\begin{theorem}\label{them profile}
		{\bf (The  case $\nu=0$)} There exists an absolute constant $\delta_{1}>0,$  sufficiently small, such that the following statements hold:

		(1) For \(a\in (1, 1+\delta_{1})\) The model \ref{PJE} with $\nu=0$ develops a finite time singularity for some \(C^{\infty}\) initial data;

		(2) For all $a\in(1-\delta_{1},1+\delta_{1}),$ the model \ref{PJE} admits a self-similar solution of the form
		\begin{align*}
			\omega(x,t) = \frac{1}{1 + c_{\omega,a}t}\omega_{a}(x),
		\end{align*}
		where $\omega_{a}$ is an odd profile and $c_{\omega,a}$ is the scaling parameter,  satisfying
		\begin{align}\label{the estimates}
			\|\omega_{a} +\sin (x)\|_{\mathcal{W}} \lesssim |1-a|, \quad |c_{\omega,a} - (1-a)|\leq \min\{C|1-a|^{2}, |1-a|\},
		\end{align}
		for some absolute constant $C > 0$. More precisely:
		\begin{itemize}
			\item for $1 < a < 1+\delta_{1}$, the scaling parameter satisfies $c_{\omega,a} < 0$, and the corresponding solution $\omega(x,t)$ blows up in finite time $T = -\frac{1}{c_{\omega,a}}$;
			\item for $a = 1$, the explicit steady state $\omega_{1} = -\sin (x)$ solves the system with $c_{\omega,a} = 0$;
			\item for $1-\delta_{1}<a<1$, one has $c_{\omega,a} > 0$, and the solution $\omega(x,t)$ exists globally with $O(t^{-1})$ decay rate as $t\to\infty$.
		\end{itemize}
	\end{theorem}

\begin{lemma}\label{le cobasis}
		$\{\tilde{e}_{k}^{(o)}, k\geq 1\}$ is a complete orthonormal basis for $\mathcal{H}$.
	\end{lemma}

In this paper, we investigate a one-parameter family of partial differential equations given by
	\begin{align}\label{gpje}
		u_{txx} + uu_{xxx} = au_{x}u_{xx}+\nu u_{xxxx}, \quad t>0, \ x\in\mathbb{T},
	\end{align}
	posed on the torus \(\mathbb{T} = [-\pi,\pi]\), where $a\in\mathbb{R}$ is a parameter and $\nu\ge0$ represents the viscosity. This equation was originally introduced in \cite{OHZ2000} for specific values of $a$ as a model derived from high-dimensional Navier-Stokes equations under certain symmetry assumptions. By defining $u_{xx}=\omega$, equation \ref{gpje} can be written as
	\begin{align}\label{PJE}
		\begin{cases}
			\omega_{t} + u\omega_{x} = a\omega u_{x}+\nu\omega_{xx}, & t >0, \ x \in \mathbb{T}, \\
			u_{xx} =\omega, & t >0, \ x \in \mathbb{T}.
		\end{cases}
	\end{align}
	When $a=1$, \eqref{PJE} is the Proudman-Johnson equation. In particular, when $a=1$ and $\nu=0$, \eqref{PJE} is the inviscid Proudman-Johnson equation, which models inviscid, incompressible flows near a boundary (see \cite{CS1989}). For general $a\in \mathbb{R}$, equation \eqref{PJE}, which is equivalent to \eqref{gpje}, is called a generalized Proudman-Johnson equation. By generalizing the Proudman-Johnson equation with a parameter
	  $a$, one can systematically study the balance between the ``convection term''($uu_{xxx}$) and the ``stretching term''($u_{x}u_{xx}$), which leads to either the creation or depletion of singularities in finite time (see \cite{OO2005, OS2008}).

 The generalized Proudman-Johnson equation encompasses several well-known models. When $a=-3$, it reduces to the classical Burgers' equation, which plays a central role in gas dynamics (see \cite{B1948}). For $a=-2$, the equation becomes the Hunter-Saxton equation (see \cite{B1991, L2007, L2008, Y2004}), which arises in the modeling of the orientation dynamics of nematic liquid crystals. The case $a=0$, the equation $u_{txx} + uu_{xxx} = 0$ appears in contexts that connect projective geometry and gravitational models (see \cite{P2001}). The inviscid form of model \eqref{PJE} also resembles the generalized Constantin-Lax-Majda (gCLM) model (see \cite{OS2008}), which takes the form
		\begin{align}\label{gCLM}
		\begin{cases}
			\omega_{t} + u\omega_{x} = a\omega u_{x}+\nu\omega_{xx}, & t >0, \ x \in \mathbb{T}, \\
			u_{x} = H\omega, & t >0, \ x \in \mathbb{T},
		\end{cases}
	\end{align}
	where $H$ represents the Hilbert transform. When $a=1$, the gCLM model \eqref{gCLM} reduces to the De Gregorio (DG) equation. Notably, the explicit steady state $(\omega,u) = (-\sin (x),\sin(x))$ is a solution to both  equation \eqref{PJE} and the DG equation on the torus $\mathbb{T}$.

