[Structure of $f(R)$ gravity in spherical symmetry]
\label{main-theo}
Consider the field equations in $f(R)$ gravity \eqref{Eq1-14}, coupled to a
possibly massive, real-valued scalar field $\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\phi)$ satisfy the following conditions:
\bel{hypo-Einstein}
\aligned
& (1)
\, &  \phi \, U'(\phi) &\geq&& 0,
\\
& (2)
\, &  U(\phi) &\geq&& 0,
\\
& (3)
\, & f'(R) &>&&0,
\\
& (4)
\, &
f(R) &\leq&& R \, f'(R).
\endaligned
\ee
Then, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations, stated in Proposition~\ref{main-propo-first-order}, below.
Indeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the sense of Definition~\ref{definition}, below. Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \to R}$ and
$U(\phi)\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.

\label{section---1}

\subsection{Purpose of this paper}

\paragraph{Theories of modified gravity}

In recent years, new observational data suggest that extensions of Einstein's field equations may be relevant for explaining the accelerated expansion of the Universe and certain galactic-scale instabilities. Many of the proposed physical theories of modified gravity exhibit undesirable features, including lack of strong hyperbolicity in commonly used gauges, or non-uniqueness of solutions, or changes in the character of the evolution equations that may alter the system's causal structure. The development of robust numerical simulations in nonlinear, physically relevant regimes is hampered by the lack of mathematically rigorous formulations.
Indeed, only a few studies address the formulation and well-posedness of these theories \cite{Avalos,Lehner2,Lehner,Cocke,Hilditch1,Felice,Figueras,LeFlochMa17a,Reall,Reall2,Pretorius,Salgado}.
More recently, alternative, well-posed theories have been proposed; cf.~\cite{Brady-Figueras,Figueras} (and the references therein).

Among the various extensions of general relativity, $f(R)$ gravity is widely regarded as a natural and physically viable alternative to Einstein's theory \cite{Felice,Ferreira}. The $f(R)$ field equations are considerably more involved than the Einstein equations: in addition to the second-order Ricci curvature terms, they contain \emph{fourth-order} metric derivatives (in particular, second-order derivatives of the scalar curvature). This partly explains why, despite their physical importance, rigorous mathematical results to date encompass only local well-posedness~\cite{Felice,LeFlochMa17a} and the global nonlinear stability in the near-Minkowski regime \cite{LeFlochMa23}.

\paragraph{Evolution in spherical symmetry.}

In this paper, we initiate the study of the global evolution of a (massive) scalar field in $f(R)$ gravity in spherical symmetry. Christodoulou \cite{Chr,Chr2,Chr3} developed, within general relativity, a framework to analyze a massless scalar field in spherical symmetry using Bondi-type coordinates. This framework was also employed to study black-hole formation via numerical methods; cf.~\cite{Goldwirth-Piran} and the references therein. These developments inspired further investigations of gravitational collapse~\cite{Brady-etal,Hilditch2,Zhang-Lu, Michel-Moss}, as well as additional mathematical results, cf.~\cite{Chae,Costa-Mena,Costa-Duarte-Mena} (classical solutions) and \cite{LeFloch-Mena} (generalized solutions). Building on Christodoulou's subsequent work using a double-null foliation~\cite{Chr-bv,Chr-naked}, many studies have also addressed the gravitational collapse toward black holes in spherical symmetry. Next, with the growing interest in modified gravity, subsequent works include \cite{Quo-Joshi, Zhang-etal-2016, Chow}, which mostly consider quadratic curvature corrections to the Einstein--Hilbert action. Despite these advances, rigorous mathematical results on the \emph{global geometry} of Cauchy developments in $f(R)$ gravity are still lacking.

\paragraph{Aim of this paper.}

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou  \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's  insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

The new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing  an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \to R}$), as well as the special case of a massless scalar field.

\subsection{A first-order formulation of $f(R)$ gravity}

\paragraph{Action of $f(R)$ modified gravity.}

Recall first that Einstein's theory is based on the Einstein--Hilbert action
\be
\Acal_{\text{EH}}[\phi, g] := \int_M \Big( {R_g  \over 16 \pi} + L[\phi,g] \Big) \, dV_g,
\ee
associated with a $(1+3)$--dimensional spacetime $({\Mcal},g)$ with signature $(-, +,+,+)$.
Hence, the functional $\Acal_{\text{EH}}[\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and
 a Lagrangian $L[\phi,g]$ describing the matter content and represented by certain fields $\phi$ defined on $\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.
As is well-known, the action $\Acal_{\text{EH}}[\phi, g]$ is (formally) critical at metrics $g$
satisfying Einstein's field equations
\be
\label{Eq1-01}
G := \Ric - {R_g \over 2} \, g = 8 \pi \, T[\phi,g]
\ee
in which the right-hand side
\bel{Eq:12}
T_{ab}[\phi,g] := -2 \, {\delta L \over \delta g^{ab}} [\phi,g]  + g_{ab}\, L[\phi,g]
\ee
is the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \ldots, 3$ denoting spacetime components).
Our emphasis in this paper is on massive scalar fields $\phi$ described by
\be
\label{stress-energy}
T_{ab} = \nabla_a \phi \nabla_b \phi- \big( {1 \over 2} \nabla^c \phi \nabla_c \phi + U(\phi) \Big) g_{ab},
\ee
in which the potential function is $U=U(\phi)$, so that the field $\phi$ satisfies the Klein-Gordon equation
\be
\Box_g \phi = U'(\phi).
\end{equation}
We study an extension of Einstein's theory, defined as follows. A function $f: \RR \to \RR$ being given, we consider the modified gravity action for the $f(R)$ theory
$$
\Acal_{\text{MG}}[\phi,g] =: \int_M \Big( {f(R) \over 16 \pi} + L[\phi, g]\Big) \, dV,
$$
whose critical points satisfy the following \emph{field equations of modified gravity}
\bel{Eq1-14}
E_{ab} :=
f'(R) \, G_{ab} - \frac{1}{2} \Big( f(R) - R f'(R) \Big) g_{ab}
+  \big( g_{ab} \, \Box_g   - \nabla_a\nabla_b \big)  \big( f'(R) \big) = 8 \pi \, T_{ab}[\phi,g].
\ee
The right-hand side is still given by \eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces
the Einstein tensor $G_{ab}$ and satisfies $\nabla^a E_{ab} = 0$,
 so that $T_{ab}$ is also divergence-free.

\paragraph{Main result.}

In local coordinates, \eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \eqref{Eq1-01} leads to second-order equations. We investigate
how to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.
The condition $f'(R) >0$ in \eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \emph{conformal metric} $\gt$ and the \emph{augmented variable} $\rho$ (in \eqref{equa-233}, below):
\be
\gt_{ab} := e^{\kappa \rho} g_{ab},
\qquad
\rho := {1 \over \kappa} \log f'(R).
\ee
Here, $\kappa>0$ is a parameter  for the definition of $\rho$, so that the general relativity limit corresponds to $f(R)\to R$ when ${\kappa \to 0}$. We summarize our results as follows.

\vskip.15cm

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

The new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \to R}$), as well as the special case of a massless scalar field.

Recall first that Einstein's theory is based on the Einstein--Hilbert action
\be
\Acal_{\text{EH}}[\phi, g] := \int_M \Big( {R_g \over 16 \pi} + L[\phi,g] \Big) \, dV_g,
\ee
associated with a $(1+3)$--dimensional spacetime $({\Mcal},g)$ with signature $(-, +,+,+)$.
Hence, the functional $\Acal_{\text{EH}}[\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and
 a Lagrangian $L[\phi,g]$ describing the matter content and represented by certain fields $\phi$ defined on $\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.
As is well-known, the action $\Acal_{\text{EH}}[\phi, g]$ is (formally) critical at metrics $g$
satisfying Einstein's field equations
\be
\label{Eq1-01}
G := \Ric - {R_g \over 2} \, g = 8 \pi \, T[\phi,g]
\ee
in which the right-hand side
\bel{Eq:12}
T_{ab}[\phi,g] := -2 \, {\delta L \over \delta g^{ab}} [\phi,g] + g_{ab}\, L[\phi,g]
\ee
is the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \ldots, 3$ denoting spacetime components).
Our emphasis in this paper is on massive scalar fields $\phi$ described by
\be
\label{stress-energy}
T_{ab} = \nabla_a \phi \nabla_b \phi- \big( {1 \over 2} \nabla^c \phi \nabla_c \phi + U(\phi) \Big) g_{ab},
\ee
in which the potential function is $U=U(\phi)$, so that the field $\phi$ satisfies the Klein-Gordon equation
\be
\Box_g \phi = U'(\phi).
\end{equation}
We study an extension of Einstein's theory, defined as follows. A function $f: \RR \to \RR$ being given, we consider the modified gravity action for the $f(R)$ theory
$$
\Acal_{\text{MG}}[\phi,g] =: \int_M \Big( {f(R) \over 16 \pi} + L[\phi, g]\Big) \, dV,
$$
whose critical points satisfy the following \emph{field equations of modified gravity}
\bel{Eq1-14}
E_{ab} :=
f'(R) \, G_{ab} - \frac{1}{2} \Big( f(R) - R f'(R) \Big) g_{ab}
+ \big( g_{ab} \, \Box_g - \nabla_a\nabla_b \big) \big( f'(R) \big) = 8 \pi \, T_{ab}[\phi,g].
\ee
The right-hand side is still given by \eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces
the Einstein tensor $G_{ab}$ and satisfies $\nabla^a E_{ab} = 0$,
 so that $T_{ab}$ is also divergence-free.

In local coordinates, \eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \eqref{Eq1-01} leads to second-order equations. We investigate
how to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.
The condition $f'(R) >0$ in \eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \emph{conformal metric} $\gt$ and the \emph{augmented variable} $\rho$ (in \eqref{equa-233}, below):
\be
\gt_{ab} := e^{\kappa \rho} g_{ab},
\qquad
\rho := {1 \over \kappa} \log f'(R).
\ee
Here, $\kappa>0$ is a parameter for the definition of $\rho$, so that the general relativity limit corresponds to $f(R)\to R$ when ${\kappa \to 0}$. We summarize our results as follows.

