\begin{thm}\label{Thm-Existence}
	For all $n\geq 2$, $0<\alpha<n$ and $\gamma>0$, there exists a positive constant $m_1$ depending on $n,\alpha$ and $\gamma$ such that for $0<m\leq m_1$, the unique minimizer (up to hyberbolic isometries) to the minimization problem \eqref{Prob-mini} with volume $m$ is given by a geodesic ball.
\end{thm}

\label{Subsec-model} The $n$-dimensional hyperbolic space $\mathbb{H}^n$ can be viewed as $\mathbb{H}^n=(\mathbb{R}^n_{+},g)$, where $\mathbb{R}^n_{+}=\{(x_1,x_2,\dots,x_n)\in\mathbb{R}^n:x_n>0\}$ is

Let $n\geq 2$, $0<\alpha<n$ and $\gamma>0$. For any measurable set $F\subset\mathbb{H}^n$, we consider the following functional:
\begin{equation}\label{defn-Functional}
	\mathcal{E}(F)=P(F)+\gamma NL_{\alpha}(F),
\end{equation}
where
\begin{equation}\label{defn-Perimeter}
 P(F)=\sup\left\{\int_{F}\mathrm{div}_g X\,dV_g:X\in C_c^1(\mathbb{H}^n,\mathrm{T}\mathbb{H}^n),|X|_g\leq 1\right\},
 \end{equation}
 denotes the perimeter of $F$ in the sense of  De Giorgi and the second term $NL_{\alpha}(F)$ is defined as
 \begin{equation}\label{defn-doubleint}
 	NL_{\alpha}(F)=\int_{\mathbb{H}^n}\int_{\mathbb{H}^n}{\frac{\chi_F(x)\chi_F(y)}{{d_g(x,y)}^{\alpha}}}\,dV_g(x)dV_g(y),
 \end{equation}
 where $\chi_F$ denotes the characteristic function of $F$. Furthermore,  in both \eqref{defn-Perimeter} and \eqref{defn-doubleint}, $g$ denotes the metric of $\mathbb{H}^n$ and in this paper, all quantities with subscript $g$ indicate that they are considered in $\mathbb{H}^n$. In the following, we investigate the minimization problem
 \begin{equation}\label{Prob-mini}
 	E(m)=\inf_{|F|_g=m}\mathcal{E}(F).
 \end{equation}
The minimization problem \eqref{Prob-mini} of functional \eqref{defn-Functional} has a physical background in Euclidean space $\mathbb{R}^n$ and has been extensively studied. The case where $\alpha=1$ in $\mathbb{R}^3$ is particularly significant, recalling Gamow's liquid drop model for atomic nuclei. In this model, the nucleus is viewed as a region $\Omega\subset\mathbb{R}^3$ densely packed with nucleons (protons and neutrons) at constant density, implying the nucleon count is proportional to the volume of $\Omega$. The perimeter term in the energy functional corresponds to the surface tension binding the nucleus and the second term represents the Coulomb repulsion between protons (here all physical constants have been normalized to 1 for simplicity). Then the minimization problem is to ask that whether stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter.

In the case of $\mathbb{R}^n$, the minimization problem of functional \eqref{defn-Functional} is highly nontrivial since by the isoperimetric inequality, the perimeter  term attains its minimum on balls for fixed volume, while the second term attains its maximum on balls by the Riesz's rearrangement inequality (cf. \cite[Theorem 3.7]{Leib-Loss01}). For set $F\subset\mathbb{R}^n$ measurable with volume $|F|=m$, we scale $F$ by $F\to m^{\frac{1}{n}}U$, then $|U|=1$ and using the scaling properties of $\mathbb{R}^n$, we have
\begin{equation}\label{Ex-scale}
	\mathcal{E}(F)=m^{\frac{n-1}{n}}\left(P(U)+\gamma m^{\frac{n+1-\alpha}{n}}NL_{\alpha}(U)\right),
\end{equation}
which suggests that the perimeter term is dominant if volume $m$ is small and the nonlocal term $NL_{\alpha}(\cdot)$ is dominant if volume $m$ is large. Hence, we expect that there exist minimizers for small $m$ and there exist no minimizers for large $m$. Indeed, we define the critical volume $m_{*}$ to be the volume such that the value of the functional $\mathcal{E}(\cdot)$ of the ball with volume $m_{*}$ equals that of two balls with both volume $m_{*}/2$, spaced infinitely far apart, i.e.,
\begin{equation*}
	\mathcal{E}\left(\left(\frac{m_{*}}{b_n}\right)^{\frac{1}{n}}B\right)=2\mathcal{E}\left(\left(\frac{m_{*}}{2 b_n}\right)^{\frac{1}{n}}B\right),
\end{equation*}
where $B$ denotes the unit ball in $\mathbb{R}^n$ and $b_n=|B|$. In the follwing of paper, we also denote $\omega_{n-1}:=|\partial B|$, which is the area of the unit sphere in $\mathbb{R}^n$. Note that by coarea formula, we have $\omega_{n-1}=n b_n$.  A direct calculation shows
\begin{equation}\label{Eq-Cri}
	m_{*}=\left(\frac{2^{\frac{1}{n}}-1}{1-2^{\frac{\alpha-n}{n}}}\frac{\omega_{n-1}}{\gamma NL_{\alpha}(B)}\right)^{\frac{n}{n+1-\alpha}}b_n.
\end{equation}
The significance of $m_{*}$ is that for $m>m_{*}$, the value of the functional $\mathcal{E}(\cdot)$ of two balls with both volume $m/2$, spaced infinitely far apart is less than that of a ball with volume $m$. Moreover, it is conjectured that
\begin{con}[cf. \cite{Choksi-Peletier11}]\label{Con-Cri}
	Given $n\geq 2$, $0<\alpha<n$ and $\gamma>0$, for $m\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\mathcal{E}(\cdot)$ among measurable sets $F\in\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.
	\end{con}
So far, Conjecture \ref{Con-Cri} has not been completely solved. Kn$\ddot{\mathrm{u}}$pfer and Muratov \cites{Knupfer-Muratov13,Knupfer-Muratov14} proved the following results:
\begin{itemize}
	\item[(a)] For every $n\geq 2$, $\alpha\in(0,n)$ and $\gamma>0$, there exists a constant $m_{c_1}>0$ depending on $n,\alpha$ and $\gamma$ such that $E(m)$ has a minimizer for every $m\leq m_{c_1}$.
	\item[(b)] For every $n\geq 2$, $\alpha\in(0,2)$ and $\gamma>0$, there exists a constant $m_{c_2}>0$ depending on $n,\alpha$ and $\gamma$ such that $E(m)$ has no minimizer for every $m>m_{c_2}$.
	\item[(c)] For fixed $\gamma>0$, if $n=2$ and $\alpha>0$ is suffciently small, then $m_{c_1}=m_{c_2}=m_{*}$, and balls are unique minimizers for $E(m)$ when $m\leq m_{*}$.
	\item[(d)] For fixed $\gamma>0$, if $n=2$ and $\alpha<2$, or if $3\leq n\leq 7$ and $\alpha<n-1$, there exists $0<m_{c_0}\leq m_{c_1}$, depending on $n,\alpha$ and $\gamma$ such that balls are unique minimizers for $E(m)$ with $m<m_{c_0}$.
\end{itemize}

Since then, much progress has been made on the basis of these results. Julin \cite{Julin14} extended the result of (d) to all $n\geq 3$ and $\alpha=n-2$. Bonacini and Cristoferi \cite{Bonacini-Cristoferi14} extended the results of (c) and (d) to all $n\geq 2$. Figalli, Fusco, Maggi, Millot and Mirini \cite{Figalli15} extended the result of (d) to to all $n\geq 2$ and $\alpha\in(0,n)$. Frank and Nam \cite{Frank-Nam21} extended the result of (a) to $m_{c_1}=m_{*}$, they also showed that (b) holds for $0<\alpha<n$ and $\alpha\leq 2$. There are also some quantitative estimates for $\alpha$, $m_{c_0}$ and $m_{c_2}$. In the case of $n=2$, Muratov and Zaleski \cite{Muratov15} showed that the result of (c) holds for $\alpha\leq 0.034$.  In the case of $n=3$ and $\alpha=1$, Frank, Killip and Nam \cite{Frank-Killp-Nam16} proved that $m_{c_2}\leq 2.38m_{*}$, while Chodosh and Ruohoniemi \cite{Chodosh-Ruohomiemi25} showed that $m_{c_0}\geq 0.28m_{*}$. We also refer readers to \cites{Alama21,Carazzato23,Carazzato25,Choksi17,Choksi22,Frank19,Frank-Lieb15,Goldman15,Goldman25,Muratov18,Novaga21,Novaga22,Pascale25} for some other interesting results on the nonlocal isoperimetric problem.

Now we turn back to the case of $\mathbb{H}^n$. In this situation, the perimeter term also attains its minimum on geodesic balls with fixed volume by the isoperimetric inequality, and the second term attains its maximum on geodesic balls with fixed volume by the Symmetrization Lemma (cf. \cite[Page 225]{Beckner93}). Hence the minimization problem \eqref{Prob-mini} in $\mathbb{H}^n$ is also highly nontrivial and we expect that there are similar results in $\mathbb{H}^n$ compared with $\mathbb{R}^n$.

Before stating our main results, we make some measure-theoretic notations. We say that a function $u\in L^1(\mathbb{H}^n)$ has bounded variation, i.e. $u\in BV(\mathbb{H}^n)$, if there holds
\begin{equation*}
	\int_{\mathbb{H}^n}{|\nabla u|_g}\,dV_g:=\sup\left\{\int_{\mathbb{H}^n}u\,\mathrm{div}_g X\,dV_g: X\in C_c^1(\mathbb{H}^n,\mathrm{T}\mathbb{H}^n), |X|_g\leq 1\right\}<\infty.
\end{equation*}
For any measurable set $F\subset\mathbb{H}^n$, we say that $F$ has finite perimeter if its characteristic function $\chi_F\in BV(\mathbb{H}^n)$ and then its perimeter can be denoted as $P(F):=\int_{\mathbb{H}^n}{|\nabla\chi_F|}\,dV_g$. The $k$-dimensional Hausdorff measure with $k\in[0,n]$ is denoted as $H^k(F)$.

