{maintheorem}

\bigskip

\begin{remark}
For the sake of exposition we will henceforth work under the stronger hypothesis that
    \[\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})\]
is 1-Lipschitz. This assumption streamlines the notations used in the proof. All arguments, however, carry over verbatim when $n-m\geq 3$ if one merely assumes that the map is area non-increasing.
\end{remark}

To prove Theorem~\ref{main;thm;1}, the approach “combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\"ock-Lichnerowicz identity for spinors.” “Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”

To prove the main theorem, the approach “combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\"ock-Lichnerowicz identity for spinors.” “Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”

\label{main;thm;1}
	Let $3\leq n\leq 7$ and $1\leq m\leq n-2$. Let $M^n$ be a closed, orientable, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $M$. Let $\psi$ be a smooth function on $M$ such that
	\[	-\Delta_M\psi -\frac{1}{2}|D_M\psi|^2 + \frac{1}{2}\Big(R_M - (n-m)(n-m-1)\Big) \geq 0.\]
Suppose that $\Phi:(M,g)\to (S^{n-m}\times \mathbb{T}^m, g_{S^{n-m}} + g_{\mathbb{T}^m})$ is a smooth map with the following properties:
	\begin{itemize}
		\item $\Phi$ has non-zero degree,
		\item $\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\geq 3$ and 1-Lipschitz when $n-m=2$.
	\end{itemize}
	Then $(M^n, g)$ is isometrically covered by $(S^{n-m}\times\mathbb{R}^m,\, g_{S^{n-m}} + g_{\mathbb{R}^m})$.,

Question:
Let $3\le n\le 7$ and $1\le m\le n-2$. Let $M^n$ be a closed, orientable, connected spin $n$-manifold with Riemannian metric $g$, and let $\psi\in C^\infty(M)$ satisfy
\[
-\Delta_M\psi-\frac12|D_M\psi|^2+\frac12\bigl(R_M-(n-m)(n-m-1)\bigr)\ge 0.
\]
Assume there is a smooth map
\[
\Phi:(M,g)\to \bigl(S^{n-m}\times \mathbb T^m,\, g_{S^{n-m}}+g_{\mathbb T^m}\bigr)
\]
of nonzero degree, where $S^{n-m}$ has its round metric and $\mathbb T^m$ its flat torus metric, such that the projection onto the sphere factor,
\[
\operatorname{pr}_{S^{n-m}}\circ \Phi:(M,g)\to (S^{n-m},g_{S^{n-m}}),
\]
is area non-increasing when $n-m\ge 3$ and is $1$-Lipschitz when $n-m=2$. Under these assumptions, which conclusion about $(M,g)$ holds?

$(M,g)$ is isometrically covered by $\bigl(S^{n-m}\times \mathbb R^m,\, g_{S^{n-m}}+g_{\mathbb R^m}\bigr)$; equivalently, there exists a covering map from $S^{n-m}\times \mathbb R^m$ onto $M$ that is a local isometry for the product of the round metric on $S^{n-m}$ and the Euclidean metric on $\mathbb R^m$.

Choices:
B. $(M,g)$ is isometric to $\bigl(S^{n-m}\times \mathbb T^m,\, g_{S^{n-m}}+g_{\mathbb T^m}\bigr)$ itself; in particular, the given map $\Phi$ can be taken to be a Riemannian isometry.
C. The universal cover of $(M,g)$ splits isometrically as $\bigl(S^{n-m}\times \mathbb R^m,\, g_{S^{n-m}}+g_{\mathbb R^m}\bigr)$.
D. $(M,g)$ is isometrically covered by $\bigl(S^{n-m}\times \mathbb R^m,\, g_{S^{n-m}}+g_{\mathbb R^m}\bigr)$ provided that $\operatorname{pr}_{S^{n-m}}\circ\Phi$ is $1$-Lipschitz; in fact the same conclusion holds without distinguishing the cases $n-m\ge 3$ and $n-m=2$.
E. $(M,g)$ is isometrically covered by $\bigl(S^{n-m}\times N^m,\, g_{S^{n-m}}+h\bigr)$ for some complete flat $m$-manifold $(N^m,h)$, but one cannot in general conclude that the cover is $S^{n-m}\times \mathbb R^m$ with the Euclidean metric on the second factor.

replace explicit isometric covering statement by universal-cover splitting

dimension-dependent hypothesis on spherical projection

rigid upward propagation forcing Euclidean torus directions


\label{thm:main}
 Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.
  Assume that
  $$
 h_\nu(\phi)<\htop(\phi)
 $$
for all non-expansive measures
$\nu \in \MMM_\phi(SM)$.   Then the geodesic flow has a unique measure $\mu$ of maximal entropy.

 Furthermore, $\mu(\mathcal{E}) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.

\begin{proof}
   For a proof of (1), (2), (3a) see \cite[Lemma III.H 3.2, Lemma III.H 3.3]{BH99}. The assertion (3b) is a consequence
   of (3a) and Lemma \ref{lem:endpts-suffice}.
\end{proof}
Gromov introduced  in \cite{mG87} a natural topology on $\partial X$ and  on $ \bar X :=X \cup \partial X$  such that  $ \bar X $ is compact (see  also \cite[Def. III.H 3.5]{BH99}).
We will give a definition in the geometric context in subsection \ref{subsec:Noconjugatepts-Ghyp}.