\vskip.15cm

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

\vskip.15cm

\vskip.15cm

In the modified field equations, there are terms that contain derivatives of the scalar
curvature, that is, third order derivatives of the metric functions $\nu$ and $\lambda$. This leads to an essential difficulty in dealing with the equations.
Following LeFloch and Ma~\cite{LeFlochMa17a}, we introduce an \emph{augmented system}
 where the relation
\be
\rho = \frac{1}{\kappa}\ln f'(R) \quad \text{(conformal formulation)}
\ee
is no longer imposed but $\rho$
 is regarded as an independent variable.
To clarify the notation, we introduce a new independent variable denoted by
\be
\rhoh \quad \text{(augmented conformal formulation)}
\ee
which plays the role of $\rho$ and will coincides with $\rho$ once the constraint below is enforced.
In view of Proposition~\ref{prop essential f(R) Bondi}, this leads us to the following system
\bel{Bondi system diff augmented1.5}
\aligned
\del_r(\nu+\lambda)
& = \frac{3}{4}\kappa^{2}r|\del_r\rhoh|^2 + 4\pi r e^{- \kappa\rhoh} \, |\del_r\phi|^2,
\\
\del_r\big(re^{\nu- \lambda}\big)
 & = \big(1 - r^{2}e^{-2\kappa\rhoh} \big( \Vstar(\rhoh) + 8\pi U(\phi) \big) \big)e^{\nu+\lambda},
 \\
D(\del_r(r\phi)) + {r \over 2} e^{\nu+\lambda- \kappa\rhoh}U'(\phi)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rhoh}\big(\Vstar(\rhoh)+8\pi U(\phi)\big)\Big)\del_r\phi \\
& \quad + \frac{\kappa r}{2}\big(D\rhoh \del_r\phi + D\phi \del_r\rhoh\big),
\\
D\big(\del_r(r\rhoh)\big) + {r \over \kappa} e^{\nu+\lambda} \Wstar(\rhoh)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rhoh}\big(\Vstar(\rhoh)+8\pi U(\phi)\big)\Big)\del_r\rhoh
\\
& - \frac{8\pi r}{3\kappa} e^{- \kappa\rhoh}\del_r\phi D\phi
 + {r \over \kappa} e^{\nu+\lambda} \frac{16\pi}{3}e^{-2\kappa\rhoh}U(\phi).
\endaligned
\ee
Observe that, as before, the last equation of \eqref{Bondi system diff augmented1.5} is equivalent to
\be
\label{box-tilde-rho}
\Box_{\gt}\rhoh - \frac{2}{\kappa} \Wstar(\rhoh)
= - {8\pi \over 3\kappa}e^{-2\kappa\rhoh}\big(\sigma+4 \, U(\phi)\big).
\ee
Now that we have defined the augmented system, the next question is whether a solution of \eqref{Bondi system diff augmented1.5} (with a certain regularity) is also a solution of the original system. We emphasize that to be a solution of the full set of field equations, a solution of \eqref{equa-three-equa}, \eqref{Bondi scalar}, and \eqref{eq:599}
must additionally satisfy the condition
\bel{Bondi system diff augmented1.5 constraint}
f'(R) = e^{\kappa\rhoh},
\ee
which we regard as a nonlinear differential constraint on the solutions.
 Conversely, any classical solution of the original $f(R)$ system with $\rho=\frac{1}{\kappa}\ln f'(R)$ satisfies \eqref{Bondi system diff augmented1.5} upon setting $\rhoh=\rho$; thus the two formulations are equivalent in the admissible class once \eqref{Bondi system diff augmented1.5 constraint} holds on the initial cone (and is then propagated by the evolution)

Under the regularity conditions at the center and the asymptotic flatness condition, we can reduce the field equations (cf.~Appendix~\ref{section---A2})) to the the following integro-differential system with the main unknowns $\phi$ and $\rho$:
\bel{notresysteme}
\aligned
D(\del_r(r\phi))
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rho}\big(\Vstar(\rho)+8\pi U(\phi)\big)\Big)\del_r\phi
\\
 & \quad + \frac{\kappa r}{2}\big(D\rho \del_r\phi + D\phi \del_r\rho\big)-{r \over 2} e^{\nu+\lambda- \kappa\rho}U'(\phi),
\\
D\big(\del_r(r\rho)\big)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rho}\big(\Vstar(\rho)+8\pi U(\phi)\big)\Big)\del_r\rho
\\
& \quad - \frac{8\pi r}{3\kappa} e^{- \kappa\rho}\del_r\phi D\phi- {r \over \kappa} e^{\nu+\lambda}\Big(\Wstar(\rho) - \frac{16\pi}{3}e^{-2\kappa\rho}U(\phi)\Big),
\endaligned
\ee
in which the metric coefficients $\nu, \lambda$ are given by
\bel{eq:403}
\aligned
\nu + \lambda
& = - \frac{3}{4}\kappa^{2} \int_r^{+ \infty}s|\del_r\rho|^2ds - 4\pi\int_r^{+ \infty} e^{- \kappa\rho} \, s|\del_r\phi|^2 \, ds,
\\
e^{\nu- \lambda}
& = {1 \over r} \int_0^r\Big(1 - s^{2}e^{-2\kappa\rho} \big( \Vstar(\rho) + 8\pi U(\phi) \big)\Big)e^{\nu+\lambda} \, ds.
\endaligned
\ee
We refer to \eqref{notresysteme} as the \emph{augmented conformal system} of modified gravity.

Suppose now that $h$ and $l$ are solutions to the system \eqref{Bondi system int augmented3} defined on $[0,u_0]\times [0,+ \infty)$. If
\bel{center.condition.2}
h,l\in C^1\big([0,u_0]\times [0,+ \infty)\big),
\ee
then the first and second regularity conditions at the center hold. In fact, it follows from the system \eqref{Bondi system int augmented3} that
\be
\label{more-conditions-center}
\hb,\,\lb,\,\nu,\,\lambda\in C^1\big([0,u_0]\times [0,+ \infty)\big).
\ee
In view of \eqref{eq fR components} and \eqref{eq Bondi components T}, we get
\be
\aligned
& r^2{e^{-\kappa\lb}}\big(E-8\pi T\big)(D,D)
\\
& = r{e^{\nu- \lambda}}\Big(\frac{e^{\nu- \lambda}}{2}\del_r(\nu+\lambda) - 2\del_u\lambda\Big)
- \frac{3r^2\kappa^{2}{ } }{2}(D\lb)^2-8\pi r^2{ }(D\hb)^2.
\endaligned
\ee
Hence, from the regularity conditions, and taking into account \eqref{more-conditions-center}, we deduce that
\be
\lim_{r\to0}{\Big(r^2e^{-\kappa\rho}\big(E-8\pi T\big)(D,D)\Big)}= 0,
\ee
since $\nu,\lambda=O(r)$ and $D\hb,D\lb$ are bounded near $r=0$. For the second regularity condition at the center, we compute
\be
r\del_r\lb = l- \lb, \qquad r\del_r\hb = h- \hb.
\ee
 This shows that $h=\partial_r(r\phi)$ and $l=\partial_r(r\rho)$ are $C^1$ on $[0,u_0]\times[0,+\infty)$, as claimed. Conversely, if the two center regularity conditions hold and $(h,l)\in C^0$, the right-hand sides of \eqref{Bondi system int augmented3} are continuous; integrating along characteristics then yields $(h,l)\in C^1$, so the two formulations are equivalent in the admissible class.

\vskip.15cm

We point out that our assumptions are quite natural since they ensure positivity and monotonicity properties that also arise in the massless case. The condition $\phi \, U'(\phi) \geq 0$ is imposed since it guarantees that the Klein-Gordon energy is \emph{defocusing}, so that the forward evolution will not be limited by the matter model.
The conditions ${U(\phi) \geq 0}$ and ${f(R) \leq R \, f'(R)}$ are required to prove that the Hawking-mass is non-negative.

\subsection{Outline of this paper.}

In Section~\ref{section---2}, we introduce Bondi coordinates and express the field equations of $f(R)$ gravity for a scalar field. We then identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center. In Section~\ref{section---A2}, we analyze the regularity at the center. In Section~\ref{section---3}, we introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations. Finally, Section~\ref{section--- 4} is devoted to the study of the Hawking mass and its monotonicity properties.

In the discussion after Theorem~\ref{main-theo}, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\phi)\ge 0$ and $f(R)\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\phi U'(\phi)\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”

In the discussion after the main theorem, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\phi)\ge 0$ and $f(R)\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\phi U'(\phi)\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”

[Structure of $f(R)$ gravity in spherical symmetry]
\label{main-theo}
Consider the field equations in $f(R)$ gravity \eqref{Eq1-14}, coupled to a
possibly massive, real-valued scalar field $\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\phi)$ satisfy the following conditions:
\bel{hypo-Einstein}
\aligned
& (1)
\, &  \phi \, U'(\phi) &\geq&& 0,
\\
& (2)
\, &  U(\phi) &\geq&& 0,
\\
& (3)
\, & f'(R) &>&&0,
\\
& (4)
\, &
f(R) &\leq&& R \, f'(R).
\endaligned
\ee
Then, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations. We first state the following proposition.

\begin{proposition}[A first-order formulation of $f(R)$ gravity]
\label{main-propo-first-order}
\bse\label{Bondi system int augmented3}
The equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns:
\be
\aligned
& Dh = \frac{1}{2r}\big(g- \gb\big)(h- \hb) +
\frac{\kappa}{2}\big(q(h- \hb) + p(l- \lb)\big)-{r \over 2} e^{\nu+\lambda- \kappa\lb}U'(\hb),
\\
& Dl = \frac{1}{2r}\big(g- \gb\big)(l- \lb)
- \frac{8\pi}{3\kappa}e^{- \kappa\lb}(h- \hb) p-{r \over \kappa} e^{\nu+\lambda}\Big(\Wstar(\lb)- \frac{16\pi}{3}e^{-2\kappa\lb}U(\hb)\Big),
\endaligned
\ee
in which the metric coefficient $\nu + \lambda$ is computed by
\be
\aligned
& \nu + \lambda = - \frac{3\kappa^{2}}{4}\int_r^{+ \infty}\frac{|l- \lb|^2}{s} \, ds - 4\pi\int_r^{+ \infty}\frac{|h- \hb|^2}{se^{\kappa\lb}} \, ds,
\endaligned
\ee
and the auxiliary variables $p$ and $q$ are defined explicitly by
\bel{p,q}
\aligned
&p = \frac{\kappa}{2r}\int_0^r\Big((l- \lb)p + (h- \hb)q\Big) \, ds +\frac{1}{2r}\int_0^r\frac{e^{\nu- \lambda}(h- \hb)}{s} \, ds
- \frac{1}{2r}\int_0^rse^{\nu+\lambda- \kappa\lb}U'(\hb) \, ds,
\\
& \kappa q
= - \frac{8\pi}{3 r}\int_0^r\frac{(h- \hb)p}{e^{\kappa\lb}} \, ds+ \frac{\kappa}{2r}\int_0^r{\frac{e^{\nu- \lambda}(l- \lb)}{s} \, ds}
 - \frac{1}{r}\int_0^r se^{\nu+\lambda}\Big (\Wstar(\lb)-
\frac{16\pi}{3} e^{-2\kappa\lb}U(\hb)\Big) \, ds.
\endaligned
\ee
\ese
\end{proposition}

Indeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the following sense.

\begin{definition}
\label{definition}
\emph{The \emph{first regularity condition at the center\footnote{In view of \eqref{eq fR components} and \eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as
\be
\label{regularity}
\lim_{r\to0} r^2e^{-\kappa\rho}\big(E - 8\pi T\big)(D,D) = 0.
\ee
The \emph{second regularity condition at the center} is defined as
\be
\label{regularity2}
\del_r(r\phi) \text{ and }\del_r(r\rho) \text{ are } C^1\text{ on } [0,u_0]\times [0,+ \infty).
\ee
\end{definition}

Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \to R}$ and
$U(\phi)\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.,

Implication
Biconditional or Equivalence

Question: Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?

For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

Choices:
B. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
C. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
D. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
E. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

center regularity needed for equivalence

equivalence and Hawking-mass monotonicity clauses removed

direction/sign of Hawking-mass monotonicity along null directions

nonnegative Hawking mass incorrectly upgraded to constancy


Question:
Let \(\rho=(1/\kappa)\log f'(R)\). Assume a spherically symmetric spacetime in \(f(R)\) gravity is coupled to a real scalar field \(\phi\), and that the defining functions satisfy \(\phi U'(\phi)\ge 0\), \(U(\phi)\ge 0\), \(f'(R)>0\), and \(f(R)\le Rf'(R)\). In the admissible class of \(C^1\) solutions satisfying the center regularity conditions
\[
\lim_{r\to 0} r^2 e^{-\kappa\rho}(E-8\pi T)(D,D)=0,
\qquad
\partial_r(r\phi),\ \partial_r(r\rho)\in C^1([0,u_0]\times[0,\infty)),
\]
which of the following statements is equivalent to the full field equations \(E_{ab}=8\pi T_{ab}\) of \(f(R)\) gravity? Here \(D=\partial_u-\tfrac12 e^{\nu-\lambda}\partial_r\), \(h=\partial_r(r\phi)\), \(l=\partial_r(r\rho)\), \(\bar h(u,r)=\frac1r\int_0^r h(u,s)\,ds\), \(\bar l(u,r)=\frac1r\int_0^r l(u,s)\,ds\), \(g=e^{\nu+\lambda}\), and \(\bar g=e^{\nu-\lambda}\).

The equations are equivalent to the first-order integro-differential system with main unknowns \(h\) and \(l\):
\[
Dh=\frac{1}{2r}(g-\bar g)(h-\bar h)+\frac{\kappa}{2}\big(q(h-\bar h)+p(l-\bar l)\big)-\frac r2 e^{\nu+\lambda-\kappa\bar l}U'(\bar h),
\]
\[
Dl=\frac{1}{2r}(g-\bar g)(l-\bar l)-\frac{8\pi}{3\kappa}e^{-\kappa\bar l}(h-\bar h)p-\frac r\kappa e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big),
\]
with metric coefficient
\[
\nu+\lambda=-\frac{3\kappa^2}{4}\int_r^{\infty}\frac{|l-\bar l|^2}{s}\,ds-4\pi\int_r^{\infty}\frac{|h-\bar h|^2}{s e^{\kappa\bar l}}\,ds,
\]
and auxiliary variables \(p\) and \(q\) given by
\[
p=\frac{\kappa}{2r}\int_0^r\big((l-\bar l)p+(h-\bar h)q\big)\,ds+\frac{1}{2r}\int_0^r\frac{e^{\nu-\lambda}(h-\bar h)}{s}\,ds-\frac{1}{2r}\int_0^r s e^{\nu+\lambda-\kappa\bar l}U'(\bar h)\,ds,
\]
\[
\kappa q=-\frac{8\pi}{3r}\int_0^r\frac{(h-\bar h)p}{e^{\kappa\bar l}}\,ds+\frac{\kappa}{2r}\int_0^r\frac{e^{\nu-\lambda}(l-\bar l)}{s}\,ds-\frac{1}{r}\int_0^r s e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big)\,ds.
\]
Thus the spherically symmetric \(f(R)\) field equations are exactly the augmented first-order coupled nonlinear hyperbolic system for \(h=\partial_r(r\phi)\) and \(l=\partial_r(r\rho)\) in this regularity class.

Choices:
B.
The equations are equivalent to the first-order integro-differential system with main unknowns \(h\) and \(l\):
\[
Dh=\frac{1}{2r}(g-\bar g)(h-\bar h)+\frac{\kappa}{2}\big(q(h-\bar h)+p(l-\bar l)\big)-\frac r2 e^{\nu+\lambda-\kappa\bar l}U'(\bar h),
\]
\[
Dl=\frac{1}{2r}(g-\bar g)(l-\bar l)-\frac{8\pi}{3\kappa}e^{-\kappa\bar l}(h-\bar h)p-\frac r\kappa e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big),
\]
with metric coefficient
\[
\nu+\lambda=-\frac{3\kappa^2}{4}\int_0^{r}\frac{|l-\bar l|^2}{s}\,ds-4\pi\int_0^{r}\frac{|h-\bar h|^2}{s e^{\kappa\bar l}}\,ds,
\]
and auxiliary variables \(p\) and \(q\) given by
\[
p=\frac{\kappa}{2r}\int_0^r\big((l-\bar l)p+(h-\bar h)q\big)\,ds+\frac{1}{2r}\int_0^r\frac{e^{\nu-\lambda}(h-\bar h)}{s}\,ds-\frac{1}{2r}\int_0^r s e^{\nu+\lambda-\kappa\bar l}U'(\bar h)\,ds,
\]
\[
\kappa q=-\frac{8\pi}{3r}\int_0^r\frac{(h-\bar h)p}{e^{\kappa\bar l}}\,ds+\frac{\kappa}{2r}\int_0^r\frac{e^{\nu-\lambda}(l-\bar l)}{s}\,ds-\frac{1}{r}\int_0^r s e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big)\,ds.
\]
Thus the spherically symmetric \(f(R)\) field equations are exactly the augmented first-order coupled nonlinear hyperbolic system for \(h=\partial_r(r\phi)\) and \(l=\partial_r(r\rho)\) in this regularity class.
C.
The full field equations imply the augmented first-order system for \(h\) and \(l\) displayed in Proposition \(\ref{main-propo-first-order}\): namely, \(Dh\) and \(Dl\) satisfy the two coupled first-order equations, \(\nu+\lambda\) is recovered from
\[
\nu+\lambda=-\frac{3\kappa^2}{4}\int_r^{\infty}\frac{|l-\bar l|^2}{s}\,ds-4\pi\int_r^{\infty}\frac{|h-\bar h|^2}{s e^{\kappa\bar l}}\,ds,
\]
and \(p,q\) are determined by the stated integral relations. In particular, every admissible \(C^1\) solution of \(E_{ab}=8\pi T_{ab}\) yields a solution of this first-order integro-differential system.
D.
The equations are equivalent to the first-order integro-differential system with main unknowns \(h\) and \(l\):
\[
Dh=\frac{1}{2r}(g-\bar g)(h-\bar h)+\frac{\kappa}{2}\big(q(h-\bar h)+p(l-\bar l)\big)-\frac r2 e^{\nu+\lambda-\kappa\bar l}U'(\bar h),
\]
\[
Dl=\frac{1}{2r}(g-\bar g)(l-\bar l)-\frac{8\pi}{3\kappa}e^{-\kappa\bar l}(h-\bar h)p-\frac r\kappa e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big),
\]
with metric coefficient
\[
\nu+\lambda=-\frac{3\kappa^2}{4}\int_r^{\infty}\frac{|l-\bar l|^2}{s}\,ds-4\pi\int_r^{\infty}\frac{|h-\bar h|^2}{s e^{\kappa\bar l}}\,ds,
\]
and auxiliary variables \(p\) and \(q\) given by
\[
p=\frac{\kappa}{2r}\int_0^r\big((l-\bar l)p+(h-\bar h)q\big)\,ds+\frac{1}{2r}\int_0^r\frac{e^{\nu-\lambda}(h-\bar h)}{s}\,ds-\frac{1}{2r}\int_0^r s e^{\nu+\lambda-\kappa\bar l}U'(\bar h)\,ds,
\]
\[
\kappa q=-\frac{8\pi}{3r}\int_0^r\frac{(h-\bar h)p}{e^{\kappa\bar l}}\,ds+\frac{\kappa}{2r}\int_0^r\frac{e^{\nu-\lambda}(l-\bar l)}{s}\,ds-\frac{1}{r}\int_0^r s e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big)\,ds.
\]
Moreover, this equivalence holds for any \(C^1\) spherically symmetric solution, without imposing the two regularity conditions at the center.
E.
The equations are equivalent to the first-order integro-differential system with main unknowns \(h\) and \(l\):
\[
Dh=\frac{1}{2r}(g-\bar g)(h-\bar h)+\frac{\kappa}{2}\big(q(h-\bar h)+p(l-\bar l)\big)-\frac r2 e^{\nu+\lambda-\kappa\bar l}U'(\bar h),
\]
\[
Dl=\frac{1}{2r}(g-\bar g)(l-\bar l)-\frac{8\pi}{3\kappa}e^{-\kappa\bar l}(h-\bar h)p-\frac r\kappa e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big),
\]
with metric coefficient
\[
\nu+\lambda=-\frac{3\kappa^2}{4}\int_r^{\infty}\frac{|l-\bar l|^2}{s}\,ds-4\pi\int_r^{\infty}\frac{|h-\bar h|^2}{s e^{\kappa\bar l}}\,ds,
\]
and auxiliary variables \(p\) and \(q\) given by
\[
p=\frac{\kappa}{2r}\int_0^r\big((l-\bar l)p+(h-\bar h)q\big)\,ds+\frac{1}{2r}\int_0^r\frac{e^{\nu-\lambda}(h-\bar h)}{s}\,ds-\frac{1}{2r}\int_0^r s e^{\nu+\lambda-\kappa\bar l}U'(\bar h)\,ds,
\]
\[
\kappa q=-\frac{8\pi}{3r}\int_0^r\frac{(h-\bar h)p}{e^{\kappa\bar l}}\,ds+\frac{\kappa}{2r}\int_0^r\frac{e^{\nu-\lambda}(l-\bar l)}{s}\,ds-\frac{1}{r}\int_0^r s e^{\nu+\lambda}\Big(\mathcal W^*(\bar l)-\frac{16\pi}{3}e^{-2\kappa\bar l}U(\bar h)\Big)\,ds.
\]
Thus this augmented first-order system is equivalent not only to the full field equations in the admissible class, but also already implies the Hawking mass is both future non-decreasing along radial directions and future non-decreasing along null directions.