For any $H^n$-measurable set $F$, its upper and lower densities at a point $x\in\mathbb{H}^n$ are defined respectively by
\begin{equation}\label{Defn-uppden}
	\overline{D}(F,x):=\limsup_{r\to 0}\frac{|F\cap B_r(x)|_g}{|B_r(x)|_g},\quad  \underline{D}(F,x):=\liminf_{r\to 0}\frac{|F\cap B_r(x)|_g}{|B_r(x)|_g},
\end{equation}
where $B_r(x)\subset\mathbb{H}^n$ is the open geodesic ball with center $x\in\mathbb{H}^n$ and radius $r$. If the upper and lower densities are equal, their common value will be called the density of $F$ at $x$, and it will be denoted by $D(F,x)$.

The essential interior $\mathring{F}^M$ of $F$ is then defined as the set of all $x\in\mathbb{H}^n$ for which $D(F,x)=1$, while the essential closure $\bar{F}^{M}$ of $F$ is defined as the sets of all points where $\overline{D}(F,x)>0$. The essential boundary $\partial^M F$ of F is defined as the set of points where $\overline{D}(F,x)>0$ and $\overline{D}(\mathbb{H}^n\setminus F,x)>0$.  For any two sets $A, B\subset\mathbb{H}^n$, we denote $A\Delta B:=(A\setminus B)\cup (B\setminus A)$ as the symmetric difference of $A$ and $B$, then by the Lebesgue–Besicovitch Differentiation Theorem (cf. \cite[Theorem 1.32]{Evans15}), we have
\begin{equation*}
	|F\Delta\mathring{F}^M|_g=|F\Delta\bar{F}^{M}|_g=0.
\end{equation*}
We also have that (cf. \cite[Page 46]{Ambrosio01})
\begin{equation}\label{Eq-Essenb}
	\partial^M F=\mathbb{H}^n\setminus(\mathring{F}^M\cup(\mathring{F}^c)^M).
\end{equation}

The reduced boundary $\partial^{*}F$ of a set of finite perimeter F is defined as a set of points $x\in\partial^M F$ such that the measure-theoretic normal exists at $x$, i.e., if the following limit exists:
\begin{equation*}
	\nu_F(x):=\lim_{r\to 0}\frac{\int_{B_r(x)}{\nabla{\chi_{F}}\,dV_g}}{\int_{B_r(x)}{|\nabla{\chi_{F}}|_g\,dV_g}}\quad \text{and}\quad |\nu_F(x)|_g=1,
\end{equation*}
where $\nabla\chi_F$ is the vector-valued Radon measure associated with the distributional derivative of $\chi_F$ and $|\nabla{\chi_{F}}|_g$ coincides with the $H^{n-1}$ measure restricted to $\partial^M F$. If $F$ is a set of finite perimeter, then by De Giorgi’s structure theorem (cf. \cite[Theorem 3.59]{Ambrosio-Fusco-Pallara00}, \cite[Theorem 15.9]{Maggi-12}), we have $P(F)=H^{n-1}(\partial^{*}F)$. Also, by a result of Federer (cf. \cite[Theorem 3.61]{Ambrosio-Fusco-Pallara00}), we have $\partial^{*} F\subset\partial^M F$ and $H^{n-1}(\partial^M F\setminus\partial^{*} F)$=0.
\begin{rem}
	For any set $A\subset\mathbb{H}^n$, we can define the relative perimeter of $F$ in $A$ as $P(F;A):=|\nabla{\chi_{F}}|(A)=H^{n-1}(\partial^{*}F\cap A)$. We say that $F$ is a set of locally finite perimeter, if for every compact set $K\subset\mathbb{H}^n$, we have $P(F;K)<\infty$.
\end{rem}

We say that a set $F$ of finite perimeter is essentially bounded, if its essential closure $\bar{F}^{M}$ is bounded. Also, we say $F$ is decomposable, if there exists a partition $(A,B)$ of $F$ with both $|A|_g>0$ and $|B|_g>0$, such that $P(F)=P(A)+P(B)$. Otherwise, we say that $F$ is indecomposable.
\begin{rem}
	All the measure-theoretic notations mentioned above suit the case of $\mathbb{R}^n$ if the corresponding quantities are considered in $\mathbb{R}^n$.
\end{rem}

Our first result says that the functional $\mathcal{E}(\cdot)$ admits geodesic ball as the unique minimizer (up to hyperbolic isometries) when the volume $m$ is small.

In the case of $\mathbb{R}^n$, the minimization problem of functional \eqref{defn-Functional} is highly nontrivial since by the isoperimetric inequality, the perimeter term attains its minimum on balls for fixed volume, while the second term attains its maximum on balls by the Riesz's rearrangement inequality (cf. \cite[Theorem 3.7]{Leib-Loss01}). For set $F\subset\mathbb{R}^n$ measurable with volume $|F|=m$, we scale $F$ by $F\to m^{\frac{1}{n}}U$, then $|U|=1$ and using the scaling properties of $\mathbb{R}^n$, we have
\begin{equation}\label{Ex-scale}
 \mathcal{E}(F)=m^{\frac{n-1}{n}}\left(P(U)+\gamma m^{\frac{n+1-\alpha}{n}}NL_{\alpha}(U)\right),
\end{equation}
which suggests that the perimeter term is dominant if volume $m$ is small and the nonlocal term $NL_{\alpha}(\cdot)$ is dominant if volume $m$ is large. Hence, we expect that there exist minimizers for small $m$ and there exist no minimizers for large $m$. Indeed, we define the critical volume $m_{*}$ to be the volume such that the value of the functional $\mathcal{E}(\cdot)$ of the ball with volume $m_{*}$ equals that of two balls with both volume $m_{*}/2$, spaced infinitely far apart, i.e.,
\begin{equation*}
 \mathcal{E}\left(\left(\frac{m_{*}}{b_n}\right)^{\frac{1}{n}}B\right)=2\mathcal{E}\left(\left(\frac{m_{*}}{2 b_n}\right)^{\frac{1}{n}}B\right),
\end{equation*}
where $B$ denotes the unit ball in $\mathbb{R}^n$ and $b_n=|B|$. In the follwing of paper, we also denote $\omega_{n-1}:=|\partial B|$, which is the area of the unit sphere in $\mathbb{R}^n$. Note that by coarea formula, we have $\omega_{n-1}=n b_n$. A direct calculation shows
\begin{equation}\label{Eq-Cri}
 m_{*}=\left(\frac{2^{\frac{1}{n}}-1}{1-2^{\frac{\alpha-n}{n}}}\frac{\omega_{n-1}}{\gamma NL_{\alpha}(B)}\right)^{\frac{n}{n+1-\alpha}}b_n.
\end{equation}
The significance of $m_{*}$ is that for $m>m_{*}$, the value of the functional $\mathcal{E}(\cdot)$ of two balls with both volume $m/2$, spaced infinitely far apart is less than that of a ball with volume $m$. Moreover, it is conjectured that
\begin{con}[cf. \cite{Choksi-Peletier11}]\label{Con-Cri}
 Given $n\geq 2$, $0<\alpha<n$ and $\gamma>0$, for $m\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\mathcal{E}(\cdot)$ among measurable sets $F\in\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.
 \end{con}
So far, Conjecture \ref{Con-Cri} has not been completely solved. Kn$\ddot{\mathrm{u}}$pfer and Muratov \cites{Knupfer-Muratov13,Knupfer-Muratov14} proved the following results:
\begin{itemize}
 \item[(a)] For every $n\geq 2$, $\alpha\in(0,n)$ and $\gamma>0$, there exists a constant $m_{c_1}>0$ depending on $n,\alpha$ and $\gamma$ such that $E(m)$ has a minimizer for every $m\leq m_{c_1}$.
 \item[(b)] For every $n\geq 2$, $\alpha\in(0,2)$ and $\gamma>0$, there exists a constant $m_{c_2}>0$ depending on $n,\alpha$ and $\gamma$ such that $E(m)$ has no minimizer for every $m>m_{c_2}$.
 \item[(c)] For fixed $\gamma>0$, if $n=2$ and $\alpha>0$ is suffciently small, then $m_{c_1}=m_{c_2}=m_{*}$, and balls are unique minimizers for $E(m)$ when $m\leq m_{*}$.
 \item[(d)] For fixed $\gamma>0$, if $n=2$ and $\alpha<2$, or if $3\leq n\leq 7$ and $\alpha<n-1$, there exists $0<m_{c_0}\leq m_{c_1}$, depending on $n,\alpha$ and $\gamma$ such that balls are unique minimizers for $E(m)$ with $m<m_{c_0}$.
\end{itemize}

Since then, much progress has been made on the basis of these results. Julin \cite{Julin14} extended the result of (d) to all $n\geq 3$ and $\alpha=n-2$. Bonacini and Cristoferi \cite{Bonacini-Cristoferi14} extended the results of (c) and (d) to all $n\geq 2$. Figalli, Fusco, Maggi, Millot and Mirini \cite{Figalli15} extended the result of (d) to to all $n\geq 2$ and $\alpha\in(0,n)$. Frank and Nam \cite{Frank-Nam21} extended the result of (a) to $m_{c_1}=m_{*}$, they also showed that (b) holds for $0<\alpha<n$ and $\alpha\leq 2$. There are also some quantitative estimates for $\alpha$, $m_{c_0}$ and $m_{c_2}$. In the case of $n=2$, Muratov and Zaleski \cite{Muratov15} showed that the result of (c) holds for $\alpha\leq 0.034$. In the case of $n=3$ and $\alpha=1$, Frank, Killip and Nam \cite{Frank-Killp-Nam16} proved that $m_{c_2}\leq 2.38m_{*}$, while Chodosh and Ruohoniemi \cite{Chodosh-Ruohomiemi25} showed that $m_{c_0}\geq 0.28m_{*}$. We also refer readers to \cites{Alama21,Carazzato23,Carazzato25,Choksi17,Choksi22,Frank19,Frank-Lieb15,Goldman15,Goldman25,Muratov18,Novaga21,Novaga22,Pascale25} for some other interesting results on the nonlocal isoperimetric problem.