\subsection{Gromov hyperbolic groups}\label{subsec:hypgroups}
\begin{definition}\label{def:Hypgroups}
Let $G$ be a finitely generated group and denote by $\CC(G, S)$ the Cayley graph of $G$ with respect to some finite generating set $S \subset G$. Then  $G$ is called Gromov hyperbolic if $\CC(G, S)$ equipped with the word metric
is a $\delta$-hyperbolic metric space for some $\delta\ge 0$
\end{definition}
\begin{remark}
\begin{itemize}
\item
If $S_1, S_2$ are finite generating sets of $G$ then the identity $\id: \CC(G, S_1) \to  \CC(G, S_2)$ is a quasi isometry.
In particular, the definition of Gromov hyperbolicity of a finitely generated group does not depend on the choice of the finite generating set.
\item Let $\Gamma$ be a group acting properly, cocompactly and isometrically on some geodesic metric space $X$.
Then $\Gamma$ is finitely generated and for any reference point $x_0 \in X$ and finite set of generators the orbit map $\Gamma \to X$ with
$\gamma \mapsto \gamma x_0$ extends to a quasi isometry between $\CC(G, S)$ and $X$ (see e.g. \cite[Prop. I.8.19]{BH99}).
In particular,  $\Gamma$ is Gromov hyperbolic if and only if $X$ is Gromov hyperbolic.
\end{itemize}
\end{remark}

Assume that $\Gamma$ is a Gromov hyperbolic group acting properly and cocompactly
by isometries on  proper metric space $X$. Consider for  $x\in X$ and $R>0$ the subset of  $\Gamma$ given by
$$
\Gamma_R(x) =  \{\gamma \in \Gamma \mid\; d(x,\gamma x) \leq R\}.
$$
 Then by a result of Coornaert \cite{mCo93} there are  constants $0 <C_1 \le C_2$ and $h>0$ such that
\begin{equation}\label{eqn:co93}
C_1 e^{hR}\le \card \ \Gamma_R(x) \le C_2 e^{hR}
\end{equation}
If $(X,g)$ is a Riemannian manifold then $h$ is the volume entropy $\hvol(g)$, i.e.
$$
h =  \lim\limits_{r \to \infty} \frac{\log \mathrm{vol} B(p,r)}{r}
$$
Define the translation length of $\gamma \in \Gamma$ by
 \begin{equation} \label{e: length}
\ell(\gamma) = \inf_{x \in X} d(x,\gamma x).
\end{equation}
Note that the infimum in
\eqref{e: length} is attained
for each $\gamma \in \Gamma$ (see
for example \cite[Prop. II.6.10]{BH99}). Furthermore, $\ell(\gamma^{-1}) = \ell(\gamma)$ and
$\ell(\alpha\gamma\alpha^{-1}) = \ell(\gamma)$ for
every isometry $\alpha$ of $X$. Therefore the length of a conjugacy class $[\gamma]$  of $\gamma$  can be defined as the translation length of a representative of  $[\gamma]$.
Using ideas developed  in \cite{gK83} and \cite{gK97} we proved in  \cite{CK02}:
\begin{theorem} \label{thm:conj-Grhyp}
Let $\Gamma$ be a group acting
properly and cocompactly by isometries on
a proper geodesic  Gromov-hyperbolic
metric space  $X$. Assume
that the Gromov boundary $\partial X$ of $X$
contains more than two points.
  For all $t \geq 0$, let
$$
P(t) = \card \{ [\gamma] \mid\; \gamma \in \Gamma \;  \; \text{is primitive and }   \; \; \ell(\gamma) \leq t \},
$$
where $ \gamma \in \Gamma$ is called primitive if it cannot be written as a proper power $\gamma = \alpha^n$ for some $\alpha \in \Gamma$ and $ n \ge 2$.

  Then there exists
constants   $A\ge 1$,  $t_0 >0$ and such that
$$
\frac{1}{A} \frac{ e^{ht} } {t} \leq P(t) \leq A e^{ht}
$$
for all $t \geq t_0$, where $h>0$  is as in \ref{eqn:co93}.
\end{theorem}
\begin{remark} \label{rem:conj-Grhyp}

This results has the following interpretation in the Riemannian setting.
Assume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group
$\pi_1(M)$. Let $\wM$ be the universal covering then $M = \wM/ \Gamma$ where  $\Gamma$   is the group of covering transformations isomorphic to $\pi_1(M)$.
As mentioned above
$h$ is the volume entropy $\hvol(g)$  of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.
By a result of Manning \cite{aM79}, the topological entropy $\htop(\phi)$ of the geodesic flow $\phi^t$ on $SM$ is larger than the volume entropy.

\end{remark}

\subsection{Closed manifolds without conjugate points and  Gromov hyperbolic fundamental groups}\label{subsec:Noconjugatepts-Ghyp}
In the following we will assume that $(M,g)$ is a closed Riemannina manifold without conjugate points and Gromov hyperbolic fundamental
group $\pi_1(M)$. As pointed out above in this case Freir\'{e}  and  Ma\~n\'e  \cite{FM82}
proved that
the volume entropy $\hvol(g)$ is equal to the topological entropy $\htop(\phi)$.