\label{Main thm}
Let $X/K$ be a smooth projective surface endowed with an elliptic fibration $\pi \colon X \rightarrow \mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/ \Spec(\mathcal{O})$ be a smooth projective model of $X$ over some open subset of $\Spec(\Oo_{K})$. Then $\Sigma(\mathcal{X})$ is a finitely generated abelian group.

\begin{thm} \label{Main thm}
Let $X/K$ be a smooth projective surface endowed with an elliptic fibration $\pi \colon X \rightarrow \Pp^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/ \Spec(\Oo)$ be a smooth projective model of $X$ over some open subset of $\Spec(\Oo_{K})$. Then $\Sigma(\mathcal{X})$ is a finitely generated abelian group.
\end{thm}

Let $X$ be a smooth projective variety over a number field $K$. A conjecture originally attributed to Swinnerton-Dyer (see \cite{zbMATH04061376}[Section 5] and also \cite{MR1086888} and \cite{MR1159204}) says that the Chow groups $\mathrm{CH}^i(X)$ are finitely generated. At present, this conjecture is widely open, and we do not even know whether the $n$-torsion subgroup $\mathrm{CH}^2(X)[n]$ is finitely generated when $X$ is a surface, although by Rojtman's theorem \cite{MR577137} this is true over the algebraic closure of $K$. The strongest evidence concerning the finiteness of $\mathrm{CH}^2(X)_{\mathrm{tors}}$ is due to \cite{CT-R}[Theorem C], where the authors show that $\mathrm{CH}^2(X)_{\mathrm{tors}}$ is finitely generated whenever $H^2(X, \Oo_X) = 0$ (see also \cite{MR1300892}). We refer the reader to \cite{MR1744949} and \cite{OTS}, which contain surveys regarding this problem and a more comprehensive account of the literature.

In general, to study $\mathrm{CH}^2(X)$ one spreads out $X$ to a smooth projective model $\mathcal{X} / \mathcal{O}$, where $\mathcal{O}$ is a localization of the ring of integers of $K$ at finitely many places, and considers the localization sequence
$$\bigoplus_{\mathfrak{p} \subset \mathcal{O}} \mathrm{Pic}(\mathcal{X}_{\Ff_{\mathfrak{p}}}) \rightarrow \mathrm{CH}^2(\mathcal{X}) \rightarrow \mathrm{CH}^2(X) \rightarrow 0$$
(note that this can be continued to the left using $K$-theory). Assume from now on that $H^1(X, \Oo_X) = 0$, so, after further localizing $\mathcal{O}$ if necessary, we have $\mathrm{Pic}(\mathcal{X}_{\Ff_{\mathfrak{p}}}) = \operatorname{NS}(\mathcal{X}_{\Ff_{\mathfrak{p}}})$ for every $\mathfrak{p} \subset \mathcal{O}$. The difficulty when $H^2(X, \Oo_X) \neq 0$ arises from the fact that the rank of $\operatorname{NS}(\mathcal{X}_{\Ff_{\mathfrak{p}}})$ can jump for infinitely many places (as for example happens for K3 surfaces), and one is not able to control the image of $\bigoplus_{\mathfrak{p} \subset \mathcal{O}} \mathrm{NS}(\mathcal{X}_{\Ff_{\mathfrak{p}}})$ in $\mathrm{CH}^2(\mathcal{X})$.

Let us call this image $\Sigma(\mathcal{X}) \cong \ker(\mathrm{CH}^2(\mathcal{X}) \rightarrow \mathrm{CH}^2(X) )$. This group appeared in \cite{MR1177312}, where the case $X = E \times E$ is studied, with $E/ \mathbb{Q}$ an elliptic curve. In particular, they show that $\Sigma(\mathcal{X})$ is finite when $K = \mathbb{Q}$ and $E$ has complex multiplication (there are only finitely many such curves over $\mathbb{Q}$). Moreover, the case where $E$ is an arbitrary elliptic curve over $\mathbb{Q}$ is also investigated. The proof starts by interpreting the new divisors modulo $\mathfrak{p}$ as graphs of the Frobenius, and then uses the theory of modular curves and their reduction modulo $\mathfrak{p}$ developed in \cite{MR337993} to control these cycles in $\mathrm{CH}^2(\mathcal{X})$. This is in fact one of the very few instances for which $H^2(X, \Oo_X) \neq 0$ and we nevertheless understand the role of vertical cycles in $\mathrm{CH}^2(\mathcal{X})$. See also \cite{MR1417619}, where these ideas are pushed further to prove more finiteness results regarding self products of elliptic curves over $\mathbb{Q}$. Our aim in this brief article is to generalize Mildenhall's finiteness to elliptic fibrations over $\mathbb{P}^1$:

Then all but finitely many primes satisfy (*), and the Tate algorithm implies that we can resolve the singularities of the Weierstrass model $\mathcal{X}'/R$ simultaneously over $R = \Oo_{\p}$. In particular, $X/K$ has good reduction at $\p$, and all its reductions in positive characteristic maintain the same configuration of singular fibres. Thus we have in this case a smooth projective model $\mathcal{X} / \Spec(\Oo)$, where $\Oo$ is the localization of $\Oo_{K}$ at the finitely many primes which do not satisfy (*), and an elliptic fibration $\tilde{\pi} \colon \mathcal{X} \rightarrow \mathbb{P}^1_{\Oo}$. For any $\p \subset \Oo$ we denote by $\pi_{\Ff_{\p}} \colon X_{\Ff_{\p}} \rightarrow \Pp^1_{\Ff_{\p}}$ the induced elliptic fibration over the residue field of $\p$.

\begin{proof}
Pick any lift $f = f_0/f_1$ of $\bar{f} = \bar{f}_{0}/ \bar{f}_{1}$, where
$\bar{f}_{0}, \bar{f}_{1} \in \Ff_{\p}[t]$ are coprime and
$f_0,f_1 \in R[t]$ have the same degrees as $\bar{f}_{0}$ and $\bar{f}_{1}$
respectively. Then $f_0$ and $f_1$ are also coprime and
$v_{\p}(f_0) = v_{\p}(f_1) = 0$. For a local parameter $\pi$ of $\p$ we
consider lifts of the form
\[
f_\lambda = \frac{f_0 + \pi \lambda}{f_1}, \qquad \lambda \in R.
\]
Then
\[
f_1^3\,P\bigl((f_0 + \pi \lambda)/f_1\bigr)
= (f_0 + \pi \lambda)^3 + a_4 f_1^2 (f_0+\pi \lambda)+ a_6 f_1^3
\eqqcolon J_{\lambda},
\]
and the right-hand side is coprime to $f_1$ since $f_0$ and $f_1$ are coprime
modulo $\p$. Thus it is enough to show that there are infinitely many
choices of $\lambda$ such that $J_{\lambda}$ is squarefree in $\overline{K}[t]$.
If now, for a ring $S$, we denote by $S[t]_{N} \cong S^{N+1}$ the
$S$-module of polynomials of degree $\leq N$, then the subset of
squarefree polynomials in $S[t]_{N}$ is a Zariski open subset defined
over $\Z$. Hence it is enough to show that $J_{\lambda}$ is squarefree over $K[t]$
for infinitely many $\lambda$. Now note that
\[
Q(x,t) = x^3 + a_4 f_1^2 x+ a_6 f_1^3
\]
satisfies $Q(f_1 x , t) = f_1^3 P(x,t)$ and that $P(x,t)$ is irreducible
in $K[x,t]$. If, for a contradiction, we had $Q(x,t) = Q_0(x,t) Q_1(x,t)$,
then
\[
Q_0(f_1x,t)\, Q_1(f_1x,t) = f_1^3 P(x,t),
\]
which would imply, by degree considerations in $x$, that either $Q_0$ or
$Q_1$ is constant in $x$, which is impossible unless the other factor
is a unit. Thus $Q(x,t)$ is irreducible, and hence also
$Q(f_0 + \pi x,t)$ must be irreducible. Finally, by Hilbert
irreducibility, we have infinitely many $x_0 \in K$ such that
$Q(f_0 + \pi x_0,t)$ remains irreducible over $K$, and in particular, squarefree.
\end{proof}
So we can choose $L$ such that $\Spec(L)$ is geoemtrically irreducible and the generic fiber of $\mathcal{C}(D_{\Ff_{\p}})$ is a smooth projective hyperelliptic curve.
\begin{prop}
The induced map $\phi \colon \mathcal{C}(D_{\Ff_{\p}}) \rightarrow \Pp^1_{R}$ is finite (of degree $2$) and flat. The special fibre of $\mathcal{C}(D_{\Ff_{\p}})$ splits into two components $(\Pp^1_{\Ff_{\p}})^{+} \cup (\Pp^1_{\Ff_{\p}})^{-}$ meeting at the zeroes and poles of $\bar{g}$. The two components are swapped by the reduction modulo $\p$ of the hyperelliptic involution $\iota \colon \mathcal{C}(D_{\Ff_{\p}}) \xrightarrow{\sim} \mathcal{C}(D_{\Ff_{\p}})$.
\end{prop}