Now we turn back to the case of $\mathbb{H}^n$. In this situation, the perimeter term also attains its minimum on geodesic balls with fixed volume by the isoperimetric inequality, and the second term attains its maximum on geodesic balls with fixed volume by the Symmetrization Lemma (cf. \cite[Page 225]{Beckner93}). Hence the minimization problem \eqref{Prob-mini} in $\mathbb{H}^n$ is also highly nontrivial and we expect that there are similar results in $\mathbb{H}^n$ compared with $\mathbb{R}^n$.

We say that a set $F$ of finite perimeter is essentially bounded, if its essential closure $\bar{F}^{M}$ is bounded. Also, we say $F$ is decomposable, if there exists a partition $(A,B)$ of $F$ with both $|A|_g>0$ and $|B|_g>0$, such that $P(F)=P(A)+P(B)$. Otherwise, we say that $F$ is indecomposable.
\begin{rem}
 All the measure-theoretic notations mentioned above suit the case of $\mathbb{R}^n$ if the corresponding quantities are considered in $\mathbb{R}^n$.
\end{rem}

Our first result says that the functional $\mathcal{E}(\cdot)$ admits geodesic ball as the unique minimizer (up to hyperbolic isometries) when the volume $m$ is small.

However, such transformations are lacking in $\mathbb{H}^n$. Fortunately, after comparing several common models of $\mathbb{H}^n$, we have adopted the $\Phi_{\lambda}$-transformation (see \S\ref{Subsec-model} for details) in the upper half-space model of $\mathbb{H}^n$, which has the advantage that by adjusting the value of parameter $\lambda$, the volume of any set in $\mathbb{H}^n$ can be scaled to a given multiple. By means of the $\Phi_{\lambda}$-transformation, we can establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability, and we can further deduce the existence of minimizers for all sufficiently small volumes $m\leq m_0$ for some positive $m_0$ depending only on $n,\alpha$ and $\gamma$.

On the other hand, by adopting similar ideas from \cite{Knupfer-Muratov14} and leveraging the $\Phi_{\lambda}$-transformation, we can prove that the functional $\mathcal{E}(\cdot)$ does not admit a minimizer for large volume $m$ with an extra assumption that $\alpha<2$.
\begin{thm}\label{Thm-Nonexistence}
 For all $n\geq 2$, $0<\alpha<2$ and $\gamma>0$, there exists a positive constant $m_2$ depending on $n,\alpha$ and $\gamma$ such that for $m\geq m_2$, the minimization problem \eqref{Prob-mini} does not admit a minimizer with volume $m$.
\end{thm}
\begin{rem}
 We find it difficult to extend Theorem \ref{Thm-Nonexistence} to the case of $\alpha=2$, as in the work of Frank and Nam \cite{Frank-Nam21}, the reason being that we are unsure how to obtain the estimate in \cite[Lemma 7]{Frank-Nam21}.
\end{rem}
\begin{rem}
 Motivated by the classical results in $\mathbb{R}^n$ (cf. \cite[Theorem 2.9]{Bonacini-Cristoferi14} and \cite[Theorem 1.5]{Figalli15}), we intend to calculate the first and second variations of the functional \eqref{defn-Functional} in our subsequent work, so as to provide explicit bounds for the volume within which geodesic balls are local minimizers. As an application, we aim to derive similar quantitative results regarding parameter $m_1$ as those presented in \cite{Chodosh-Ruohomiemi25}.
\end{rem}
The paper is organized as follows: In \S\ref{Sec-pre}, we collect some preliminaries which will be used later in this paper, including the upper-half space model of $\mathbb{H}^n$, some isperimetric-type inequalities in $\mathbb{H}^n$, two notions of quasiminimizer of the perimeter, and Fuglede’s type estimates of $P(E)$ and $NL_{\alpha}(E)$ for a nearly spherical set $E\subset\mathbb{H}^n$. In \S\ref{Sec-Basic es}, we prove some basic properties for the minimizer of the functional $\mathcal{E}(\cdot)$, including the regularity result, the boundedness, the indecomposability and then prove a uniform lower density bound. During this process, we also derive a general criterion on a set of finite perimeter being not a minimizer. In \S\ref{Sec-Existence}, we prove that there exists a minimizer if the volume $m$ is small and it is contained in a geodesic ball with small radius. In \S\ref{Sec-Ball}, we prove that the functional $\mathcal{E}(\cdot)$ admits geodesic ball as unique minimizers (up to hyperbolic isometries) when volumes $m$ are small. In \S\ref{Sec-Nonex}, we firstly prove that both the the minimization value \eqref{Prob-mini} and the diameter of the essential closure of a minimizer can be two-sided controlled by the volume $m$, and then complete the proof that there exist no minimizers for large volumes.

Then we are left to prove that for $m\leq m_0$, the minimization problem \eqref{Prob-mini} admits a minimizer $E$ with volume $m$ which is contained in a geodesic ball of radius $(1+C_3 r^{\frac{1}{2n}})r$. Firstly, we can choose a sequence of sets of finite perimeter $\{F_k\}$ with $|F_k|_g=m$ such that
\begin{equation*}
 \lim_{k\to\infty}\mathcal{E}(F_k)=\inf_{|F|_g=m}{\mathcal{E}(F)}.
\end{equation*}
By the argument above, we can further choose a minimizing sequence $\{G_k\}$, such that up to isometries, $G_k\subset B_{(1+C_3 r^{\frac{1}{2n}})r}(p)$ for some fixed $p\in\mathbb{H}^n$. Then by the lower semicontinuity of perimeter (cf. \cite[Proposition 12.15]{Maggi-12}) and the compactness of perimeter (cf. \cite[Theorem 12.26]{Maggi-12}), there exists a set of finite perimeter $E\subset\mathbb{H}^n$, a fixed point $p\in\mathbb{H}^n$ and a subsequence, which we also denote as $\{G_k\}$ such that
\begin{align}
 G_k\to& E,\quad E\subset B_{(1+C_3 r^{\frac{1}{2n}})r}(p),\\
 P(E)&\leq\liminf_{k\to\infty} P(G_k).\label{Eq-peri}
\end{align}
Since $G_k\to E$, we have $m=|G_k|_g\to|E|_g$ and hence $|E|_g=m$. Furthermore, by Lemma \ref{Lem-NL}, we have
\begin{equation}\label{Eq-NLa}
 \lim_{k\to\infty}{NL_{\alpha}(G_k)}=NL_{\alpha}(E).
\end{equation}
Combining \eqref{Eq-peri} with \eqref{Eq-NLa} gives
\begin{align*}
 \inf_{|F|_g=m}{\mathcal{E}(F)}\leq\mathcal{E}(E)&=P(E)+\gamma NL_{\alpha}(E)\\
 &\leq \liminf_{k\to\infty} P(G_k)+\gamma\lim_{k\to\infty}{NL_{\alpha}(G_k)}\\
 &=\liminf_{k\to\infty}\mathcal{E}(G_k)\\
 &\leq \liminf_{k\to\infty}\mathcal{E}(F_k)=\inf_{|F|_g=m}{\mathcal{E}(F)},
\end{align*}
which implies that $E$ is a minimizer of the functional $\mathcal{E}(\cdot)$ with volume $m$.
\end{proof}
\section{Geodesic ball as the minimizer}\label{Sec-Ball}
In this section, we give the proof of Theorem \ref{Thm-Existence}. By Proposition \ref{Prop-Quasi}, we know that for any given $r_0>0$, the minimizer to the minimization problem \eqref{Prob-mini} is an $(\omega,r_0)$-minimizer for the perimeter with some constant $\omega>0$. Using Theorem \ref{Lem-Min}, we can calculate which quantities $\omega$ precisely depends on provided $m\leq m_0$.
\begin{prop}\label{Prop-Qp}
 Let $n\geq 2$, $\alpha\in(0,n)$ and $\gamma>0$. Assume that $E$ is a minimizer with $|E|_g=|B_r|_g=m$ for $m\leq m_0$, then there exist two constants $\Lambda_1$ and $\Lambda_2$ depending only on $n$, $\alpha$ and $\gamma$ such that
\begin{equation}\label{In-Qp}
 P(E)\leq P(F)+\left(\frac{\Lambda_1}{r}+\Lambda_2\right)|E\Delta F|_g
\end{equation}
for every $F\subset\mathbb{H}^n$. Here $\Lambda_1$ and $\Lambda_2$ can be chosen as
\begin{align}
 \Lambda_1&=\frac{6C_4}{b_n},\label{Defin-Lamb1}\\
 \Lambda_2&=\max\left\{\frac{6C_4}{b_n}+14\gamma c_3,\gamma c_3\left(2C_5^{\frac{2(\alpha+1-n)}{n-1}}+C_5+1\right)\right\},\label{Defin-Lamb2}
\end{align}
where $C_4$ is the constant defined in \eqref{Defn-C_4},
\begin{equation}\label{Defn-C5}
 C_5=\left(\frac{2C_4}{nb_n}\right)^{\frac{n}{n-1}},
\end{equation}
and $c_3$ is the constant defined in Lemma \ref{Lem-Bound} depending on $n$, $\alpha$ and $C_5 m_0$ (we choose $\bar{m}=C_5 m_0$).
\end{prop}
\begin{proof}
 In the following of proof, we always assume that $P(F)\leq P(E)$, since otherwise \eqref{In-Qp} holds trivially. We claim that we can reduce to prove \eqref{In-Qp} in the case that
 \begin{equation}\label{In-twos}
 \frac{1}{2}\leq\frac{|F|_g}{m}\leq C_5.
 \end{equation}
 Indeed, if we compare $E$ with $B_r$, by \eqref{Eq-Min0}, we have
 \begin{equation}\label{Qp1}
 \mathcal{E}(E)=P(E)+\gamma NL_{\alpha}(E)\leq C_4(r^{n-1}+r^n).
 \end{equation}
 Then if $|F|_g<\frac{1}{2}m$, we have
 \begin{equation*}
 |E\Delta F|_g\geq |E|_g-|F|_g>\frac{1}{2}m\geq\frac{b_n}{2} r^n.
 \end{equation*}
 Combined with the definition of $\Lambda_1$ and $\Lambda_2$ gives \eqref{In-Qp}. Also, if $|F|_g>C_5 m\geq C_5 b_n r^n$, then by \eqref{Eq-HEin} and $m\leq m_0$ (which implies $r\leq 1$ by \eqref{Defn-m0}), we have
 \begin{equation*}
 P(F)\geq nb_n^{\frac{1}{n}}|F|_g^{\frac{n-1}{n}}\geq nb_n C_5^{\frac{n-1}{n}}r^{n-1}=2C_4 r^{n-1}\geq P(E),
 \end{equation*}
 which also gives \eqref{In-Qp}.