First we recall the following theorem which is foundational in the theory
of manifolds without conjugate points.
\begin{theorem}(Hadamard-Cartan)\label{thm: Hadamard-Cartan}\\
Let $(M,g)$ be a complete $n$-dimensional Riemannian manifold with
no conjugate points. Then for all $p \in M$ the exponential map
$\exp_p: T_pM \to M$ is a covering map.
In particular complete simply connected manifolds without conjugate points are diffeomorph to
$\R^n$.
\end{theorem}
\begin{remark}
\begin{enumerate}
\item[{\rm (a)}] A complete Riemannian manifold $(M,g)$ has no conjugate points iff for any pair of points on the universal cover $\wM$ there is a unique connecting geodesic geodesics with respect to the lifted metric. In particular all geodesics are minimizing.
\item[{\rm (b)}] The topology of those manifolds is to a large extend determined by the
fundamental group since the contractibility of the universal cover implies that the higher
homotopy groups are vanishing, i.e., $\pi_k(M) = 0$ for $k \ge 2$.
\item[{\rm (c)}] Manifolds of non-positive sectional curvature form an important subclass of manifolds with no conjugate points. Simply connected complete manifolds of nonpositive curvature are called Hadamard manifolds.

\end{enumerate}
\end{remark}

We assume that $(M,g)$ satisfies the divergence property, i.e.
 for any pair of geodesics  $c_1 \neq c_2$ in $( \wM, g)$ with $c_1(0) =c_2(0)$
we have
\begin{equation}\label{eqn:divergence}
\lim\limits_{t \to \infty} d(c_1(t), c_2(t)) = \infty.
\end{equation}
\begin{remark}
\begin{itemize}
\item
By a result of E. Hopf \cite{eH48} all closed non-flat surfaces without conjugate points have genus at least two. Since such surfaces
carry a metric with negative curvature their fundamental groups are Gromov hyperbolic. Furthermore,  Green \cite{wG56} showed that surfaces without conjugate have the divergence property.
\item
Until now there is no example of a closed manifold without conjugate points known where the divergence property does not hold. A sufficient condition for the divergence property is the continuity of the stable Jacobi-tensors (see \cite{ES76}). In particular, this assumption holds if $(M,g)$ has non-positive curvature, or more generally no focal points.
\end{itemize}
\end{remark}
  The following notion was introduced by Eberlein  in \cite{pEb72} and  Eberlein and  O'Neill \cite{EO73}.
\begin{definition}
A simply connected Riemannian manifold $\wM$ without conjugate points
is a \emph{(uniform) visibility manifold} if for every $\epsilon>0$ there exists $L>0$ such that whenever a geodesic $c\colon [a, b] \to\wM$ stays at distance at least $L$ from some point $p\in\wM$, then the angle sustained by $c$ at $p$ is less than $\epsilon$, that is
  \begin{equation*}
    \angle_p(c)=\sup_{a\leq s,t\leq b} \angle_p(c(s),c(t))<\epsilon.
  \end{equation*}
\end{definition}
The following Theorem is due to Ruggiero \cite{rR03},
  \begin{theorem}\label{thm:visibility}
Let $(M,g)$ be a closed manifold without conjugate points and Gromov hyperbolic fundamental group. Then $( \wM, g)$ is a visibility manifold if and only if $(M,g)$ has the divergence property.
\end{theorem}
\begin{remark}
Under the stronger assumption that $(M,g)$ is a closed manifold without conjugate points admitting a background metric of negative curvature this has been proved by Eberlein  \cite{pEb72}.
\end{remark}
Since $\wM$ is Gromov hyperbolic,  Lemma \ref{lem:GB} and the divergence property implies that the map
$f_p\colon S_p\wM \to \ideal$ defined by $f_p(v) = c_v(\infty)$ is a bijection.
The topology (sphere-topology) on $\partial \wM$ is defined such that $f_p$
becomes a homeomorphism.
Since for all $q \in \wM$ the map $f_q^{-1} f_p \colon S_p\wM \to S_q\wM$ is a homeomorphism, see \cite{pEb72},
the topology is independent on the reference point $p$.
The topologies on $\partial \wM$ and $\wM$
extend naturally
 to  $\cl (\wM): =  \wM\cup \partial \wM$
by requiring that the map
$\varphi\colon B_1(p) = \{v \in T_p \wM:   \|v\| \le 1\} \to \cl(\wM)$
defined by
\[
\varphi(v) = \begin{cases}
 \exp_p\left(\frac{v}{1-\|v\|}\right) & \|v\| < 1\\
f_p(v) & \|v\| = 1
\end{cases}
\]
is a homeomorphism. This topology, called the cone topology, was introduced by Eberlein
and O'Neill \cite{EO73} in the case of Hadamard manifolds and by Eberlein   \cite{pEb72} in the case of visibility manifolds. In
particular, $\cl( \wM) $ is homeomorphic to a closed ball in
$\mathbb{R}^n$. The relative topology on $\ideal$ coincides with the sphere topology, and the relative topology on $\wM$ coincides with the topology of $\wM$.

Note that for the case which we are considering
this compactification agrees with the compactification of  $\delta$-hyperbolic spaces do to Gromov \cite{mG87}.