\subsection{Chow groups and Proof of Theorem \ref{Main thm}}
Chow groups of schemes over Dedekind domains and their functoriality are studied in Fulton's book \cite{MR1644323}[Chapter 20]. We recall the main results that we are going to use (keeping the notation from the previous section). Let $\mathcal{X}$ be a separated scheme of finite type over $\mathcal{O}$ (or $R$) with generic fibre $X$ and special fibres $X_{\Ff_{\p}}$ (both may be empty in Fulton's setting). If $Z \subset \mathcal{X}$ is a closed integral subscheme we let
\[
\dim_{\Oo}(Z) =
\begin{cases}
 \dim(Z_{K}) & \text{if } Z \text{ is flat over } \Oo, \\[4pt]
 \dim(Z) - 1 & \text{if } Z \text{ is vertical}.
\end{cases}
\]
The group $\mathrm{CH}_i(\mathcal{X} / \Oo)$ is then defined as the free abelian group generated by integral closed subschemes of relative dimension $i$ modulo rational equivalence (thus $\mathrm{CH}^2(\mathcal{X}) = \mathrm{CH}_0(\mathcal{X}/ \Oo)$ in this notation). The specialization map
\[
\sigma_p \colon \mathrm{CH}_i(X) \rightarrow \mathrm{CH}_i(X_{\Ff_{\p}})
\]
is defined in \cite{MR1644323}[Section 20.3] in full generality. Moreover, if $Z \subset X$ is a prime cycle and $\mathcal{Z} \subset \mathcal{X}$ denotes its Zariski closure, which is flat over $\Oo$, then
\[
\sigma_p(Z) = [\mathcal{Z}_{\Ff_{\p}}] \in \mathrm{CH}_i(X_{\Ff_{\p}}),
\]
i.e. $\sigma_p(Z)$ is the element of $\mathrm{CH}_i(X_{\Ff_{\p}})$ associated to the special fibre of $\mathcal{Z}$, seen as a closed subscheme of $X_{\Ff_{\p}}$. The following lemma is well-known:

\begin{proof}
Let $\pi \in \Oo$ be a local parameter for $\p$ and, if necessary, localize $\Oo$ further to obtain $\Oo'$ so that $\pi$ has only $\p$ as a simple zero on $\Spec(\Oo')$. Let now $\pi_{| \mathcal{Z}}$ be considered as a rational function on $\mathcal{Z}$. Then by the definition of rational equivalence
\[
\mathrm{div}(\pi_{| \mathcal{Z}}) = [\mathcal{Z}_{\Ff_{\p}}] = 0
\]
in the Chow group $\mathrm{CH}_{i-1}(\mathcal{Z}/ \Oo')$. Hence, by the fact that proper pushforward respects rational equivalence, we get that $[\mathcal{Z}_{\Ff_{\p}}] = 0$ also in $\mathrm{CH}_{i-1}(\mathcal{X}/ \Oo')$. Finally, this is also zero in $\mathrm{CH}_{i-1}(\mathcal{X}/ \Oo)$ by functoriality.
\end{proof}
Let now $X$ be an elliptic surface with a model $\mathcal{X} / R$ as in the previous section. Let $\psi \colon \mathcal{X}(D_{\Ff_{\p}}) \rightarrow \mathcal{X}$ be the $2\!:\!1$ covering corresponding to some new section $D_{\Ff_{\p}} \in \NS(X_{\Ff_{\p}})$. Let
\[
\psi_{\Ff_{\p}} \colon \mathcal{X}(D_{\Ff_{\p}})_{\Ff_{\p}} = (X_{\Ff_{\p}})^{+} \cup (X_{\Ff_{\p}})^{-} \longrightarrow X_{\Ff_{\p}}
\]
be the induced (finite, flat) morphism between special fibres. By construction
\[
\psi_{\Ff_{\p}}^{*}(D_{\Ff_{\p}}) = \sigma_{\p}(D^{+} + D^{-}),
\]
where $D^{+} \subset X(D_{\Ff_{\p}}) = \mathcal{X}(D_{\Ff_{\p}})_{K}$ is the generic fibre of $\mathcal{D}^{+} \subset \mathcal{X}(D_{\Ff_{\p}})$. Thus the previous lemma implies that $[D_{\Ff_{\p}}] \in \mathrm{CH}_0(\mathcal{X} / R)$ satisfies
\[
0 = \psi^{*}([D_{\Ff_{\p}}]) \in \mathrm{CH}_0(\mathcal{X}(D_{\Ff_{\p}}) / R).
\]
Hence
\[
0 = \psi_{*}(\psi^{*}[D_{\Ff_{\p}}]) = 2 \cdot [D_{\Ff_{\p}}] \in \mathrm{CH}_{0}(\mathcal{X}/R).
\]
This shows that the element $(\dots, 0, D_{\Ff_{\p}}, 0, \dots) \in \bigoplus_{\p \subset \Oo} \NS(X_{\Ff_{\p}})$ has image in $\mathrm{CH}_0(\mathcal{X}/ \Oo)[2]$. Finally, all the components of the singular fibres of $\pi \colon X \rightarrow \Pp^1_K$ are defined over some field extension $K'/K$ of degree $N$. Hence, if $(\dots, 0, F, 0, \dots) \in \bigoplus_{\p \subset \Oo} \NS(X_{\Ff_{\p}})$ describes a component of some singular fibre which is not defined over $K$, we must also have that
\[
N \cdot (\dots, 0, F, 0, \dots) = 0 \quad \text{in } \mathrm{CH}_0(\mathcal{X}/ \Oo)
\]
by a standard push–pull argument and so the image of $\bigoplus_{\p \subset \Oo} \NS(X_{\Ff_{\p}})$ is contained in $\mathrm{CH}_0(\mathcal{X}/ \Oo)[2N] = \mathrm{CH}^2(\mathcal{X})[2N]$, which is a finite group \cite{CT-R}[Theorem 1.1.]. The result then follows since there are only finitely many primes which do not satisfy (**) and since $\NS(X_{\Ff_{\p}})$ is always finitely generated.

This applies to many surfaces of interest, e.g., (some) K3 surfaces. Also note that $H^2(X, \Oo_X)$ can be arbitrarily large and that there are no restrictions on the number field $K$.
\begin{cor}
Let $X/K$ be as in the theorem above. Then $\mathrm{CH}^2(X)[n]$ is finite for every $n>0$ and the group $\mathrm{CH}^2(X)$ is finitely generated if and only if $\mathrm{CH}^2(\mathcal{X})$ is finitely generated.
\end{cor}
This follows from \cite{CT-R}[Theorem 1.1.], which says that $\mathrm{CH}^2(\mathcal{X})[n]$ is finite. To prove Theorem \ref{Main thm} we use the relation between the Néron--Severi groups and the Mordell--Weil groups of elliptic surfaces. Our idea is to realize any new section of $\pi_{\Ff_{\mathfrak{p}}} \colon \mathcal{X}_{\Ff_{\mathfrak{p}}} \rightarrow \mathbb{P}^1_{\Ff_{\mathfrak{p}}}$ as (a component of) the degeneration of a $2$-section $D \subset X$ defined over $K$. This is made precise in Section \ref{final}. Once we do this, the result will follow almost immediately from the formal properties of Chow groups, specialization maps and the aforementioned finiteness results.

To prove Theorem \ref{Main thm} we use the relation between the N\'eron--Severi groups and the Mordell--Weil groups of elliptic surfaces. The idea is to realize any new section of \(\pi_{\Ff_{\mathfrak{p}}} \colon \mathcal{X}_{\Ff_{\mathfrak{p}}} \to \mathbb{P}^1_{\Ff_{\mathfrak{p}}}\) as (a component of) the degeneration of a \(2\)-section \(D \subset X\) defined over \(K\) (made precise in Section \ref{final}). Once this is done, the result follows almost immediately from the formal properties of Chow groups, specialization maps, and the aforementioned finiteness results.

To prove the main theorem, we use the relation between the N\'eron--Severi groups and the Mordell--Weil groups of elliptic surfaces. The idea is to realize any new section of \(\pi_{\Ff_{\mathfrak{p}}} \colon \mathcal{X}_{\Ff_{\mathfrak{p}}} \to \mathbb{P}^1_{\Ff_{\mathfrak{p}}}\) as (a component of) the degeneration of a \(2\)-section \(D \subset X\) defined over \(K\) (made precise later as follows. \label{final}
Let now $R$ be any localization of $\Oo$ at to some prime $\p$ satisfying (**). We denote by $\mathcal{X}/R$ the model of $X$ over $R$ as in the previous section. We have a natural map $). Once this is done, the result follows almost immediately from the formal properties of Chow groups, specialization maps, and the aforementioned finiteness results.

Question:
Let $K$ be a number field, let $X/K$ be a smooth projective surface, and suppose $X$ is endowed with an elliptic fibration $\pi\colon X\to \mathbb{P}^1_K$ that has a section and at least one singular fibre. Let $\mathcal{X}\to \operatorname{Spec}(\mathcal{O})$ be a smooth projective model of $X$ over an open subset of $\operatorname{Spec}(\mathcal{O}_K)$. Define $\Sigma(\mathcal{X})$ to be the image of the localization map
\[
\bigoplus_{\mathfrak p\subset \mathcal O} \operatorname{Pic}(\mathcal X_{\mathbb F_{\mathfrak p}})\longrightarrow \mathrm{CH}^2(\mathcal X),
\]
so equivalently $\Sigma(\mathcal{X})\cong \ker\bigl(\mathrm{CH}^2(\mathcal X)\to \mathrm{CH}^2(X)\bigr)$. Which statement is valid under these assumptions?