\begin{rem}
	It remains an open problem to provide  explicit lower bounds on $m_1$.
\end{rem}
\begin{rem}
	Through a more refined analysis of the constants appearing in the estimates, we can further show that if $0<\alpha\leq\bar{\alpha}$ and $0<\gamma\leq\bar{\gamma}$ for two fixed $\bar{\alpha}\in(0,n)$ and $\bar{\gamma}>0$, then the positive constant $m_1$ in Theorem \ref{Thm-Existence} can be chosen to depend only on $n$, $\bar{\alpha}$ and $\bar{\gamma}$.
\end{rem}
The proof of this theorem in $\mathbb{H}^n$ differs significantly from that in $\mathbb{R}^n$. In $\mathbb{R}^n$, scaling transformations possess particularly convenient geometric properties. In particular, they allow one to rescale the volume to an arbitrary prescribed value while preserving the shape of the underlying set, and geodesic balls are mapped to geodesic balls under such transformations. Consequently, the analysis of the shape of minimizers in the small-volume regime can be reduced to the study of the corresponding minimization problem for a fixed reference volume $b_n$, as presented in \cites{Bonacini-Cristoferi14,Knupfer-Muratov13,Knupfer-Muratov14,Figalli15}.

However, such transformations are lacking in $\mathbb{H}^n$. Fortunately, after comparing several common models of $\mathbb{H}^n$, we have adopted the $\Phi_{\lambda}$-transformation (see \S\ref{Subsec-model} for details) in the upper half-space model of $\mathbb{H}^n$, which has the advantage that by adjusting the value of parameter $\lambda$, the volume of any set in $\mathbb{H}^n$ can be scaled to a given multiple. By means of the $\Phi_{\lambda}$-transformation, we can establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability, and we can further deduce the existence of minimizers for all sufficiently small volumes $m\leq m_0$ for some positive $m_0$ depending only on $n,\alpha$ and $\gamma$.

When proving that geodesic balls are the unique minimizers for small volumes, we encounter a new difficulty. Since our $\Phi_{\lambda}$-transformation cannot map geodesic balls to geodesic balls, the argument for a fixed volume of $b_n$ in $\mathbb{R}^n$ is inapplicable. A key observation is that if the minimizer $E$ satisfies $|E|_g=|B_r|_g=m$ for $m\leq m_0$ and some positive $r$, then $E$ is contained in a geodesic ball of radius $(1+C_3 r^{\frac{1}{2n}})r$ for some uniform constant $C_3$. Combining this observation with a notion of quasiminimizer of the perimeter, we are able to show that $\partial E$ can be viewed as a radial graph over $\mathbb{S}^{n-1}$ with the radial function close to $r$ provided the volume $m$ is small enough. Then by means of proof by contradiction and Fuglede's type estiamtes for the perimeter and the nonlocal term, we are able to prove that for small volumes $m$, geodesic balls are unique minimizers up to hyperbolic isometries.

On the other hand, by adopting similar ideas from \cite{Knupfer-Muratov14} and leveraging the $\Phi_{\lambda}$-transformation, we can prove that the functional $\mathcal{E}(\cdot)$ does not admit a minimizer for large volume $m$ with an extra assumption that $\alpha<2$.
\begin{thm}\label{Thm-Nonexistence}
	For all $n\geq 2$, $0<\alpha<2$ and $\gamma>0$, there exists a positive constant $m_2$ depending on $n,\alpha$ and $\gamma$ such that for $m\geq m_2$, the minimization problem \eqref{Prob-mini} does not admit a minimizer with volume $m$.
\end{thm}
\begin{rem}
	We find it difficult to extend Theorem \ref{Thm-Nonexistence} to the case of $\alpha=2$, as in the work of Frank and Nam \cite{Frank-Nam21}, the reason being that we are unsure how to obtain the estimate in \cite[Lemma 7]{Frank-Nam21}.
\end{rem}
\begin{rem}
	Motivated by the classical results in $\mathbb{R}^n$ (cf. \cite[Theorem 2.9]{Bonacini-Cristoferi14} and \cite[Theorem 1.5]{Figalli15}), we intend to calculate the first and second variations of the functional \eqref{defn-Functional} in our subsequent work, so as to provide explicit bounds for the volume within which geodesic balls are local minimizers. As an application, we aim to derive similar quantitative results regarding parameter $m_1$ as those presented in \cite{Chodosh-Ruohomiemi25}.
\end{rem}
The paper is organized as follows: In \S\ref{Sec-pre}, we collect some preliminaries which will be used later in this paper, including the upper-half space model of $\mathbb{H}^n$, some isperimetric-type inequalities in $\mathbb{H}^n$, two notions of quasiminimizer of the perimeter, and Fuglede’s type estimates of $P(E)$ and $NL_{\alpha}(E)$ for a nearly spherical set $E\subset\mathbb{H}^n$. In \S\ref{Sec-Basic es}, we prove some basic properties for the minimizer of the functional $\mathcal{E}(\cdot)$, including the regularity result, the boundedness, the indecomposability and then prove a uniform lower density bound. During this process, we also derive a general criterion on a set of finite perimeter being not a minimizer. In \S\ref{Sec-Existence}, we prove that there exists a minimizer if the volume $m$ is small and it is contained in a geodesic ball with small radius. In \S\ref{Sec-Ball}, we prove that the functional $\mathcal{E}(\cdot)$ admits geodesic ball as unique minimizers (up to hyperbolic isometries) when volumes $m$ are small. In \S\ref{Sec-Nonex}, we firstly prove that both the the minimization value \eqref{Prob-mini} and the diameter of the essential closure of a minimizer can be two-sided controlled by the volume $m$, and then complete the proof that there exist no minimizers for large volumes.

\begin{ack}
	The research was supported by NSFC Grant No.12471047 and  China Postdoctoral Science Foundation No.2024M751605.
\end{ack}

Compared to $\mathbb{R}^n$, the proof of Theorem~\ref{Thm-Existence} in $\mathbb{H}^n$ “differs significantly” because “such transformations are lacking in $\mathbb{H}^n$.” The authors instead “adopted the $\Phi_{\lambda}$-transformation … in the upper half-space model,” which lets one “by adjusting the value of parameter $\lambda$, [scale] the volume of any set in $\mathbb{H}^n$ … to a given multiple.”

Using the $\Phi_{\lambda}$-transformation, they “establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability,” and “further deduce the existence of minimizers for all sufficiently small volumes $m\leq m_0$.”

For uniqueness of geodesic balls at small volume, they note a key obstruction: “our $\Phi_{\lambda}$-transformation cannot map geodesic balls to geodesic balls,” so the Euclidean fixed-volume reduction is “inapplicable.” A “key observation” is that if a minimizer $E$ has $|E|_g=|B_r|_g=m\leq m_0$, then “$E$ is contained in a geodesic ball of radius $(1+C_3 r^{\frac{1}{2n}})r$.” Combining this with “a notion of quasiminimizer of the perimeter,” they show “$\partial E$ can be viewed as a radial graph over $\mathbb{S}^{n-1}$ with the radial function close to $r$ provided the volume $m$ is small enough.” Finally, “by means of proof by contradiction and Fuglede's type estiamtes for the perimeter and the nonlocal term,” they conclude that “for small volumes $m$, geodesic balls are unique minimizers up to hyperbolic isometries.”

Compared to $\mathbb{R}^n$, the proof in $\mathbb{H}^n$ “differs significantly” because “such transformations are lacking in $\mathbb{H}^n$.” The authors instead “adopted the $\Phi_{\lambda}$-transformation … in the upper half-space model,” which lets one “by adjusting the value of parameter $\lambda$, [scale] the volume of any set in $\mathbb{H}^n$ … to a given multiple.”

Using the $\Phi_{\lambda}$-transformation, they “establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability,” and “further deduce the existence of minimizers for all sufficiently small volumes $m\leq m_0$.”

For the uniqueness of geodesic balls at small volume (which is the conclusion in establishing the main theorem), they note a key obstruction: “our $\Phi_{\lambda}$-transformation cannot map geodesic balls to geodesic balls,” so the Euclidean fixed-volume reduction is “inapplicable.” A “key observation” is that if a minimizer $E$ has $|E|_g=|B_r|_g=m\leq m_0$, then “$E$ is contained in a geodesic ball of radius $(1+C_3 r^{\frac{1}{2n}})r$.” Combining this with “a notion of quasiminimizer of the perimeter,” they show “$\partial E$ can be viewed as a radial graph over $\mathbb{S}^{n-1}$ with the radial function close to $r$ provided the volume $m$ is small enough.” Finally, “by means of proof by contradiction and Fuglede's type estiamtes for the perimeter and the nonlocal term,” they conclude that “for small volumes $m$, geodesic balls are unique minimizers up to hyperbolic isometries.”