For simply connected manifolds $X$ without conjugate points for $v\in SX$, the limit
\[
b_{v}(q) := \lim_{t\to\infty} \left( d(q, c_{v}(t)) - t\right)
\]
exists and is called the \emph{Busemann function} associated to $v$. In \cite{gK85} it was shown that Busemann functions are of class $C^{1,1}$ provided the sectional curvature is uniformly bounded from below.
\begin{definition}\label{def:busemann}
Let $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group.
Then according to Lemma \ref{lem:GB} and the divergence property,
for each $p\in X$ and $\xi \in \partial X$ there exists a uniquely determined vector $v\in S_p X$  such that $c_{v}(\infty) = \xi$.
We call $b_\xi(q,p) := b_{v}(q)$ the Busemann function based at $\xi$
and normalized by $b_\xi(p,p) =0$.
\end{definition}

\begin{remark}\label{rem:minmal}
The isometric action of $\Gamma=\pi_1(M)$ on $\wM$ extends to a continuous action on $\ideal$.
Since by \cite{pEb72} the geodesic flow is topologically transitive, every $\Gamma$-orbit in $\ideal$ is dense, i.e. the action on  $\ideal$ is minimal.
\end{remark}

\begin{proposition}\label{cor:busemann}
Let $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group.
Then the following holds
\begin{enumerate}
\item
For $p,q \in \wM$ and $\xi \in \ideal$ we have
$$
\lim\limits_{z \to \xi }d(q,z) -d(p,z) = b_p(q, \xi)
$$
\item
For all $p,q,z\in \wM$ we have
$$b_{q}(z,\xi) = b_p(z,\xi) - b_p(q,\xi)$$
In particular $b_{q}(z,\xi) = -  b_z(q,\xi)$
\end{enumerate}
\end{proposition}

\begin{proof}
For a proof see e.g. \cite{CKW21}
\end{proof}

\begin{MainThm}\label{thm:expansiveMME}
 Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov
 hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \delta$, where $\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow
 has a unique measure of maximal entropy $\mu$.

Furthermore, $\mu(\EE) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.
\end{MainThm}

\begin{MainThm}\label{thm:main}
 Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.
  Assume that
  $$
 h_\nu(\phi)<\htop(\phi)
 $$
for all non-expansive measures
$\nu \in \MMM_\phi(SM)$.   Then the geodesic flow has a unique measure $\mu$ of maximal entropy.

 Furthermore, $\mu(\EE) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.

\end{MainThm}

\label{sec:intro}
If $f: X \to X$ is a homeomorphism of a compact metric space $X$,
the topological entropy $h_{\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an
exponential scale. On the other hand, measure-theoretic entropy $h_{\mu}(f)$ is an invariant of
measurable dynamics reflecting the average complexity of the system relative
to a measure $\mu$ contained in the space of $f$-invariant Borel probability measure $\MMM_f(X)$ on $X$.
The relation between these notions is provided by the variational principle which implies (see e.g. \cite{pW82} for details)
$$
h_{\mathrm{top}} (f) = \sup \{h_\mu (f) \;|\; \mu \in \MMM_f(X)\}.
$$
A measure $\mu \in \MMM_f(X)$ such that $h_{\mu}(f) =h_{\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).
For continuous flows $\phi^t: X \to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\phi := \phi^1$.

In the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.
If $(M,g)$ is a closed Riemannian manifold without conjugate points
(all geodesics are globally minimizing if lifted to the universal cover $\widetilde{M}$) then by a theorem of Freir\'{e} and Ma\~n\'e \cite{FM82} the topological entropy $\htop (\phi)$ of the geodesic
flow $\phi^t: SM \to SM$ agrees with the volume entropy $\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \subset \widetilde{M}$ in the universal covering $\widetilde{M}$ of $M$ and is given by
$$
\hvol(g) = \lim\limits_{r \to \infty} \frac{\log \mathrm{vol} B(p,r)}{r}
$$
For manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points,
an easier proof of the result of Freir\'{e} and Ma\~n\'e were previously given by Manning \cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points
is not a local condition. Properties like the convexity of distance functions which
 hold for manifolds of non-positive sectional curvature and even no focal points
are not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds
of non-positive sectional curvature, fail under the assumption of no conjugate points \cite{kB92}.

In this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.
Hyperbolicity implies the existence of a non-trivial expansive set $\mathcal{E} \subset SM$ consisting of flow lines
which diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \in SM$ is contained in $\mathcal{E}$ if there is some
 $\epsilon >0$ such that for all $w \in SM $ with $d(c_v(t), c_w(\mathbb{R})) < \epsilon$ for all $t \in \mathbb{R}$,
the geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set
is invariant under the geodesic flow.

Geodesic flows are called expansive if $SM =\mathcal{E}$.
In case that $(M,g)$ has no conjugate points it was proved by Ruggiero \cite{rR94} that the expansiveness of the geodesic
flow
implies that the fundamental group $\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \infty) \to \widetilde{M} $ on the universal cover we have
$\sup_{t \ge 0} d(c_1(t),c_2(t)) = \infty$.
While negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.

In the following we call an invariant Borel probability measure $\nu \in \MMM_\phi(SM)$ of the geodesic flow $\phi^t: SM \to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\nu(\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.

In the following we call an invariant Borel probability measure $\nu \in \MMM_\phi(SM)$ of the geodesic flow $\phi^t: SM \to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\nu(\EE) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.