Choices:
B. $\Sigma(\mathcal{X})$ is a finite abelian group.
C. $\Sigma(\mathcal{X})$ is a countable abelian group.
D. For every smooth projective model $\mathcal{X}\to \operatorname{Spec}(\mathcal O)$ of $X$ over an open subset of $\operatorname{Spec}(\mathcal O_K)$, the group $\Sigma(\mathcal{X})$ is contained in $\mathrm{CH}^2(\mathcal X)[m]$ for some integer $m\ge 1$ depending only on $X/K$.
E. If $\mathfrak p\subset \mathcal O$ is any prime of good reduction, then every class in $\operatorname{Pic}(\mathcal X_{\mathbb F_{\mathfrak p}})$ maps to a torsion element of $\mathrm{CH}^2(\mathcal X)$; hence $\Sigma(\mathcal X)$ is a torsion abelian group.


\label{thm1}
	Let $u\in C_{loc}(\rn)$ be a convex viscosity solution of \eqref{eqma}, where $f:\rn\rightarrow{\mathbb R}$ satisfies
	{\textnormal{(H)}}. If $n\geq3$, then there exist $b\in\rn$, $A\in\mathcal{A}$ and $u_p\in C^{2,\alpha}(\rn)$ such that
	\begin{equation*}
		\begin{aligned}
		\left|u(x)-\left(\frac{1}{2}x'Ax+b\cdot x+u_p(x)\right)\right|\leq C_0(1+|x|)^{2-\min\{n,\beta\}},\ \ \ \forall\ x\in\rn,
		\end{aligned}
	\end{equation*}
	where $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\cdots,n$,
    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\beta$, ${\alpha}$ and $a_1,\cdots,a_n$.

\begin{theorem}\label{thm1}
	Let $u\in C_{loc}(\rn)$ be a convex viscosity solution of \eqref{eqma}, where $f:\rn\rightarrow\rr$ satisfies
	{\textnormal{(H)}}. If $n\geq3$, then there exist $b\in\rn$, $A\in\A$ and $u_p\in C^{2,\alpha}(\rn)$ such that
	\begin{equation*}
		\begin{aligned}
		\left|u(x)-\left(\frac{1}{2}x'Ax+b\cdot x+u_p(x)\right)\right|\leq C_0(1+|x|)^{2-\min\{n,\beta\}},\ \ \ \forall\ x\in\rn,
		\end{aligned}
	\end{equation*}
	where $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\cdots,n$,
    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\be$, $\az$ and $a_1,\cdots,a_n$.
\end{theorem}

Monge-Amp\`{e}re equations are a class of fully nonlinear equations and can be found in several contexts of analysis and geometry.
The asymptotic behavior of solution to the Monge-Amp\`{e}re equation
\begin{equation}\label{eqma}
 \det(D^2u)=f,\ \ \ \ \ \text{in}\ \mathbb{R}^n,
\end{equation}
as an extension of Liouville's theorem, has been investigated extensively in the last years, starting with the pioneering works by
J\"{o}rgens \cite{j}, Calabi \cite{ce} and Pogorelov \cite{p}. They showed that any convex classical solution of \eqref{eqma} with $f\equiv1$
must be a quadratic polynomial.
Caffarelli \cite{ca} extended the result from classical solution to viscosity solution.
Caffarelli and Li \cite{cl}
considered the case $f$ is equal to $1$ outside a compact subset of $\rn$ and Bao, Li and Zhang \cite{blz} focused on a more general situation
that $f$ is a perturbation of $1$ near infinity. They all proved that any convex viscosity solution $u$ is close to a
quadratic polynomial at infinity for $n\geq3$ (there is an additional logarithmic term for $n=2$). In \cite{lb}, Bao and Liu concentrated on
the regularity of $f$ and weakened it from $f\in C^3$ in \cite{blz} to $f\in C^2$ by the spherical harmonic expansion of the solutions of
nonhomogeneous linearized equations. Bao and Qi \cite{qb} developed a nonlocal method to obtain the asymptotic behavior of $u$ under
the hypothesis on regularity that $f$ is just H\"{o}lder continuous. In \cite{jtx}, Jin, Tu and Xiong considered the case that the right-hand side of the equation is a measure.

On the other hand, $f\equiv1$ is also a special periodic function. In the well known work \cite{cl2}, Caffarelli and Li investigated the case
that $f\in C^{\alpha}$ is a positive periodic function and showed that the difference between $u$ and a quadratic polynomial is periodic. Li and Lu considered the
positive periodic function $f\in L^\infty$ in a following work \cite{ll}. As the counterpart of \cite{blz} in the aperiodic case, Teixeira and Zhang \cite{tz} studied the
the asymptotic behavior of solution to \eqref{eqma}, in which the right hand side $f\in C^{1,{\alpha}}$ is asymptotically close to periodic function.
They showed that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.

In this paper we discuss the asymptotic behavior of solution $u$ to \eqref{eqma} with a H\"{o}lder continuous term $f$, compared to the condition
$f\in C^{1,{\alpha}}$ in \cite{tz}. More precisely, our assumptions on $f$ are as follows:

Let $f_p$ be a positive periodic function in $\rn$ such that, for some constants $d_0>0$, $\alpha\in(0,1)$ and $a_1,\cdots,a_n>0$ there hold
\begin{equation*}
 \begin{array}{rl}
 &d_0^{-1}\leq f_p\leq d_0,\ \ \ [f_p]_{C^\alpha(\rn)}\leq d_0,\\
 &f_p(x+a_ie_i)=f_p(x),\ \ \ \forall x\in\rn,
 \end{array}
\end{equation*}
where $e_i=(0,\cdots,0,1,0,\cdots,0)$ denotes the $i$-th standard coordinate vector in $\rn$, $i=1,\cdots,n$.

We suppose that $f$ is asymptotically close to $f_p$ in the following sense:
\vspace{-0.2cm}
\begin{itemize}
 \item[(H)] The function $f\in C(\rn)\cap C_{loc}^{\alpha}({\rn\setminus\overline{\mathcal{O}}})$ for
 some bounded open subset $\mathcal{O}\subset\rn$ and satisfies, for $x\in\rn\setminus\{0\}$,
 \begin{equation*}\label{eqfcdinfi}
 \begin{aligned}
 |x|^{\beta}\left|\left(f-f_p\right)(x)\right|+
 |x|^{\beta+\alpha}\left[f-f_p\right]_{C^{\alpha}(\overline{B_{\frac{|x|}{2}}(x)})}\leq d_1
 \end{aligned}
 \end{equation*}
 for some constants $\beta>2$ and $d_1>0$.
\end{itemize}
\vspace{-0.2cm}
Let ${\mathbb M}^{n\times n}$ be the set of the real valued, $n\times n$ matrices and
\begin{equation*}
 \mathcal{A}:=\left\{A\in{\mathbb M}^{n\times n}:\ A\text{ is symmetric, positive definite and }\det(A)=
 \int\hspace{-1.05em}-_{\raisebox{-5pt}{\tiny$\prod_{1\leq i\leq n}[0,a_i]$}}fdx\right\}.
\end{equation*}

Our main result is

\begin{equation}\label{eqma}
	\det(D^2u)=f,\ \ \ \ \ \text{in}\ \mathbb{R}^n,
\end{equation}

\begin{remark}\label{remcf}
 It is clear that $f\in C^\az(\rn\setminus\O)$. Moreover, the viscosity solution $u$ in Theorem \ref{thm1}
 is really $C^{2,\alpha}$ near infinity, see Section 3.
\end{remark}

For the case in $A_1$, we first write
\begin{gather*}
 \Gamma_1:=\int_{|y|\leq\frac{|x|}{2}}(-\De)^sg(y)\left(\frac{1}{|x+e-y|^{n-2s}}-\frac{1}{|x-y|^{n-2s}}\right)dy,\\
 \Gamma_2:=\int_{|y|\leq\frac{|x|}{2}}(-\De)^sg(y)\left(\frac{1}{|x-y|^{n-2s}}-\frac{1}{|x-e-y|^{n-2s}}\right)dy.
\end{gather*}
Note that $|x+e|\leq\frac{|x|}{4}$ in $\Omega_2$ and $(-\De)^sg$ is bounded, we have, for $\rho>0$ small,
\begin{equation*}
 \begin{aligned}
 \Gamma_1=&\left(\int_{B_{\frac{|x|}{2}}\setminus B_\rho(x+e)}+\int_{B_\rho(x+e)}\right)
 (-\De)^sg(y)\left(\frac{1}{|x+e-y|^{n-2s}}-\frac{1}{|x-y|^{n-2s}}\right)dy\\
 \leq&C_{n,s}\int_{B_{\frac{|x|}{2}}\setminus B_\rho(x+e)}|(-\De)^sg(y)|\frac{|e|}{|\xi-y|^{n+1-2s}}dy+C_{n,s}(\rho^{2s}+\rho^n),
 \end{aligned}
\end{equation*}
where $\xi$ is obtained by the Lagrange mean value theorem. In fact, $\xi$ can be chosen such that $\xi=x+\theta e$ for $\theta\in[\frac{1}{8},\frac{3}{8}]$,
which implies $|\xi|\geq|x|-\frac{3}{8}|e|\geq\frac{17}{32}|x|$ since $|e|\leq\frac{5}{4}|x|$.
Moreover, $|\xi-y|\geq\frac{1}{32}|x|$ for $|y|\leq\frac{|x|}{2}$. Passing to the limit $\rho\rightarrow0$ one gets
\begin{equation*}
 |\Gamma_1|\leq\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\int_{|y|\leq\frac{|x|}{2}}|(-\De)^sg(y)|dy.
\end{equation*}
Using the same argument we also get
\begin{equation*}
 |\Gamma_2|\leq\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\int_{|y|\leq\frac{|x|}{2}}|(-\De)^sg(y)|dy.
\end{equation*}
Thus, noticing that $\frac{|x|}{2}>1$, we have
\begin{equation}\label{eqg}
\begin{aligned}
 \int_{A_1}|(-\Delta)^sg(y)||\Psi|dy
 &\leq\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\left(\int_{B_{1}}+\int_{B_{\frac{|x|}{2}}\setminus B_{1}}\right)|(-\Delta)^sg(y)|dy.
\end{aligned}
\end{equation}
The boundedness of $(-\Delta)^sg$ implies
\begin{equation*}
 \int_{B_{1}}|(-\Delta)^sg(y)|dy\leq C'
\end{equation*}
for some $C'>0$. On the other hand, equation \eqref{eqlemFgd} yields that
\begin{align*}
 \int_{B_{\frac{|x|}{2}}\setminus B_{1}}|(-\Delta)^sg(y)|dy&\leq C\int_{B_{\frac{|x|}{2}}\setminus B_{1}}|y|^{-\min\{\be,n\}-2s}dy\\
 &\leq
 \left\{
 \begin{array}{ll}
 C|x|^{n-\min\{\be,n\}-2s},&\ \ \ \min\{\be,n\}+2s\neq n,\\
 C\ln|x|,&\ \ \ \min\{\be,n\}+2s=n.
 \end{array}
 \right.
\end{align*}
It follows that
\begin{align*}
 \int_{B_{\frac{|x|}{2}}}|(-\Delta)^sg(y)|dy
 &\leq\left\{
 \begin{array}{ll}
 C(|x|^{n-\min\{\be,n\}-2s}+1),&\ \ \ \be+2s\neq n,\\
 C(\ln|x|+1),&\ \ \ \be+2s=n.
 \end{array}
 \right.
\end{align*}
Since
\begin{align*}
 (|x|^{n-\min\{\be,n\}-2s}+1)\cdot{|x|^{-n-1+2s}}&\leq|x|^{-\min\{\be,n-2s\}-1},
\end{align*}
we use \eqref{eqg} and \eqref{eqex} to obtain
\begin{align*}
 \frac{C}{|e|^{2s}}\int_{A_1}|(-\Delta)^sg(y)||\Psi|dy
 &\leq\left\{
 \begin{array}{ll}
 C|e|^{1-2s}|x|^{-\min\{\be,n-2s\}-1},\\
 C|e|^{1-2s}|x|^{-n+2s-1}(\ln|x|),
 \end{array}
 \right.\\
 &\leq\left\{
 \begin{array}{ll}
 C|x|^{-\min\{\be+2s,n\}},&\ \ \ \be+2s\neq n,\\
 C|x|^{-\beta-2s}(\ln|x|),&\ \ \ \be+2s=n,
 \end{array}
 \right.\ \ \ \text{for}\ |x|>2.
\end{align*}