\label{Thm-Existence}
	For all $n\geq 2$, $0<\alpha<n$ and $\gamma>0$, there exists a positive constant $m_1$ depending on $n,\alpha$ and $\gamma$ such that for $0<m\leq m_1$, the unique minimizer (up to hyberbolic isometries) to the minimization problem
\begin{equation}\label{Prob-mini}
 	E(m)=\inf_{|F|_g=m}\mathcal{E}(F).
 \end{equation}
 with volume $m$ is given by a geodesic ball.,

Question:
Let \(n\ge 2\), \(0<\alpha<n\), and \(\gamma>0\). For a measurable set \(F\subset \mathbb H^n\) with hyperbolic volume \(|F|_g\), consider the energy
\[
\mathcal E(F)=P(F)+\gamma\,NL_{\alpha}(F),
\]
where \(P(F)\) is the hyperbolic perimeter and \(NL_{\alpha}(F)\) is the corresponding nonlocal interaction term. For each prescribed volume \(m>0\), define
\[
E(m)=\inf\{\mathcal E(F): |F|_g=m\}.
\]
Which statement holds about the minimizers of this fixed-volume problem in \(\mathbb H^n\)?

There exists a constant \(m_1>0\), depending only on \(n\), \(\alpha\), and \(\gamma\), such that for every \(m\) with \(0<m\le m_1\), the problem \(E(m)=\inf_{|F|_g=m}\mathcal E(F)\) has a minimizer, and that minimizer is a geodesic ball of volume \(m\), unique up to hyperbolic isometries.

Choices:
B. There exists a constant \(m_1>0\), depending only on \(n\), \(\alpha\), and \(\gamma\), such that for every \(m\) with \(0<m\le m_1\), the problem \(E(m)=\inf_{|F|_g=m}\mathcal E(F)\) has a minimizer, and every minimizer is a geodesic ball of volume \(m\), unique without passing to hyperbolic isometries.
C. There exists a constant \(m_1>0\), depending only on \(n\), \(\alpha\), and \(\gamma\), such that for every \(m\) with \(0<m\le m_1\), the problem \(E(m)=\inf_{|F|_g=m}\mathcal E(F)\) has a minimizer in \(\mathbb H^n\).
D. There exists a constant \(m_1>0\), depending only on \(n\), \(\alpha\), and \(\gamma\), such that for every \(m>0\), the problem \(E(m)=\inf_{|F|_g=m}\mathcal E(F)\) has a minimizer, and that minimizer is a geodesic ball of volume \(m\), unique up to hyperbolic isometries.
E. For every prescribed volume \(m>0\), there exists a constant \(m_1>0\), depending only on \(n\), \(\alpha\), \(\gamma\), and \(m\), such that whenever \(0<m\le m_1\), the problem \(E(m)=\inf_{|F|_g=m}\mathcal E(F)\) has a minimizer, and that minimizer is a geodesic ball of volume \(m\), unique up to hyperbolic isometries.

uniqueness only modulo hyperbolic isometries

geodesic-ball characterization and uniqueness up to isometries

uniform dependence of threshold only on n,\alpha,\gamma

The stem only sets up the variational problem and asks which conclusion holds; it does not reveal the small-volume threshold, geodesic-ball characterization, or uniqueness up to isometries.

The correct option is essentially a direct theorem statement about small-volume minimizers being geodesic balls, so the item functions largely as theorem recall rather than requiring selection among substantively different mathematical conclusions.

There is some reasoning pressure in distinguishing exact quantifiers and the role of uniqueness up to isometries versus absolute uniqueness, but overall the task mainly tests memory of the theorem's precise wording. Generative pressure is further weakened because option C is a weaker true statement.

Several distractors are plausible and target common errors (dropping the small-volume restriction, mishandling uniqueness modulo isometries, changing parameter dependence), but the presence of C as a weaker true statement makes the distractor set flawed for a single-correct MCQ.

No answer leakage, but the item is mostly a theorem-restatement question with only moderate reasoning demand. Distractors are conceptually relevant, yet the inclusion of a weaker true alternative undermines single-answer quality.


\begin{theorem}\label{theorem: intro}
	Consistently from a measurable cardinal, there are two $\sigma$-complete ultrafilters $\mathcal{V}, \mathcal{W}$ such that $\mathcal{V}$ is Lipschitz below $\mathcal{W}$, but $\mathcal{V}$ and $\mathcal{W}$ are Ketonen-incomparable. Furthermore, the same is also consistent with the assumption that the Weak Ultrapower Axiom holds.
\end{theorem}

The study and analysis of the various connections between $\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts
that for every pair of $\sigma$-complete ultrafilters, $U,W$, there are $\sigma$-complete ultrafilters $W^*\in M_U$ and $U^*\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$.\footnote{For a $\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\colon V\to M_U$ denotes the corresponding elementary embedding. If $U\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \colon N\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.}
Goldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.

In his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:

\begin{definition}[The Ketonen order\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \cite{Ketonen1972strongcompactnesscardinalsins}.}]\label{Def: Ketonen order}
	Let $U,W$ be $\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:
	\begin{enumerate}
		\item There exists $I\in W$ and a sequence $\la U_\xi \colon \xi\in I \ra$ of $\sigma$-complete ultrafilters, such that each $U_\xi$ concentrates on $\xi$, and, for every $X\subseteq \kappa$ (where $\kappa$ is the underlying ordinal of $U$),
		$$X\in U \iff \{ \xi\in I \colon X\cap \xi \in U_\xi \}\in W.$$
		\item There exists a $\sigma$-complete ultrafilter $U^*\in M_W$ and an elementary embedding $k\colon M_U\to M^{M_W}_{U^*}$ such that $k\circ j_U = j^{M_W}_{U^*}\circ j_W$, and $k\left( [Id]_U \right)< j_{U^*}\left( [Id]_W \right)$.
	\end{enumerate}

\end{definition}

\begin{definition}[The Lipschitz order on ultrafilters]\label{Def: Lipschitz order on ultrafilters}
Let $U,W\subseteq \mathcal{P}(\kappa)$ be ultrafilters on $\kappa$. We say that $U$ is Lipschitz below $W$, and denote $U<_L W$, if one of the following equivalent conditions hold:
\begin{itemize}
	\item player I has a winning strategy in $G_{\kappa}(W,U)$.
	\item there exists a super-Lipschitz function $f\colon \mathcal{P}(\kappa)\to \mathcal{P}(\kappa)$ that reduces $U$ to $W$, in the sense that $f^{-1}[W] = U$.
\end{itemize}

\end{definition}

\begin{abstract}
In his study of the Ultrapower Axiom (UA), Goldberg revealed a connection between UA and the determinacy of certain games that witness Lipschitz reducibility between ultrafilters. In particular, he analyzed the relationship between the Ketonen and Lipschitz orders—two natural extensions of the Mitchell order from normal measures to arbitrary $\sigma$-complete ultrafilters—and proved that the Lipschitz order extends the Ketonen order. He further observed that under UA the two orders coincide. Goldberg asked if it's consistent that the orders differ from each other. We show that the answer is positive. In fact, even the Weak Ultrapower Axiom does not imply that the Ketonen and Lipschitz orders coincide.
\end{abstract}

The study and analysis of the various connections between $\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts
that for every pair of $\sigma$-complete ultrafilters, $U,W$, there are $\sigma$-complete ultrafilters $W^*\in M_U$ and $U^*\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$.\footnote{For a $\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\colon V\to M_U$ denotes the corresponding elementary embedding. If $U\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \colon N\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.}
Goldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.

In his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:

The \textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$. Models of the Weak UA+$\neg$UA were constructed in \cite{benhamouGoldberg2024applications} and more recently in \cite{kaplan2025number}. We will rely on the construction from \cite{kaplan2025number} in the proof of Theorem \ref{theorem: intro}.

\begin{definition}[The Ketonen order\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \cite{Ketonen1972strongcompactnesscardinalsins}.}]\label{Def: Ketonen order}
 Let $U,W$ be $\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:
 \begin{enumerate}
 \item There exists $I\in W$ and a sequence $\la U_\xi \colon \xi\in I \ra$ of $\sigma$-complete ultrafilters, such that each $U_\xi$ concentrates on $\xi$, and, for every $X\subseteq \kappa$ (where $\kappa$ is the underlying ordinal of $U$),
 $$X\in U \iff \{ \xi\in I \colon X\cap \xi \in U_\xi \}\in W.$$
 \item There exists a $\sigma$-complete ultrafilter $U^*\in M_W$ and an elementary embedding $k\colon M_U\to M^{M_W}_{U^*}$ such that $k\circ j_U = j^{M_W}_{U^*}\circ j_W$, and $k\left( [Id]_U \right)< j_{U^*}\left( [Id]_W \right)$.
 \end{enumerate}

The study of the Ketonen order is especially interesting in the context of Goldberg's Ultrapower Axiom. Recall that the \textit{Ultrapower Axiom} (UA) is the assertion that for every pair of $\sigma$-complete ultrafilters, $U,W$, there are $\sigma$-complete ultrafilters $U^*\in M_W$ and $W^*\in M_U$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$, and $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$. The \textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$.

We proceed and define the Lipschitz order between $\sigma$-complete ultrafilters. For that, we describe the Lipschitz game of length $\kappa$, $G_{\kappa}(W,U)$, where $W,U$ are subsets of $\mathcal{P}(\kappa)$ for an ordinal $\kappa$. The game is being held between two players, I and II, and consists of $\kappa$ stages. On the $i$-th stage ($i<\kappa$), Player I chooses a bit $a(i)\in \{0,1\}$, and Player II responds with a bit $b(i)\in \{0,1\}$. Since Player I moves first at any stage, they are aware of the sequences $\la a(j) \colon j<i \ra, \la b(j) \colon j<i \ra$ constructed so far; Player II moves second, being aware of the values of $\la a(j) \colon j\leq i \ra, \la b(j) \colon j<i \ra$. After $\kappa$ rounds, the players have constructed a pair of subsets of $\kappa$,
$$ a = \{ i<\kappa \colon a(i) = 1 \} $$
$$ b = \{ i<\kappa \colon b(i) = 1 \}. $$
Player II wins if $a\in W\leftrightarrow b\in U$. Otherwise, Player I wins.