In this section we prove first that $\mu$ is the unique MME under the following conditions.
\begin{theorem}\label{thm:uniqueMME-inj}
Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and with Gromov hyperbolic fundamental group. Let $\delta $ the Gromov hyperbolicity constant of the universal cover $\wM$ and assume that the injectivity radius $\inj(M)$ of $M$ is larger than $16 \delta$. Assume that at least one of the following two
conditions is fulfilled:
\begin{enumerate}
\item
The entropy of non-expansive measures is strictly smaller than the topological entropy, i.e.
$$ h_\nu(\phi)< \htop(\phi) \; \; \text{for all} \; \; \nu \in \MMM_\phi(SM) \; \; with \; \;\nu(\EE)=0$$
\item
The expansive set $\EE \subset SM$ has non-empty interior and the geodesic flow $\phi^t: SM \to SM$ is entropy expansive
at some scale larger than $8 \delta$.
\end{enumerate}
Then the measure $\mu$ is the unique measure of maximal entropy of the geodesic flow.
\end{theorem}

Now we can prove Theorem \ref{thm:main} and Theorem \ref{thm:expansiveMME} using Theorem \ref{thm:uniqueMME-inj}
as follows.
\begin{proof}
Let $(M,g)$ be a closed manifold Riemannian manifold without conjugate points and divergence property.
 Furthermore, assume that the fundamental group of $M$ is residually finite and Gromov hyperbolic fundamental group with Gromov hyperbolicity.
 By Proposition \ref{prop:large-inj} there is a smooth Riemannian manifold $N$ and is for some
$k \in \N$ a locally isometric $k$ to $1$ covering map $p\colon N\to M$ such that the injectivity radius of $N$ is larger
than $8 \delta$. In particular $N$ has no conjugate, the divergence property
and the geodesic $\phi^t_{SM}: SM \to SM$ is $k$ to $1$ factor of the
geodesic flow $\phi^t_{SN} : SN \to SN$ as well. If $\mu$ is a invariant Borel probability measure on $SM$
we denote by $\tmu$ the canonical lift defined by
$$
\tmu(A) = \int_{SM} \frac{1} {k}\card \{dp^{-1} (v) \cap A \}d\mu(v),
$$
where $dp :SN \to SM$ is the differential of $p$.
Obviously is $\tmu$ is $\phi^t_{SN}$ invariant Borel probability measure with $dp_*\tmu =\mu$ and
$h_{\tmu}(\phi_{SN}) = h_{\mu}(\phi_{SM})$.
Since the expansive set of $SN$ is the lift of the expansive set in $SM$
and the expansivity constants of $\phi^t_{SM}: SM \to SM$ and $\phi^t_{SN} : SN \to SN$ agree
the assumptions of Theorem \ref{thm:main} or Theorem \ref{thm:expansiveMME} imply that
$\phi^t_{SN} : SN \to SN$ fullfils the assumptions
of Theorem \ref{thm:uniqueMME-inj}. Hence $\phi^t_{SN} : SN \to SN$ has a unique measure of maximal entropy.
 If $\mu_1$ and $\mu_2$ are measures of maximal entropy for $\phi^t_{SM}$
then their lifts $\tmu_1$ and $\tmu_2$ are measures of maximal entropy for $\phi^t_{SN}$ as well.
In particular $\tmu_1 = \tmu_2$ therefore $\mu_1 = \mu_2 = \mu$.