The local well-posedness of the generalized  Proudman Johnson equation \ref{gpje} in the periodic setting was established in \cite{OHZ2000, OH2009}. In particular, \cite{OHZ2000} derived a sufficient condition for finite time blow-up of solutions in the case $a=\infty$,
	\begin{align*}
		u_{txx} - u_{x}u_{xx} = \nu u_{xxxx}, \quad t>0, \ x \in \mathbb{T}.
	\end{align*}
	Moreover, the authors in \cite{OHZ2000, OH2009} investigated parameter conditions on $a$ that ensure the global existence of solutions for general initial data. Further analytical and numerical results concerning blow-up phenomena were also presented in \cite{OHZ2000}. For $a=1$, more refined blow-up criteria have been derived, as discussed in \cite{CS1989}, where the authors employed various trajectory-based analytical techniques.

	  In \cite{CW2010}, the authors proved the existence of a class of global-in-time solutions to the generalized inviscid Proudman-Johnson equation for parameters of the form $a = -\frac{n+3}{n+1}$, where $n\in\mathbb{N}$. Subsequently, in \cite{CW2012}, global existence was established for $a\in [-2,-1)$ in the inviscid setting, using the method of characteristics. In \cite{SS2013,SS2015}, various global existence and blow-up scenarios are investigated based on a representation formula for the velocity gradient along particle trajectories. In contrast, singularity formation for $a>1$ remains less understood. In \cite{OH2009}, the authors construct non-smooth self-similar solutions of the form $u(x,t)=\frac{F(x)}{T-t}$, which blow-up at time $T$. These solutions are obtained by solving the nonlinear ordinary differential equation
	\begin{align}\label{Fequ}
		F''(x) + F(x)F'''(x) - aF'(x)F''(x) = 0, \quad x\in \mathbb{T},
	\end{align}
	for $a>1$. However, in this regime, equation \eqref{Fequ} admits only non-smooth solutions. More recently, Kogelbauer \cite{KF2020} established new criteria for global existence and finite time blow-up in the inviscid generalized Proudman-Johnson equation, depending on the parameter $a$. In particular, \cite{KF2020} demonstrated that even smooth initial data may lead to finite time singularities when $a>1$. Additionally, \cite{KF2020} provides a physical derivation of the generalized equation for arbitrary values of the parameter $a$.

	In this paper, we will investigate model \ref{PJE} on the torus $\mathbb{T}$ for parameter values $a$ near 1, which can be interpreted as a slight perturbation of the Proudman-Johnson case $(a=1)$. For the inviscid model \ref{PJE}, this regime is particularly interesting for smooth initial data, as it captures two distinct dynamical scenarios:
	\begin{itemize}
		\item When $a>1$, the advection term is slightly weaker than the vortex stretching term, leading to finite time blow-up;
		\item When $a<1$, the advection term slightly dominates the vortex stretching term, resulting in different long-time behavior.
	\end{itemize}
	  Moreover, we establish finite time singularity formation for the inviscid model \ref{PJE} with  H\"{o}lder continuous  data in the case $a=1$ and the finite-time blow-up for the viscous case with $a>1$.