The Lipschitz order is a strict partial order on $\sigma$-complete ultrafilters (see \cite[Corollary 3.4.29]{GoldbergUABook}). To the best of our knowledge, it is still not known whether the Lipschitz order on $\sigma$-complete ultrafilters must be well-founded (but it is well-founded under UA, see Corollary \ref{Corollary: UA implies that Ketonen is Lipschitz} below). We remark that the Lipschitz order on arbitrary subsets of $\mathcal{P}(\kappa)$ is ill-founded in models of AC (see \cite[Theorem 3.4.30]{GoldbergUABook}); however, in this paper, we focus only on the Lipschitz order restricted to $\sigma$-complete ultrafilters. We freely refer to it as the “Lipschitz order,” without mentioning the restriction to pairs of ultrafilters.
We proceed and prove that the Lipschitz order extends the Ketonen order. We provide a proof directly from the definitions given above (for an alternative justification, see Proposition \ref{Proposition: equivalent characterization of Lipschitz and Ketonen}).

The \textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$. Models of the Weak UA+$\neg$UA were constructed in \cite{benhamouGoldberg2024applications} and more recently in \cite{kaplan2025number}. We will rely on the construction from \cite{kaplan2025number} in the proof of Theorem \ref{theorem: intro}.

The structure of this paper is as follows:
\begin{itemize}
	\item In Section 2 we define the Ketonen and Lipschitz orders and outline their basic properties; all the results in this section are due to Goldberg.
	\item In Section 3 we show that, consistently, the Ketonen and Lipschitz orders do not coincide (however, this is not done in a model of the Weak UA). See Theorem \ref{Theorem: Separating Lipschitz from Ketonen the Easy way}.
	\item In Section 4 we sketch the basic properties of forcing with  nonstationary support products.
	\item In Section 5, we separate the Ketonen and Lipschitz orders in a model of the Weak UA, thereby completing the proof of Theorem \ref{theorem: intro}.
\end{itemize}

\textbf{Acknowledgments:} The author deeply thanks Gabe Goldberg and Eilon Bilinsky for many conversations on the subject of this paper.

We will rely on the construction from \cite{kaplan2025number} in the proof of Theorem \ref{theorem: intro}. The paper then proceeds by: (1) defining the Ketonen and Lipschitz orders and outlining their basic properties (Section 2); (2) showing, consistently, that the Ketonen and Lipschitz orders do not coincide (not in a model of the Weak UA) (Section 3, Theorem \ref{Theorem: Separating Lipschitz from Ketonen the Easy way}); (3) sketching basic properties of forcing with nonstationary support products (Section 4); and (4) separating the Ketonen and Lipschitz orders in a model of the Weak UA, "thereby completing the proof of Theorem \ref{theorem: intro}" (Section 5).

We will rely on the construction from \cite{kaplan2025number} in the proof of the main theorem. The paper then proceeds by: (1) defining the Ketonen and Lipschitz orders and outlining their basic properties; (2) showing, consistently, that the Ketonen and Lipschitz orders do not coincide (not in a model of the Weak UA). We first prove the following theorem. \begin{theorem}\label{Theorem: Separating Lipschitz from Ketonen the Easy way}
	Assume $V= L[U]$, $\kappa$ is the unique measurable cardinal, $\po$ is the forcing notion described above and $G,G^*, U_0,U_1$ are as above. Denote $\vc = U_0$ and $\wc$ the measure derived from $j^{V[G]}_{(U_1)^2}$ using the ordinal $j_U(\kappa)+\kappa$ as a seed. Then $\vc<_{L} \wc$ but $\vc, \wc$ are Ketonen incomparable.
\end{theorem} (3) sketching basic properties of forcing with nonstationary support products; and (4) separating the Ketonen and Lipschitz orders in a model of the Weak UA, thereby completing the proof of the main theorem.

Question: Assume the consistency of a measurable cardinal. For $\sigma$-complete ultrafilters $U,W$ on ordinals, write $U<_L W$ if player I has a winning strategy in the length-$\kappa$ game $G_{\kappa}(W,U)$ (where at stage $i<\kappa$ player I chooses $a(i)\in\{0,1\}$, player II chooses $b(i)\in\{0,1\}$, and player II wins iff $a=\{i:a(i)=1\}\in W\leftrightarrow b=\{i:b(i)=1\}\in U$), and write $U<_k W$ if there is a $\sigma$-complete ultrafilter $U^*\in M_W$ and an elementary embedding $k\colon M_U\to M^{M_W}_{U^*}$ such that $k\circ j_U=j^{M_W}_{U^*}\circ j_W$ and $k([\mathrm{Id}]_U)<j_{U^*}([\mathrm{Id}]_W)$. The Weak Ultrapower Axiom says that for every pair of $\sigma$-complete ultrafilters $U,W$, there are $\sigma$-complete ultrafilters $U^*\in M_W$ and $W^*\in M_U$ such that $M^{M_U}_{W^*}=M^{M_W}_{U^*}$. Which statement is consistent under that large-cardinal assumption?

It is consistent that there exist two $\sigma$-complete ultrafilters $\mathcal V,\mathcal W$ such that $\mathcal V<_L\mathcal W$, but $\mathcal V$ and $\mathcal W$ are Ketonen-incomparable (that is, neither $\mathcal V<_k\mathcal W$ nor $\mathcal W<_k\mathcal V$); moreover, it is also consistent that such a situation occurs in a model satisfying the Weak Ultrapower Axiom.

Choices:
B. It is consistent that whenever $\mathcal V,\mathcal W$ are $\sigma$-complete ultrafilters with $\mathcal V<_L\mathcal W$, one has $\mathcal V<_k\mathcal W$; in particular, in every model satisfying the Weak Ultrapower Axiom, the Lipschitz and Ketonen orders coincide on $\sigma$-complete ultrafilters.
C. It is consistent that there exist two $\sigma$-complete ultrafilters $\mathcal V,\mathcal W$ such that $\mathcal V<_L\mathcal W$, but $\mathcal V\not<_k\mathcal W$.
D. It is consistent that there exist two $\sigma$-complete ultrafilters $\mathcal V,\mathcal W$ such that $\mathcal V<_L\mathcal W$, but $\mathcal W<_k\mathcal V$; moreover, it is also consistent that such a situation occurs in a model satisfying the Weak Ultrapower Axiom.
E. It is consistent that there exist two $\sigma$-complete ultrafilters $\mathcal V,\mathcal W$ such that $\mathcal V<_L\mathcal W$, $\mathcal V$ and $\mathcal W$ are Ketonen-incomparable, and this already follows in every model satisfying the Weak Ultrapower Axiom.

dropped reverse non-comparability and Weak UA refinement

incomparability replaced by reverse Ketonen comparability

consistency in some Weak-UA model strengthened to consequence of all Weak-UA models

The stem gives definitions of the Lipschitz order, Ketonen order, and Weak Ultrapower Axiom, but it does not explicitly or implicitly reveal which of the listed consistency statements is correct.

This is not a mere restatement of a definition or theorem. The respondent must distinguish among several nearby consistency claims involving separation, incomparability, and the role of Weak UA.

The item does require some reasoning about logical strength and the interaction between the two orders. However, the reasoning pressure is weakened because one distractor is a weaker statement entailed by the marked correct answer, so the task becomes ambiguous rather than cleanly generative.

The distractors are not all functioning well. In particular, choice C is a weaker statement implied by A, so it is also consistent if A is consistent. That creates answer ambiguity. The other distractors are distinct, but the presence of an actually true weaker option is a major flaw.

A mathematically sophisticated and non-tautological item, but it is significantly weakened by poor option design: the marked correct answer entails another choice, so the MCQ does not have a uniquely correct answer as written.


\label{thmA}
Let $M$ be a type $\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\theta \in \operatorname{Aut}(M)$ be a state preserving automorphism. Then the following conditions are equivalent:
\begin{enumerate}
	\item $\theta$ has infinite order in $\mathrm{Out}(M)$;
	\item there exists $u \in \mathcal U(M)$ such that $\operatorname{Ad}(u)\circ\theta$ is state preserving and ergodic.
\end{enumerate}

\begin{thmA}\label{thmA}
Let $M$ be a type $\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\theta \in \Aut(M)$ be a state preserving automorphism. Then the following conditions are equivalent:
\begin{enumerate}
	\item $\theta$ has infinite order in $\mathrm{Out}(M)$;
	\item there exists $u \in \mathcal U(M)$ such that $\Ad(u)\circ\theta$ is state preserving and ergodic.
\end{enumerate}
\end{thmA}

Cocycle perturbations of group actions provide a useful method for modifying dynamical
properties while preserving the underlying von Neumann algebraic structure.
In the case of type $\mathrm{II}_1$ factors, Marrakchi and Vaes \cite{MV23} proved that for every
outer action of a countable amenable group, one can find a 1-cocycle such that the
perturbed action becomes ergodic, and moreover that such cocycles form a dense $G_\delta$
subset of the cocycle space.
We refer to such a cocycle as an \emph{ergodic cocycle}.
This settles the ergodicity problem for cocycle perturbations in the type $\mathrm{II}_1$ setting,
and in particular implies that for a single automorphism, the existence of an ergodic cocycle
is equivalent to the outerness of all its nonzero powers.

The purpose of this paper is to extend the above framework to the \emph{state preserving}
setting for type $\mathrm{III}$ factors, where state preserving means that the action
leaves a faithful normal state invariant.
While many concrete examples of ergodic actions on type $\mathrm{III}$ factors are known,
it has remained open whether an outer action admits an ergodic cocycle.
The state preserving assumption ensures that the action extends to an isometry on the
GNS Hilbert space, which underlies the topological arguments used in \cite{MV23} and
in the present work.
We also note that Marrakchi and Vaes \cite[Open problem~4.6]{MV23} explicitly asked whether their
results could extend to type $\mathrm{III}$ factors and suggested that the state preserving
case might be tractable.