For the construction of Green bundles we need to define stable and unstable Jacobi tensors. Since $(M.g)$ has no conjugate points
for each $r > 0$, there exists orthogonal Jacobi tensors along $c_v$ with
\begin{align*}
S_{v,r}(0) = \Id_{v^\bot}, &\quad S_{v,r}(r) = 0, \\
U_{v,r}(0) = \Id_{v^\bot}, &\quad U_{v,r}(-r) = 0.
\end{align*}
Note that we have $U_{v,r}(t) = S_{-v,r}(-t)$.
The stable and unstable Jacobi tensor along $c_v$ are defined via the following initial conditions
$$ S_v(0) = \Id_{v^\bot}, \quad S_v'(0) = \lim_{r \to \infty} S_{v,r}'(0), $$
and similarly
$$ U_v(0) = \Id_{v^\bot}, \quad U_v'(0) = \lim_{r \to \infty} U_{v,r}'(0). $$
The existence of $\lim_{r \to \infty }S_{v,r}'(0)$ follows from the monotonicity $S'_{v,r}(0) <S'_{v,s}(0)$ for $r < s$ and the upper bound
$S'_{v,r}(0) < U'_{v,l}(0)$ for all $r, l > 0$. Moreover, we introduce the symmetric endomorphisms
$$ S(v) = S'_v(0) \quad \text{and} \quad U(v) = U'_v(0). $$
In particular $U(v) -S(v) \ge 0$. Since $U_{v,r}(t) = S_{-v,r}(-t)$ for all $t \in \R$ and $r>0$ we obtain $U_{v}(t) = S_{-v}(-t)$ and therefore
$$
U(v) = U'_v(0) = - S'_{-v}(0) = -S(-v).
$$
Furthermore,
\begin{equation}\label{eqn:invGbundles}
U(\phi^tv) = U'_v(t) U^{-1}_v(t) \; \text{and} \;S(\phi^tv) = S'_v(t) S^{-1}_v(t)
\end{equation}
 \begin{definition}\label{def:Green bundles}
 Let $(M,g)$ be a manifold without conjugate points. Then the subbundles $E^s$ and $E^$ of $TSM$ defined by
 $$
 E^s(v) =\{(x, U(v) x) \mid x \in T_{\pi (v)}M, x\perp v \}
 $$
 and
 $$
 E^u(v) =\{(x, U(v) x) \mid x \in T_{\pi (v)}M, x\perp v \}
 $$
 are called the stable and unstable Green bundles
 \end{definition}
 \begin{remark}
 From the equations \ref{eqn:invGbundles} follows the invariance of the Green bundles under the geodesic flow $\phi^t$, i.e.
 $$
 D\phi^t(v)E^s(v) = E^s(\phi^t(v)) \; \text{and} \; D\phi^t(v)E^u(v) = E^u(\phi^t(v))
 $$
 for all $t \in \R$.
 \end{remark}
 For the metric
entropy of the geodesic flow,
Freir\'{e} and Ma\~n\'e \cite{FM82}
 obtained, based on Ruelle's inequality \cite{dR78} and Pesin's \cite{jPe77} formula the following estimate.
 \begin{theorem}\label{thm: FM}
Let $(M,g)$ be a compact manifold without conjugate points and $\mu \in \MMM_\phi (SM) $ be a
$\phi^t$-invariant
 probability measure for the geodesic flow.
Then
$$
h_\mu(\phi^t) \le \int\limits_{SM} \tr U d \mu.
$$
Furthermore equality holds if $\mu$
is given by the Liouville measure.
\end{theorem}
\subsection{Positivity of the Liouville entropy}
In the \cite{gK85} we showed that for manifolds without conjugate points and continuous asymptote positivity the metric entropy
of the Liouville measure provided the topological entropy is positive as well.
Since the paper is not easily accessible we provide the proof in this paper.
\begin{theorem}\label{Kn-thesis}
Let $(M,g)$ be a compact manifold without conjugate points and continuous Green bundles.
If the the geodesic flow $\phi^t$ has positive topological entropy $\htop(\phi) $ then the metric entropy $h_\lambda(\phi)$ of the geodesic flow with respect to the Liouville measure $\lambda$.
\end{theorem}
\begin{proof}
Assume that $\htop(\phi) >0$. Then by the variational principle there exists a measure $\mu \in \MMM_\phi (SM) $ such that
$$
h_\mu(\phi) \ge \frac{1}{2} \htop(\phi) >0
$$
Let $F: SM \to SM$ be the flip map given by $F(v) = -v$. Then the push forward measure $ \mu_F$ given
by $ \mu_F(A) = \mu (F(A))$ for all measurable sets $A \subset SM$ is flow-invaraint as well since
$$
 \mu_F(\phi^t A) = \mu ( F(\phi^t(A)) = \mu ( \phi^{-t} F(A)) = \mu(F(A)) = \mu_F(A)
 $$
 Consider the measure
 $\nu = \frac{1}{2}( \mu + \mu_F) $. The affine property of the metric entropy
 yields
 $$
 h_\nu (\phi) = \frac{1}{2}h_\mu (\phi) + \frac{1}{2}h_{\mu_F} (\phi) >0
 $$
 From theorem \ref{thm: FM} we obtain $h_\nu (\phi_t) \le \int\limits_{SM} \mathrm{tr} U(v) d \nu $.
 Since $\tr U(v) = -\tr S(-v)$ the flip invariance of $\nu$ implies
 $$
 \int\limits_{SM} \tr U(v) d \nu = \int\limits_{SM} -\tr S(v) d \nu
$$
Hence
$$
0 < h_\nu (\phi_t) \le \frac{1}{2}\int\limits_{SM} \tr(U(v) - S(v)) d \nu
$$
Since $\tr(U(v) - S(v)) \ge 0$ and by the continuity of the Green Bundles $v \mapsto U(v)-S(v)$ is continuous the set
$$
Q:= \{v \in SM \mid \mathrm{tr} (U(v) - S(v) )> 0 \}
$$
is open and non-empty by $ 0 < h_\nu (\phi_t)$.
In particular $\lambda(Q) >0$ . Using that
$ h_\lambda (\phi_t) = \int\limits_{SM} \mathrm{tr} U(v) d \lambda$ holds
the flip invariance of the Liouville measure $ \lambda$ yields
$$
h_\lambda (\phi_t) = \frac{1}{2}\int\limits_{SM} \mathrm{tr} (U(v) - S(v) )d \lambda >0.
$$