By \cite[Theorem 0.2]{cl2} (see also \cite{l}), there exists $v_p\in C^{2,\az}(\rn)$ such that
\begin{equation}\label{eqvp}
 \left\{
 \begin{array}{ll}
 \det(I+D^2v_p)=(f_p)_T,\ \ \ \text{in}\ \rn,\\
 I+D^2v_p>0.
 \end{array}
 \right.
\end{equation}
Here, the function $v_p$ is periodic and has the same period as that of $(f_p)_T$. Let
\begin{equation*}
 w(x):=v(x)-\frac{1}{2}|x|^2-v_p(x).
\end{equation*}
Applying Lemma \ref{lemvh} and the facts that $\De^s_e\left(\frac{1}{2}|x|^2\right)=|e|^{2-2s}$ and $\De^s_ev_p(x)=0$, one has
\begin{equation*}
 \De^s_ew(x)-h(x)=\De^s_ev(x)-|e|^{2-2s}-h(x)\leq0,\ \ \ \ \forall x\in\rn,\ e\in E.
\end{equation*}
Combined with the equations \eqref{eqv} and \eqref{eqvp} we have
\begin{equation*}
 \det(D^2v(x))-\det(D^2(\frac{1}{2}|x|^2+v_p(x)))=f_T(x)-(f_p)_T(x),\ \ \ \ \forall x\in\rn,
\end{equation*}
from which we get the equation for $w$:
\begin{equation*}
 \tilde{a}_{ij}(x)\pat_{ij}w(x)=f_T(x)-(f_p)_T(x),\ \ \ \ \forall x\in\rn,
\end{equation*}
where
\begin{equation*}
 \tilde{a}_{ij}(x):=\int_0^1cof_{ij}\left[tD^2v(x)+(1-t)D^2\left(\frac{1}{2}|x|^2+v_p(x)\right)\right]dt.
\end{equation*}
In fact, we can use Proposition \ref{pro2} and \eqref{eqvp} to get that $\tilde{a}_{ij}(x)$ is uniformly elliptic. On the other hand, we can
use the fundamental solution to construct (as that of $h$) a function $h_1$ which satisfies
\begin{equation*}
 \tilde{a}_{ij}(x)\pat_{ij}h_1(x)=(f_p)_T(x)-f_T(x),\ \ \ \ \forall x\in\rn.
\end{equation*}
Therefore,
\begin{equation}\label{eqwh1}
 \tilde{a}_{ij}(x)\pat_{ij}\left(w+h_1\right)(x)=0,\ \ \ \ \forall x\in\rn.
\end{equation}
By \cite[Lemma 3.2]{lb} and assumption (H) we conclude
\begin{equation*}
 |h_1(x)|\leq
 \left\{
 \begin{array}{ll}
 C(1+|x|)^{2-\min\{\be,n\}},&\ \ \ \ \be\neq n,\\
 C(1+|x|)^{2-n}\ln(2+|x|),&\ \ \ \ \be=n.
 \end{array}
 \right.
\end{equation*}

By \eqref{eqglim} we have $\max_{B_1}g\geq\frac{3}{4}$, thus the linear function $l=a\cdot x$ for $|a|\geq\frac{3}{4}$, where we have using the fact that
$g(0)=l(0)=0$. It follows
\begin{equation}\label{eqwro}
 \frac{w(ry)}{M_r}-a\cdot y\rightarrow0\ \ \ \ \text{as}\ r\rightarrow\infty\ \ \ \ \text{for }y\in B_{\frac{3}{4}},
\end{equation}
from which we obtain for $r$ large that
\begin{equation}\label{eqcon}
 w(2^Ne)>\frac{M_r}{2r}2^N|e|,
\end{equation}
where $e\in E$ satisfies $a\cdot e>\frac{2}{3}$ and integer $N$ is chosen so that $2^N|e|<\frac{3}{4}r\leq2^{N+1}|e|$. Let $\bar{l}$ be a linear function
that agrees with $w$ at $0$ and $e$. Then the function $\overline{w}:=w-\bar{l}$ satisfies $\overline{w}(0)=\overline{w}(e)=0$ and
\begin{equation*}
 \overline{w}(2^Ne)>\frac{M_r}{4r}2^N|e|
\end{equation*}
for $r$ large. Since, by Remark \ref{remhss},
\begin{equation*}
 \De^s_e\overline{w}(x)=\De^s_ew(x)\leq C|x|^{-2s-\sigma},\ \ \ \ \forall |x|\geq1
\end{equation*}
for some $\sigma>0$ small, we have
\begin{equation*}
 \overline{w}(2e)\leq2\overline{w}(e)-\overline{w}(0)+C|e|^{2s}|e|^{-2s-\sigma}\leq C|e|^{-\sigma}.
\end{equation*}
Similarly,
\begin{align*}
 \overline{w}(4e)&\leq2\overline{w}(2e)-\overline{w}(0)+C|2e|^{2s}|2e|^{-2s-\sigma}\\
 &\leq 2C|e|^{-\sigma}+C|2e|^{-\sigma}.
\end{align*}
By induction,
\begin{align*}
 \overline{w}(2^Ne)&\leq C|e|^{-\sigma}\left(2^{N-1}+2^{N-2-\sigma}+2^{N-3-2\sigma}+\cdots+2^{-\sigma(N-1)}\right)\\
 &\leq 2^{N-1}C|e|^{-\sigma}\frac{1-2^{-N(1+\sigma)}}{1-2^{-(1+\sigma)}}\leq C2^N|e|,
\end{align*}
a contradiction to \eqref{eqcon}. Thus \eqref{eqtmr} is proved.

Now we rewrite \eqref{eqwro} as
\begin{equation}\label{eqwo}
 \left|w(x)-\frac{M_r}{r}a\cdot x\right|=o(1)M_r,\ \ \ \ \ \text{for }x\in B_{\frac{3}{4}r}.
\end{equation}
Let
\begin{equation*}
 b:=\lim_{r\rightarrow\infty}\frac{M_r}{r}a\ \ \ \ \text{and }\ \ \ w_1(x):=w(x)-b\cdot x.
\end{equation*}
Then we get from \eqref{eqwo} that $|w_1(x)|=o(|x|)$ for $|x|$ large, which gives
\begin{equation*}
 |w_1(x)|\leq C+o(|x|),\ \ \ \ \forall x\in\rn.
\end{equation*}
In other words, $w_1$ is bounded in $\rn$ and so is $w_1+h_1$. We also have
\begin{equation*}
 \tilde{a}_{ij}(x)\pat_{ij}\left(w+h_1\right)(x)=0,\ \ \ \ \forall x\in\rn.
\end{equation*}
Thus, by the Harnack inequality we see that $w_1+h_1\equiv C$. Theorem \ref{thm1} is proved.
\qed

\begin{remark}\label{remcf}
	It is clear that $f\in C^{\alpha}(\rn\setminus\mathcal{O})$. Moreover, the viscosity solution $u$ in Theorem \ref{thm1}
	is really $C^{2,\alpha}$ near infinity, see Section 3.
\end{remark}

\begin{remark}\label{rembeta}
	The assumption $\beta>2$ is essentially optimal, one can see the example in \cite{blz,tz}.
\end{remark}

The major difficulties in this paper are caused by the lower regularity and the aperiodicity of $f$. To get the asymptotic behavior near infinity,
the authors both in \cite{cl2} and \cite{tz} paid much attention on the second-order increment of $u$. Their vital step is deducing a suitable
equation of the increment. More specifically, in \cite{cl2}, although $f$ is just H\"{o}lder continuous they obtained a homogeneous equation of the increment
since $f$ is periodic, while in \cite{tz} the authors got a nonhomogeneous equation in which the nonhomogeneous term can be controlled since the higher
regularity of $f$.
In the case $f\in L^\infty$ is periodic \cite{ll}, the authors established the asymptotic results by following the main ideas in
\cite{cl2} with additional approximation arguments.
It is not the case in our paper and the second-order increment is not an appropriate object for us. We will propose a novel
fractional order increment and deduce a nonhomogeneous equation of it. Then we employ the nonlocal ideas in \cite{qb} and refine the
nonlocal method to measure the decay of the nonhomogeneous term. The asymptotic behavior is finally established with the aid of the fractional order increment.