	Before stating our main results, we introduce a class of weighted Sobolev spaces that will serve as the functional framework for our analysis.
	\begin{definition}[Weighted norms and spaces]
	  Define the singular weight $\rho = \frac{1}{4\pi\sin^{2} (\frac{x}{2})}$ and the weighted norms  $\|\cdot\|_{\mathcal{H}}$ and  $\|\cdot\|_{\mathcal{W}}$ as follows
	  \begin{align}
	  	\|f\|_{\mathcal{H}}^{2} = \frac{1}{4\pi}\int_{\mathbb{T}}\frac{|f_{x}|^{2}}{\sin^{2} (\frac{x}{2})} \, \mathrm{d}x,
	  \end{align}
	  \begin{align}
	  	\|f\|_{\mathcal{W}} = \|f\|_{\mathcal{H}}^{2} + \int_{\mathbb{T}} |f_{xx}|^{2}\cos^{2} (\frac{x}{2}) \, \mathrm{d}x.
	  \end{align}
	  The associated Hilbert spaces are defined as
	  \begin{align*}
	  	\mathcal{H} = \{f\in H^{1}(\mathbb{T}) |\, \text{f is odd},\ \|f\|_{\mathcal{H}} < + \infty\},
	  \end{align*}
	  \begin{align*}
	  	\mathcal{W} = \{f\in H^{2}(\mathbb{T}) |\, \text{f is odd},\ \|f\|_{\mathcal{W}} < +\infty\},
	  \end{align*}
	  which are equipped with inner products naturally induced by the corresponding norms.
	\end{definition}
	Define $\tilde{e}_{k}^{(o)}=\frac{\sin((k+1)x)}{k+1}-\frac{\sin (kx)}{k}$. In fact, $\{\tilde{e}_{k}^{(o)},k\geq 1\}$ forms a complete orthonormal basis of $\mathcal{H}$ (see Lemma \ref{le cobasis}).
   The norm and inner product in $\mathcal{H}$  were first introduced in \cite{LL2020} for stability analysis. The $\mathcal{W}$ norm can be found in \cite{C2021} for the purpose of higher order derivative estimates.

    Throughout this paper,  $\|\cdot\|_{L^{p}},\|\cdot\|_{H^{m}}$ and $\|\cdot\|_{L^{\infty}}$ represent the norms in the spaces $L^{p}(\mathbb{T}), H^{m}(\mathbb{T})$ and $L^{\infty}(\mathbb{T})$, respectively. Similarly, $\left\langle\cdot, \cdot\right\rangle$ denotes the usual inner product in $L^{2}(\mathbb{T})$, defined as
   \begin{align*}
   	\left\langle f, g\right\rangle = \int_{\mathbb{T}}fg \, \mathrm{d}x.
   \end{align*}
   We use $C,C_{i}$ to denote absolute constants and $C(A,B,\cdots,Z)$ to denote constants depending on $A,B,\cdots,Z$. And we also employ the notation $A\lesssim B$ to indicate that there exists a constant $C$ such that $A\leq CB$.

Now we are ready to state our main results.   Our first main result establishes the existence of a family of self-similar solutions to the inviscid model \ref{PJE}.

In this paper, we investigate a one-parameter family of partial differential equations given by
 \begin{align}\label{gpje}
 u_{txx} + uu_{xxx} = au_{x}u_{xx}+\nu u_{xxxx}, \quad t>0, \ x\in\mathbb{T},
 \end{align}
 posed on the torus \(\mathbb{T} = [-\pi,\pi]\), where $a\in\mathbb{R}$ is a parameter and $\nu\ge0$ represents the viscosity. This equation was originally introduced in \cite{OHZ2000} for specific values of $a$ as a model derived from high-dimensional Navier-Stokes equations under certain symmetry assumptions. By defining $u_{xx}=\omega$, equation \ref{gpje} can be written as
 \begin{align}\label{PJE}
 \begin{cases}
 \omega_{t} + u\omega_{x} = a\omega u_{x}+\nu\omega_{xx}, & t >0, \ x \in \mathbb{T}, \\
 u_{xx} =\omega, & t >0, \ x \in \mathbb{T}.
 \end{cases}
 \end{align}
 When $a=1$, \eqref{PJE} is the Proudman-Johnson equation. In particular, when $a=1$ and $\nu=0$, \eqref{PJE} is the inviscid Proudman-Johnson equation, which models inviscid, incompressible flows near a boundary (see \cite{CS1989}). For general $a\in \mathbb{R}$, equation \eqref{PJE}, which is equivalent to \eqref{gpje}, is called a generalized Proudman-Johnson equation. By generalizing the Proudman-Johnson equation with a parameter
 $a$, one can systematically study the balance between the ``convection term''($uu_{xxx}$) and the ``stretching term''($u_{x}u_{xx}$), which leads to either the creation or depletion of singularities in finite time (see \cite{OO2005, OS2008}).

The generalized Proudman-Johnson equation encompasses several well-known models. When $a=-3$, it reduces to the classical Burgers' equation, which plays a central role in gas dynamics (see \cite{B1948}). For $a=-2$, the equation becomes the Hunter-Saxton equation (see \cite{B1991, L2007, L2008, Y2004}), which arises in the modeling of the orientation dynamics of nematic liquid crystals. The case $a=0$, the equation $u_{txx} + uu_{xxx} = 0$ appears in contexts that connect projective geometry and gravitational models (see \cite{P2001}). The inviscid form of model \eqref{PJE} also resembles the generalized Constantin-Lax-Majda (gCLM) model (see \cite{OS2008}), which takes the form
 \begin{align}\label{gCLM}
 \begin{cases}
 \omega_{t} + u\omega_{x} = a\omega u_{x}+\nu\omega_{xx}, & t >0, \ x \in \mathbb{T}, \\
 u_{x} = H\omega, & t >0, \ x \in \mathbb{T},
 \end{cases}
 \end{align}
 where $H$ represents the Hilbert transform. When $a=1$, the gCLM model \eqref{gCLM} reduces to the De Gregorio (DG) equation. Notably, the explicit steady state $(\omega,u) = (-\sin (x),\sin(x))$ is a solution to both equation \eqref{PJE} and the DG equation on the torus $\mathbb{T}$.