\begin{remark}
\begin{itemize}
\item Closed Riemannian manifolds without conjugate points and Gromov hyperbolic fundamental group have positive topological
entropy (see \cite{CK02} and Remark  \ref{rem:conj-Grhyp}).
\item It follows from the work of Newhouse \cite{sN89} that for smooth geodesic flows $\phi^t$
there exists a measure $\nu \in \MMM_\phi(SM)$ of maximal entropy, i.e. $h_\nu(\phi)= \htop(\phi)$.
Hence, the assumption of our theorem forces  $\nu(\mathcal{E})$ to be positive. In particular, the expansive set is not empty.
\item If $(M,g)$ is a closed and non-flat  surface without conjugate points, then all assumptions of Theorem \ref{thm:main}
hold (see \cite{CKW21}). Therefore such surfaces have a unique MME.
\item If $(M,g)$ is a closed smooth Riemannian manifold without conjugate points and expansive geodesic flow (i.e $\mathcal{E} =SM$)
 the uniqueness of the MME has been obtained by Bosch\'e in \cite{aB18}.
As we mentioned above a closed manifold without conjugate points and expansive geodesic flow has the divergence
property and Gromov hyperbolic fundamental group. Since for expansive geodesic flows non-expansive measures obviously do not exist,   the result of Bosch\'e follows from our theorem as a special case.
\item If $(M,g)$ is a closed rank 1 manifold of non-positive curvature the rank 1 (regular)  set consists of orbits of the geodesic flow which do not have non-trivial parallel Jacobi-fields orthogonal to the geodesic (see e.g. \cite{gK97} or \cite{gK98}).
It is a consequence of the flat strip theorem that the regular set is contained in the expansive set.
In  \cite{gK98} we showed that for closed rank 1 manifolds the geodesic flow has a unique MME. Furthermore, the measure has full weight
on the regular, and hence, on the expansive set. A different proof of the uniqueness of the MME was later given in  \cite{BCFT}.
Since there are examples of closed rank 1 manifolds whose fundamental group has $(\mathbb{Z}^2,+)$ as a subgroup, the
fundamental group is generally not Gromov hyperbolic (see \cite{gK98}).
\end{itemize}
\end{remark}
In \cite{CKW21} Climenhaga, War and the author proved Theorem \ref{thm:main} under the more special condition of a background
metric of negative curvature and
the slightly stronger entropy gap assumption
$$
 \sup\{h_\nu(\phi) : \nu\in \MMM_\phi(SM), \nu(\mathcal{E})=0\} < \htop(\phi).
 $$
 for non-expansive measures.
 The entropy gap would follow from our condition that non-expansive measure do not have maximal
 entropy provided the expansive set is open.  Namely, if $\mathcal{E}$ is open and the entropy gap would not hold there would exist a sequence of measures
 $\nu_n \in \MMM_\phi(SM)$   with $\nu_n(\mathcal{E})=0 $ converging weakly to $\nu$ and $\lim_{n \to \infty}h_{\nu_n}(\phi) = \htop(\phi)$. Since  $\mathcal{E}$ is open, $\nu(\mathcal{E}) \le \liminf_{n \to \infty}\nu_n(\mathcal{E})= 0$ and by upper semi-continuity of entropy this would yield $h_{\nu}(\phi) = \htop(\phi)$. But this contradicts our assumption that  $h_{\nu}(\phi) <\htop(\phi)$ for non-expansive measures.
 However, to our knowledge openness of the expansive set is not known in our setting.

The assumption of a background
metric of negative curvature implies that the fundamental group is Gromov hyperbolic (see subsection \ref{subsec:hypgroups}).
Due to the uniformization theorem for surfaces and the proof of the geometrization conjecture in dimension three
Gromov hyperbolicity of the fundamental group implies the existence of a metric with even constant negative curvature.
However,  for closed manifolds of dimension bigger than three,  Gromov hyperbolicity of the fundamental group and existence of a metric without conjugate points might not be enough to yield a metric of negative curvature. In any case, to provide a solution to this question is a very difficult problem.

In \cite{CKW21} the proof of the uniqueness of the measure of maximal entropy used the background metric  of negative curvature
to establish with the help of the Morse Lemma a coarse specification property
for the geodesic flow.  Applying the work of Climenhaga and Thompson \cite{CT16}, the specification property was used
to prove the uniqueness of the MME.

 However, the proof of the above theorem does not require specification but relied on methods derived in a paper
 of the author  \cite{gK98} on the uniqueness of the measure of maximal entropy for geodesic flows on non-positively curved rank 1 manifolds.\\

 There is a interesting and quite flexible notion due to Bowen \cite{rBo72}, called entropy expansiveness  which hold for many dynamical systems for which expansiveness fails (see section \ref{sec:e-expansiveness}).

 Given a closed Riemannian manifold $(M,g)$ without conjugate points and   $(\widetilde{M},g)$
 be the universal cover with the lifted Riemannian metric denoted again by $g$. For $v \in S\widetilde{M}$ and $\rho >0$ we define
 the set
 $$
 Z_{\rho} (v) = \{w \in S\widetilde{M} \mid d(c_v(t), c_w(t)) \le \rho, t \in \mathbb{R} \}.
$$
 The geodesic flow $\phi^t: SM \to SM$  is called entropy expansive at scale $\rho >0$
if $\htop(\tilde \phi, Z_\rho(v))  = 0$, where $\tilde \phi^t$ is the geodesic flow lifted to $S \widetilde{M}$.
Due to the flat strip theorem geodesic flows on manifolds of non-positive curvature or more generally no focal points are entropy expansive at any scale (see \cite[proposition 3.3]{gK98}).
This also holds for non-flat surfaces without conjugate points \cite[Lemma 4.5]{GKOS14} even so the flat strip theorem fails in this case \cite{kB92}. As far as we know, there is no example of a closed Riemannian manifold with metric without conjugate known for which the
geodesic flow is not entropy expansive for all or even some $\rho >0$. Alternatively to Theorem \ref{thm:main}, we obtain
the uniqueness of the MME under the following conditions.
\begin{MainThm}\label{thm:expansiveMME}
 Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov
 hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \delta$, where $\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow
 has a unique measure of maximal entropy $\mu$.

Furthermore, $\mu(\mathcal{E}) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.
\end{MainThm}

 Using the results of Climenhaga, War and the author proved in \cite[Theorem 1.2]{CKW21}
 together with Theorem \ref{thm:conj-Grhyp} and Remark \ref{rem:conj-Grhyp},
 we can conclude that the measure of maximal entropy is given by the limiting distribution of closed orbits.