Our main theorem, stated below, gives an affirmative answer for type $\mathrm{III}_1$ factors under the assumption of \emph{trivial bicentralizer}, see Subsection~\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\mathrm{III}_1$ setting is settled in essentially all cases.

\label{Preliminaries}

	Let $M$ be a von Neumann algebra and $\varphi\in M_\ast$ a faithful state. The \textit{modular operator, conjugation}, and \textit{action} are denoted by $\Delta_\varphi$, $J_\

Our main theorem, stated below, gives an affirmative answer for type $\mathrm{III}_1$ factors under the assumption of \emph{trivial bicentralizer}, see Subsection~\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\mathrm{III}_1$ setting is settled in essentially all cases.

When $M$ is amenable, the state preserving assumption can be removed by using the classification of automorphisms of the amenable type $\mathrm{III}_1$ factor \cite{KST89}.
This yields the following corollary.

\begin{thmA}\label{thmC}
Let $\alpha \colon \Gamma \actson M$ be a state preserving outer action of a countably infinite discrete group $\Gamma$ on a type $\mathrm{III}_1$ factor with separable predual.
Assume that $\Gamma$ is amenable and that $M$ has trivial bicentralizer.
If $\Gamma_{\mathrm{mod}}$ is trivial, then there exists an $\alpha$-cocycle
$u \colon \Gamma \to \mathcal U(M)$ such that the perturbed action
is state preserving and weakly mixing.
\end{thmA}

\begin{lem}\label{lem-relative-bicentralizer}
 Let $\Gamma \actson M$ be a state preserving outer action of a countable group on a von Neumann algebra. Assume that:
\begin{itemize}
 \item $\Gamma_{\mathrm{mod}}$ is trivial;
 \item $M$ is a type $\mathrm{III}_1$ factor with separable predual and has trivial bicentralizer.
\end{itemize}
Then we have $\mathrm{BC}(M \subset M\rtimes_\alpha \Gamma, \varphi)=\mathbb C$ for all faithful state $\varphi\in M_\ast$.
\end{lem}

\begin{lem}\label{lem-cocycle3.6}
 Let $\alpha\colon \Gamma \actson (M,\varphi)$ be a state preserving outer action of a countably infinite discrete group on a diffuse factor with separable predual and with a faithful normal state.
Assume that $\Gamma$ is amenable and that $\mathrm{BC}(M\subset M\rtimes \Gamma,\varphi)=\C$.
Then for every $\varepsilon>0$ and every finite subset $F\subset \Gamma$, there exist $h\in \Gamma$ and $U\in \mathcal U(M^\omega)$ such that:
\begin{itemize}
 \item $\varphi^\omega(U)=0$, \quad $\|U\varphi^\omega-\varphi^\omega U\|<\varepsilon$,\quad and \quad $\|\alpha_g^\omega(U)-U\|_{\varphi^\omega}<\varepsilon$ for all $g\in F$;
 \item $M$, $U$, and $\alpha_h^\omega(U)$ are $\ast$-free with respect to $\varphi^\omega$.
\end{itemize}
\end{lem}

\begin{lem}\label{lem-cocycle3.5}
Let $\alpha\colon \Gamma \actson (M,\varphi)$ be a state preserving outer action of a countably infinite discrete group on a diffuse factor with separable predual. Assume that $\Gamma$ is amenable and that $\mathrm{BC}(M\subset M\rtimes \Gamma,\varphi)=\C$. For any $x\in M\setminus\C$, any finite subset $E\subset M\setminus\C$, and any $\varepsilon>0$, the pair $(1,\varphi)\in\mathcal C_{\mathrm{state}}(\alpha)$ can be approximated by elements of $\mathcal U(x)\cap \mathcal V( E,\varepsilon)$, where $1\in\mathcal C(\alpha)$ is the trivial cocycle.
\end{lem}
\begin{proof}
 Fix $\varepsilon>0$, $x\in M\setminus\C$, finite subsets $F\subset\Gamma$ and $E\subset M$. We will show that for every sufficiently small $\delta>0$, there exists $u\in\mathcal U(M)$ satisfying the following conditions:
\begin{itemize}
 \item[(a)] $\|u-\alpha_g(u)\|_\varphi<\delta$ for all $g\in F$, and $\|u\varphi-\varphi u\|<\delta$;
 \item[(b)] there exists $h\in\Gamma$ such that
 \[
 \|\alpha_h(uxu^*)-uxu^*\|_\varphi>\|uxu^*-\varphi(uxu^*)\|_\varphi;
 \]
 \item[(c)] there exists $h\in\Gamma$ such that
 \[
 \sum_{a,b\in E}\bigl|
 \varphi(\alpha_h(uau^*)ubu^*)-\varphi(uau^*)\varphi(ubu^*)
 \bigr|<\varepsilon.
 \]
\end{itemize}
If such a unitary $u$ exists, define an $\alpha$-cocycle $v_g:=u^*\alpha_g(u)$ for $g\in\Gamma$ and set $\psi:=u^*\varphi u$. Then $ \alpha^v$ preserves $\psi$, and by (a) the distance between $(1,\varphi)$ and $(v,\psi)$ is small. Moreover, (b) and (c) ensures that $(v,\psi)\in\mathcal U(x) \cap \mathcal V(E,\varepsilon)$. Thus to prove the lemma, it suffices to construct such a unitary $u$.

\begin{thm}\label{thm-case1}
Let $\alpha \colon \Gamma \actson M$ be a state preserving outer action satisfying the following assumptions:
\begin{itemize}
 \item $\Gamma$ is amenable and countably infinite;
 \item $\Gamma_{\mathrm{mod}} = \{e\}$;
 \item $M$ is a type ${\rm III_1}$ factor with separable predual and has trivial bicentralizer.
\end{itemize}
Then both $\mathcal C_{\mathrm{erg}}(\alpha)$ and $\mathcal C_{\mathrm{wm}}(\alpha)$ are dense $G_\delta$ subsets of $\mathcal C_{\mathrm{state}}(\alpha)$.
\end{thm}

\begin{cor}
Let $M$ be a type $\mathrm{III}_\lambda$ factor $(0<\lambda<1)$ with separable predual, and let
$\varphi\in M_\ast$ be a faithful state such that $\sigma^\varphi$ has period
$T$, where $T=-2\pi/\log(\lambda)$.
Let $\alpha \colon \Gamma \actson (M,\varphi)$ be a state preserving outer action
satisfying the following assumptions:
\begin{itemize}
 \item $\Gamma$ is amenable and countably infinite;
 \item $\Gamma_{\mathrm{mod}} = \{e\}$.
\end{itemize}
Then both $\mathcal C_{\mathrm{erg}}(\alpha,\varphi)$ and
$\mathcal C_{\mathrm{wm}}(\alpha,\varphi)$ are dense $G_\delta$ subsets
of $\mathcal C(\alpha,\varphi)$.
\end{cor}

\label{Preliminaries}

	Let $M$ be a von Neumann algebra and $\varphi\in M_\ast$ a faithful state. The \textit{modular operator, conjugation}, and \textit{action} are denoted by $\Delta_\varphi$, $J_\

When $M$ is amenable, the state preserving assumption can be removed by using the classification of automorphisms of the amenable type $\mathrm{III}_1$ factor \cite{KST89}.
This yields the following corollary.

\begin{corA}\label{corB}
Every automorphism of the amenable type $\mathrm{III}_1$ factor $R_\infty$ that has infinite order in $\mathrm{Out}(R_\infty)$ admits an ergodic cocycle.
\end{corA}

For a von Neumann algebra $M$ with a faithful state $\varphi \in M_\ast$, define
\[
\mathrm{Mod}(M)
:= \{\, \operatorname{Ad}(u)\circ\sigma_t^{\varphi} \in \operatorname{Aut}(M) \mid u \in \mathcal U(M),\ t \in \mathbb R \,\}.
\]
This is a normal subgroup of $\operatorname{Aut}(M)$ containing $\mathrm{Int}(M)$.
For a discrete group action $\alpha \colon \Gamma \to \operatorname{Aut}(M)$, we define the normal subgroup
\[
\Gamma_{\mathrm{mod}} := \alpha^{-1}(\mathrm{Mod}(M)) \le \Gamma.
\]
For outer actions on type $\mathrm{II}$ factors, one always has $\Gamma_{\mathrm{mod}}=\{e\}$.
One of the major differences in the type $\mathrm{III}$ setting is that $\Gamma_{\mathrm{mod}}$
may be nontrivial.
The proof of Theorem~\ref{thmA} will be divided according to whether
$\Gamma_{\mathrm{mod}}$ is trivial or not.

\bigskip
\noindent
{\bf Case 1: $\Gamma_{\mathrm{mod}}$ is trivial.}
In the case where $\Gamma_{\mathrm{mod}}$ is trivial, we prove the following theorem,
which is a direct generalization of \cite{MV23} to the type III setting.

\begin{thmA}\label{thmC}
Let $\alpha \colon \Gamma \curvearrowright M$ be a state preserving outer action of a countably infinite discrete group $\Gamma$ on a type $\mathrm{III}_1$ factor with separable predual.
Assume that $\Gamma$ is amenable and that $M$ has trivial bicentralizer.
If $\Gamma_{\mathrm{mod}}$ is trivial, then there exists an $\alpha$-cocycle
$u \colon \Gamma \to \mathcal U(M)$ such that the perturbed action
is state preserving and weakly mixing.
\end{thmA}

Up to stabilizing by another type $\mathrm{III}_1$ factor, we obtain the following consequence.

\begin{corA}\label{corD}
Let $\alpha \colon \Gamma \curvearrowright M$ be a state preserving outer action of a countably infinite discrete amenable group on a type $\mathrm{III}_1$ factor with separable predual. Let $N$ be another type $\mathrm{III}_1$ factor with separable predual. Then the action $\alpha \,\overline\otimes\, \id_N \colon \Gamma \curvearrowright M \,\overline\otimes\, N$ admits a cocycle perturbation that is state preserving and weakly mixing.
\end{corA}

The original proof of \cite{MV23} in the case of $\mathrm{II}_1$ factors relies heavily on Popa's free independence result \cite{Po95} (and its variant in \cite{PSV18}). This free independence theorem has been generalized to the setting of type III factors under appropriate hypotheses in \cite{HI14}. Therefore, if a state preserving action $\alpha\colon \Gamma \curvearrowright (M,\varphi)$ in Theorem~\ref{thmC} further satisfies $M_\varphi' \cap M=\mathbb C$, then by making use of \cite{HI14}, the proof becomes a rather direct adaptation of the argument in \cite{MV23}.