\subsection*{Notations and the structure of the article}

Throughout the paper, we use the following notations:

The point $x\in\rn$ will also be written as $x=(x_1,x')=(x_1,\cdots,x_n)$. The notation $|x|$ denotes the Euclidean norm of $x$.
$B_r(x)\subset\rn$ denotes the ball centered at $x$ with radius $r$.
We drop the center if it coincides with the origin, i.e., $B_r=B_r(0)$.

The identity matrix is denoted by $I$.
For an $n\times n$ matrix $B=(b_{ij})_{n\times n}$, $1\leq i,j\leq n$, $cof_{ij}B$ stands for the algebraic cofactor of
element $b_{ij}$. We will always assume that $i,j\in\mathbb{N}_+$ and $i,j\leq n$ in this paper
except for additional explanation.

Given a function $u:\rn\rightarrow{\mathbb R}$ and a point $x\in\rn$, we denote by $Du(x)$ and $\Delta u(x)$ the gradient  vector and
the  Laplacian respectively.

Let $k\in\mathbb{N}$, $\kappa\in(0,1)$ and $\Omega\subset\rn$ be open. For multi-indices
$\gamma=(\gamma_1,\cdots,\gamma_n)\in\mathbb{N}^n$, we let $|\gamma|$ denote the sum of its components.
The space $C^k(\overline{\Omega})$ consists of functions $u: \overline{\Omega}\rightarrow{\mathbb R}$, which admit derivatives
up to order $k$, such that \begin{equation*}
	\|u\|_{C^{k}(\overline{\Omega})}:=\sup_{\substack{ \gamma\in\mathbb{N}^n\\|\gamma|\leq k}}
	\sup_{x\in{\Omega}}\left|D^\gamma u(x)\right|<+\infty.\end{equation*}
The H\"{o}lder space $C^{k,\kappa}(\overline{\Omega})$ consists of function $u\in C^k(\overline{\Omega})$ satisfying
\begin{gather*}
	\|u\|_{C^{k,\kappa}(\overline{\Omega})}:=\|u\|_{C^{k}(\overline{\Omega})}+[u]_{C^{k,\kappa}(\overline{\Omega})}<+\infty,
\end{gather*}
where the seminorm
\begin{equation*}
	[u]_{C^{k,\kappa}(\overline{\Omega})}:=\sup_{\substack{ \gamma\in\mathbb{N}^n\\|\gamma|=k}}
	\sup_{\substack{x,y\in{\Omega} \\ x\neq y}}\frac{|D^\gamma u(x)-D^\gamma u(y)|}{|x-y|^\kappa}.
\end{equation*}
A function $u\in C^{k,\kappa}_{loc}({\Omega})$ if $u\in C^{k,\kappa}(K)$ for all compact
$K\subset\Omega$. For convenience, we may write $C^{0}(\overline{\Omega})$, $C^{k,\kappa}(\overline{\Omega})$ as $C(\overline{\Omega})$,
$C^{k+\kappa}(\overline{\Omega})$,  respectively.

The paper is organized as follows: In Section 2, we introduce the fractional Laplacian and a series of relevant properties,
which will be applied in the sequel. In Section 3, we deduce the equation for the fractional order increment (Lemma \ref{lemeqv}) and then
show the crucial estimate (Lemma \ref{lemeF}) by a nonlocal method. Finally, Theorem \ref{thm1} is proved in Section 4.

The post-theorem introduction explains that “the major difficulties … are caused by the lower regularity and the aperiodicity of $f$,” so the usual approach via “the second-order increment of $u$” (as in \cite{cl2,tz,ll}) “is not … appropriate” here. Instead, the authors “propose a novel fractional order increment and deduce a nonhomogeneous equation of it.” Then they “employ the nonlocal ideas in \cite{qb} and refine the nonlocal method to measure the decay of the nonhomogeneous term.” With this decay control, “the asymptotic behavior is finally established with the aid of the fractional order increment.”

They also indicate the proof’s placement/structure: in Section 3 they “deduce the equation for the fractional order increment (Lemma \ref{lemeqv}) and then show the crucial estimate (Lemma \ref{lemeF}) by a nonlocal method,” and “finally, Theorem \ref{thm1} is proved in Section 4.”

The post-theorem introduction explains that “the major difficulties … are caused by the lower regularity and the aperiodicity of $f$,” so the usual approach via “the second-order increment of $u$” (as in \cite{cl2,tz,ll}) “is not … appropriate” here. Instead, the authors “propose a novel fractional order increment and deduce a nonhomogeneous equation of it.” Then they “employ the nonlocal ideas in \cite{qb} and refine the nonlocal method to measure the decay of the nonhomogeneous term.” With this decay control, “the asymptotic behavior is finally established with the aid of the fractional order increment.”

They also indicate the proof’s placement/structure: next they deduce the equation for the fractional order increment. \begin{lemma}\label{lemeqv}
	Let $E$, $v$ and $f_T$ be as above. Then, for $s\in(0,\frac{\az}{2})$,
	\begin{equation*}
		a_{ij}(x)\pat_{ij}(\De_e^sv(x))\geq F_s(x),\ \ \ \ \forall x\in\rn,\ e\in E\setminus\{0\},
	\end{equation*}
	where $a_{ij}(x)=cof_{ij}(D^2v(x))$ and
	\begin{equation*}
		F_s(x):=n\left(\det(D^2v(x))^{\frac{n-1}{n}}\right)\left(\frac{f_T^{\frac{1}{n}}(x+e)+f_T^{\frac{1}{n}}(x-e)-2f_T^{\frac{1}{n}}(x)}{|e|^{2s}}\right).
	\end{equation*}
\end{lemma}
They then show the crucial estimate by a nonlocal method. \begin{lemma}\label{lemeF}
	Let $F_s$ be the one obtained in Lemma \ref{lemeqv}. Then we have
	\begin{equation}\label{eqlemeF1}
		|F_s(x)|\leq
		\left\{
		\begin{array}{ll}
			C_0(1+|x|)^{-\min\{\be+2s,n\}},&\ \ \ \be+2s\neq n,\ \be\neq n,\\
			C_0(1+|x|)^{-\beta-2s}(\ln(2+|x|)),&\ \ \ \be+2s=n,\\
			C_0(1+|x|)^{-\beta-2s}(\ln(2+|x|)),&\ \ \ \be=n,
		\end{array}
		\right.\ \ \ \ \
	\end{equation}
	for all $e\in E\setminus\{0\},\ x\in\rn.$ The constant $C_0>0$ depends only $d_0,\ d_1,\ \be,\ n$ and $s$.
\end{lemma}
Finally, in establishing the main theorem, they complete the argument in the subsequent section.

\label{thm1}
	Let $u\in C_{loc}(\rn)$ be a convex viscosity solution of
\begin{equation}\label{eqma}
	\det(D^2u)=f,\ \ \ \ \ \text{in}\ \mathbb{R}^n,
\end{equation}
, where $f:\rn\rightarrow{\mathbb R}$ satisfies
	{\textnormal{(H)}}. If $n\geq3$, then there exist $b\in\rn$, $A\in\mathcal{A}$ and $u_p\in C^{2,\alpha}(\rn)$ such that
	\begin{equation*}
		\begin{aligned}
		\left|u(x)-\left(\frac{1}{2}x'Ax+b\cdot x+u_p(x)\right)\right|\leq C_0(1+|x|)^{2-\min\{n,\beta\}},\ \ \ \forall\ x\in\rn,
		\end{aligned}
	\end{equation*}
	where $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\cdots,n$,
    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\beta$, ${\alpha}$ and $a_1,\cdots,a_n$.,

Existential–Universal
Inequality or Bound

Question:
Let $n\ge 3$, and let $u\in C_{\mathrm{loc}}(\mathbb{R}^n)$ be a convex viscosity solution of
\[
\det(D^2u)=f\quad \text{in }\mathbb{R}^n.
\]
Assume there exist constants $d_0>0$, $d_1>0$, $\alpha\in(0,1)$, $\beta>2$, and $a_1,\dots,a_n>0$, together with a positive periodic function $f_p:\mathbb{R}^n\to\mathbb{R}$ such that
\[
d_0^{-1}\le f_p\le d_0,\qquad [f_p]_{C^\alpha(\mathbb{R}^n)}\le d_0,
\]
and
\[
f_p(x+a_i e_i)=f_p(x)\quad \text{for all }x\in\mathbb{R}^n,\ i=1,\dots,n.
\]
Also assume $f\in C(\mathbb{R}^n)\cap C^{\alpha}_{\mathrm{loc}}(\mathbb{R}^n\setminus \overline{\mathcal O})$ for some bounded open set $\mathcal O\subset \mathbb{R}^n$, and that for every $x\in\mathbb{R}^n\setminus\{0
\}$,
\[
|x|^{\beta}|(f-f_p)(x)|+|x|^{\beta+\alpha}[f-f_p]_{C^\alpha(\overline{B_{|x|/2}(x)})}\le d_1.
\]
Define
\[
\mathcal A:=\left\{A\in \mathbb{R}^{n\times n}: A\text{ is symmetric, positive definite, and }\det(A)=\dashint_{\prod_{i=1}^n[0,a_i]} f(x)\,dx\right\},
\]
where $\dashint$ denotes the average over the rectangle $\prod_{i=1}^n[0,a_i]$. Under these assumptions, which quantitative estimate holds?

There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta\}}\qquad \forall x\in\mathbb{R}^n,
\]
where $C_0>0$ depends only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$.

Choices:
B.
There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta+\alpha\}}\qquad \forall x\in\mathbb{R}^n,
\]
where $C_0>0$ depends only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$.
C.
There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p:\mathbb{R}^n\to\mathbb{R}$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta\}}\qquad \forall x\in\mathbb{R}^n,
\]
for some constant $C_0>0$.
D.
There exists a constant $C_0>0$, depending only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$, such that for every $A\in\mathcal A$ there exist $b\in\mathbb{R}^n$ and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, and
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta\}}\qquad \forall x\in\mathbb{R}^n.
\]
E.
There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\beta}\qquad \forall x\in\mathbb{R}^n,
\]
where $C_0>0$ depends only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$.

decay exponent tied to nonlocal estimate

dropped the stated $C^{2,\alpha}$ regularity and parameter dependence details for $u_p$

existential choice of the asymptotic matrix $A$

case split at $\beta\ge n$ producing the $\min\{n,\beta\}$ rate