Furthermore, as in \cite[Theorem 1.2]{CKW22}, an estimate on the growth of pairwise non-free-homotopic closed geodesics,
obtained by Margulis \cite{gM69} in the case of negative curvature, follows. More precisely:

 \begin{MainThm}\label{thm:closed geodesics}
	Let $(M,g)$ be a closed Riemannian manifold such that the assumption in Theorem \ref{thm:main}
	or Theorem \ref{thm:expansiveMME} hold. Denote for $T>0$  by $\mathcal{P}(T)$ be any maximal set of pairwise non-free-homotopic closed geodesics of minimal length in the free homotopy classes and $P(T) =\mathrm{card} \mathcal{P}(T) $ its cardinality. Consider the measures
\begin{equation*}\label{eqn:nut}
\mu_T= \frac {1}{P(T)} \sum_{c\in \mathcal{P}(T)} \frac{\Leb_c}{T},
\end{equation*}
where $\Leb_c$ is Lebesgue measure (length) along the curve $\dot{c}$ in the unit tangent bundle $SM$. Then
\begin{enumerate}
\item \label{eqn:distr}

The measures $\mu_T$ converge  in the weak* topology  as $T\to \infty$ to the measure of maximal entropy.

\item
Furthermore,
\begin{equation*}\label{eqn:margulis}
P(T) \sim \frac{e^{hT}}{hT},
\end{equation*}
which means that the ratio of $P(T)$ and  $\frac{e^{hT} }{hT} $ converges $1$ as $T \to \infty$.
\end{enumerate}

\end{MainThm}

In \cite{CKW21} the uniqueness of the MME in Theorem~\ref{thm:main} was proved (under a background metric of negative curvature) by using the Morse Lemma to establish a \emph{coarse specification property} for the geodesic flow, and then applying the work of Climenhaga--Thompson \cite{CT16} to deduce uniqueness of the MME from specification.

The text also explains why their (slightly stronger) \emph{entropy gap} assumption
\[
\sup\{h_\nu(\phi): \nu\in\MMM_\phi(SM),\,\nu(\mathcal E)=0\}<h_{\mathrm{top}}(\phi)
\]
would follow from the present hypothesis “$h_\nu(\phi)<h_{\mathrm{top}}(\phi)$ for all non-expansive measures” \emph{if} the expansive set $\mathcal E$ were open: otherwise one could take measures $\nu_n$ with $\nu_n(\mathcal E)=0$ and $h_{\nu_n}(\phi)\to h_{\mathrm{top}}(\phi)$; by openness, the weak* limit $\nu$ still has $\nu(\mathcal E)=0$, and by upper semi-continuity of entropy one gets $h_\nu(\phi)=h_{\mathrm{top}}(\phi)$, contradicting the assumption.

For the present theorem, the text says that its proof “does not require specification” and instead “relied on methods derived in” \cite{gK98} (developed for uniqueness of the MME for geodesic flows on non-positively curved rank 1 manifolds).

Question:
Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points, and let $\phi^t:SM\to SM$ be its geodesic flow on the unit tangent bundle. Write $\phi=\phi^1$, so that $h_\nu(\phi)$ denotes the metric entropy of the time-$1$ map and $h_{\mathrm{top}}(\phi)$ its topological entropy. Assume that geodesic rays on the universal cover $\widetilde M$ have the divergence property: for every pair of geodesic rays $c_1,c_2:[0,\infty)\to\widetilde M$, one has $\sup_{t\ge 0} d(c_1(t),c_2(t))=\infty$. Assume also that the fundamental group $\pi_1(M)$ is Gromov hyperbolic and residually finite. Define the expansive set $\mathcal E\subset SM$ by declaring that $v\in\mathcal E$ if there exists $\varepsilon>0$ such that for every $w\in SM$ with $d(c_v(t),c_w(\mathbb R))<\varepsilon$ for all $t\in\mathbb R$, the geodesics $c_v$ and $c_w$ agree up to a time shift. Call a $\phi^t$-invariant Borel probability measure $\nu$ non-expansive if $\nu(\mathcal E)=0$. Suppose that every non-expansive measure $\nu\in\mathcal M_\phi(SM)$ satisfies
$$
h_\nu(\phi)<h_{\mathrm{top}}(\phi).
$$
Which conclusion about the geodesic flow holds?

There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, $\mu(\mathcal E)=1$, $\mu$ is mixing for the geodesic flow, and $\operatorname{supp}(\mu)=SM$.

Choices:
B. There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, every $\phi^t$-invariant Borel probability measure gives full weight to the expansive set, $\nu(\mathcal E)=1$ for all $\nu\in\mathcal M_\phi(SM)$, and $\mu$ is mixing with $\operatorname{supp}(\mu)=SM$.
C. There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, $\mu(\mathcal E)=1$.
D. There exists a measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$ with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$, and every measure of maximal entropy is mixing for the geodesic flow and fully supported on $SM$; however, the hypotheses do not force uniqueness of the measure of maximal entropy or the conclusion $\mu(\mathcal E)=1$.
E. There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, $\mu$ is ergodic and supported on the expansive set $\mathcal E$, but one cannot conclude that $\mu$ is mixing or that $\operatorname{supp}(\mu)=SM$.

dropped mixing and full-support conclusions

uniqueness of MME and full-weight expansivity of the MME

mixing and full support strengthened down to mere ergodicity/support on expansive set