However, the theorem also covers the case where $M_\varphi=\mathbb C$, in which \cite{HI14} cannot be applied directly. The key point of our proof is the Connes--St\o rmer transitivity theorem for type $\mathrm{III}_1$ factors, which ensures that all faithful states become unitarily conjugate at the level of ultrapowers. As a consequence, $(M^\omega)_{\varphi^\omega}$ is sufficiently large for every $\varphi$, and this fact is the key technical ingredient of our proof.

\bigskip
\noindent
{\bf Case 2: $\Gamma_{\mathrm{mod}}$ is nontrivial.}
When $\Gamma_{\mathrm{mod}}$ is nontrivial, it is not clear how to deal with this situation in full generality. We therefore restrict ourselves to the case $\Gamma=\mathbb Z$, that is, to single automorphisms. In this case $\Gamma_{\mathrm{mod}}$ has the form $p\mathbb Z \leq \mathbb Z$ for some $p>0$, and the action restricted to $p\mathbb Z$ agrees with a modular action. By \cite{MV23}, one obtains an ergodic cocycle for $p\mathbb Z$. We then analyze how such a cocycle can be lifted appropriately to the entire group $\mathbb Z$. This step does not appear in \cite{MV23} and requires a new argument, which will be carried out in Theorem~\ref{thm-case2}.
\bigskip

In the final section, we collect several related results, including the case of type $\mathrm{III}_\lambda$ factors and additional properties of actions and cocycles on ultrapowers.

\tableofcontents

The proof of Theorem~\ref{thmA} is split depending on whether \(\Gamma_{\mathrm{mod}}:=\alpha^{-1}(\mathrm{Mod}(M))\) is trivial or not.

\noindent\textbf{Case 1: \(\Gamma_{\mathrm{mod}}\) is trivial.} One proves Theorem~\ref{thmC}, described as “a direct generalization of \cite{MV23} to the type III setting,” producing an \(\alpha\)-cocycle \(u\colon\Gamma\to\mathcal U(M)\) so that the perturbed action is “state preserving and weakly mixing.” The discussion explains that if additionally \(M_\varphi'\cap M=\mathbb C\), then using the type III free independence result \cite{HI14}, “the proof becomes a rather direct adaptation of the argument in \cite{MV23}.” In the remaining situation “where \(M_\varphi=\mathbb C\), in which \cite{HI14} cannot be applied directly,” the “key point” is the Connes--St\o rmer transitivity theorem for type \(\mathrm{III}_1\) factors: it “ensures that all faithful states become unitarily conjugate at the level of ultrapowers,” so “\((M^\omega)_{\varphi^\omega}\) is sufficiently large for every \(\varphi\), and this fact is the key technical ingredient of our proof.”

\noindent\textbf{Case 2: \(\Gamma_{\mathrm{mod}}\) is nontrivial.} The text says this is not handled “in full generality,” so it restricts to \(\Gamma=\mathbb Z\) (single automorphisms). Then \(\Gamma_{\mathrm{mod}}=p\mathbb Z\) for some \(p>0\) and the restriction to \(p\mathbb Z\) “agrees with a modular action.” Using \cite{MV23}, one “obtains an ergodic cocycle for \(p\mathbb Z\),” and then “analyze[s] how such a cocycle can be lifted appropriately to the entire group \(\mathbb Z\).” This lifting step “does not appear in \cite{MV23} and requires a new argument,” to be carried out later (Theorem~\ref{thm-case2}).

The proof of the main theorem is split depending on whether \(\Gamma_{\mathrm{mod}}:=\alpha^{-1}(\mathrm{Mod}(M))\) is trivial or not.

\noindent\textbf{Case 1: \(\Gamma_{\mathrm{mod}}\) is trivial.} We first prove the following theorem.
\begin{thmA}\label{thmC}
Let $\alpha \colon \Gamma \actson M$ be a state preserving outer action of a countably infinite discrete group $\Gamma$ on a type $\mathrm{III}_1$ factor with separable predual.
Assume that $\Gamma$ is amenable and that $M$ has trivial bicentralizer.
If $\Gamma_{\mathrm{mod}}$ is trivial, then there exists an $\alpha$-cocycle
$u \colon \Gamma \to \mathcal U(M)$ such that the perturbed action
is state preserving and weakly mixing.
\end{thmA}
This theorem is described as “a direct generalization of \cite{MV23} to the type III setting,” producing an \(\alpha\)-cocycle \(u\colon\Gamma\to\mathcal U(M)\) so that the perturbed action is “state preserving and weakly mixing.” The discussion explains that if additionally \(M_\varphi'\cap M=\mathbb C\), then using the type III free independence result \cite{HI14}, “the proof becomes a rather direct adaptation of the argument in \cite{MV23}.” In the remaining situation “where \(M_\varphi=\mathbb C\), in which \cite{HI14} cannot be applied directly,” the “key point” is the Connes--St\o rmer transitivity theorem for type \(\mathrm{III}_1\) factors: it “ensures that all faithful states become unitarily conjugate at the level of ultrapowers,” so “\((M^\omega)_{\varphi^\omega}\) is sufficiently large for every \(\varphi\), and this fact is the key technical ingredient of our proof.”

\noindent\textbf{Case 2: \(\Gamma_{\mathrm{mod}}\) is nontrivial.} The text says this is not handled “in full generality,” so it restricts to \(\Gamma=\mathbb Z\) (single automorphisms). Then \(\Gamma_{\mathrm{mod}}=p\mathbb Z\) for some \(p>0\) and the restriction to \(p\mathbb Z\) “agrees with a modular action.” Using \cite{MV23}, one “obtains an ergodic cocycle for \(p\mathbb Z\),” and then “analyze[s] how such a cocycle can be lifted appropriately to the entire group \(\mathbb Z\).” This lifting step “does not appear in \cite{MV23} and requires a new argument,” to be carried out later. We next prove the following theorem.
\begin{thm}\label{thm-case2}
Let $M$ be a type ${\rm III_1}$ factor with separable predual, $\varphi\in M_\ast$ a faithful state, and $\alpha\colon \mathbb Z \actson (M,\varphi)$ a state preserving outer action such that $\Gamma_{\mathrm{mod}} = p\mathbb Z$ for some $p\geq 1$.
Then there exists a unitary $u\in \mathcal U(M)$ such that $\alpha^u$ is an ergodic state preserving action.
\end{thm}

Question: Let \(M\) be a type \(\mathrm{III}_1\) factor with separable predual and trivial bicentralizer, and let \(\theta\in\operatorname{Aut}(M)\) be state preserving, meaning that \(\theta\) leaves some faithful normal state on \(M\) invariant. Write \(\mathrm{Out}(M)=\operatorname{Aut}(M)/\operatorname{Inn}(M)\), so that saying \(\theta\) has infinite order in \(\mathrm{Out}(M)\) means that \([\theta]^n\neq e\) for every nonzero integer \(n\). Also, \(\mathcal U(M)\) denotes the unitary group of \(M\), \(\operatorname{Ad}(u)(x)=uxu^*\), and an automorphism \(\beta\) of \(M\) is ergodic if its fixed-point algebra \(\{x\in M:\beta(x)=x\}\) is \(\mathbb C1\). Which statement holds?

The following are equivalent: (i) \(\theta\) has infinite order in \(\mathrm{Out}(M)\); (ii) there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is state preserving and ergodic.

Choices:
B. The following are equivalent: (i) \(\theta\) has finite order in \(\mathrm{Out}(M)\); (ii) there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is state preserving and ergodic.
C. If there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is state preserving and ergodic, then \(\theta\) has infinite order in \(\mathrm{Out}(M)\).
D. The following are equivalent: (i) \(\theta\) has infinite order in \(\mathrm{Out}(M)\); (ii) there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is ergodic.
E. The following are equivalent: (i) every nonzero power \(\theta^n\) is outer; (ii) for every faithful normal state \(\psi\) on \(M\), there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) preserves \(\psi\) and is ergodic.

dropped the converse implication from infinite order to existence of an ergodic state-preserving perturbation

state-preserving requirement in the perturbation conclusion

existential state in the hypothesis replaced by uniform preservation for every faithful normal state

The stem does not explicitly reveal the correct option or paraphrase it closely enough to make the answer trivial. It asks for an equivalent characterization, but the key property in the correct choice (existence of a unitary perturbation that is ergodic) is not given away in the question itself.

This is very close to a direct theorem-recall item: it asks which statement is equivalent to a given property, and the correct answer is essentially the target equivalence itself. The task is mainly to identify the exact formulation rather than derive a new conclusion.

Some reasoning is needed to compare nearby variants: weaker existence without ergodicity, stronger conditions like all powers ergodic, and quantifier changes over states. However, the item still primarily tests recognition of the precise equivalence rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful. They differ by common failure modes: dropping ergodicity, strengthening to weak mixing or all powers ergodic, and changing an existential statement to a universal one over preserved states.

A technically well-constructed recall/comparison MCQ with strong distractors, but it is largely a direct restatement of a theorem and only moderately tests reasoning.