The local well-posedness of the generalized Proudman Johnson equation \ref{gpje} in the periodic setting was established in \cite{OHZ2000, OH2009}. In particular, \cite{OHZ2000} derived a sufficient condition for finite time blow-up of solutions in the case $a=\infty$,
 \begin{align*}
 u_{txx} - u_{x}u_{xx} = \nu u_{xxxx}, \quad t>0, \ x \in \mathbb{T}.
 \end{align*}
 Moreover, the authors in \cite{OHZ2000, OH2009} investigated parameter conditions on $a$ that ensure the global existence of solutions for general initial data. Further analytical and numerical results concerning blow-up phenomena were also presented in \cite{OHZ2000}. For $a=1$, more refined blow-up criteria have been derived, as discussed in \cite{CS1989}, where the authors employed various trajectory-based analytical techniques.

In \cite{CW2010}, the authors proved the existence of a class of global-in-time solutions to the generalized inviscid Proudman-Johnson equation for parameters of the form $a = -\frac{n+3}{n+1}$, where $n\in\mathbb{N}$. Subsequently, in \cite{CW2012}, global existence was established for $a\in [-2,-1)$ in the inviscid setting, using the method of characteristics. In \cite{SS2013,SS2015}, various global existence and blow-up scenarios are investigated based on a representation formula for the velocity gradient along particle trajectories. In contrast, singularity formation for $a>1$ remains less understood. In \cite{OH2009}, the authors construct non-smooth self-similar solutions of the form $u(x,t)=\frac{F(x)}{T-t}$, which blow-up at time $T$. These solutions are obtained by solving the nonlinear ordinary differential equation
 \begin{align}\label{Fequ}
 F''(x) + F(x)F'''(x) - aF'(x)F''(x) = 0, \quad x\in \mathbb{T},
 \end{align}
 for $a>1$. However, in this regime, equation \eqref{Fequ} admits only non-smooth solutions. More recently, Kogelbauer \cite{KF2020} established new criteria for global existence and finite time blow-up in the inviscid generalized Proudman-Johnson equation, depending on the parameter $a$. In particular, \cite{KF2020} demonstrated that even smooth initial data may lead to finite time singularities when $a>1$. Additionally, \cite{KF2020} provides a physical derivation of the generalized equation for arbitrary values of the parameter $a$.

Throughout this paper, $\|\cdot\|_{L^{p}},\|\cdot\|_{H^{m}}$ and $\|\cdot\|_{L^{\infty}}$ represent the norms in the spaces $L^{p}(\mathbb{T}), H^{m}(\mathbb{T})$ and $L^{\infty}(\mathbb{T})$, respectively. Similarly, $\left\langle\cdot, \cdot\right\rangle$ denotes the usual inner product in $L^{2}(\mathbb{T})$, defined as
 \begin{align*}
 \left\langle f, g\right\rangle = \int_{\mathbb{T}}fg \, \mathrm{d}x.
 \end{align*}
 We use $C,C_{i}$ to denote absolute constants and $C(A,B,\cdots,Z)$ to denote constants depending on $A,B,\cdots,Z$. And we also employ the notation $A\lesssim B$ to indicate that there exists a constant $C$ such that $A\leq CB$.

Now we are ready to state our main results. Our first main result establishes the existence of a family of self-similar solutions to the inviscid model \ref{PJE}.

Our next result concerns the formation of finite time singularities in the generalized Proudman-Johnson model with H\"{o}lder continuous initial data.
 \begin{theorem}\label{the holder}
 {\bf (The case $\nu=0$ and $a=1$)} There exists a constant $\delta_{2}>0$, such that for all $\alpha\in(1-\delta_{2},1),$ the model \ref{PJE} with $\nu=0$ and $a=1$ develops a finite time singularity for some $C^{\alpha}$ initial data. Moreover, there exists a $C^{\alpha}$ self-similar profile, analogous to the setting in Theorem \ref{them profile}.
 \end{theorem}

For a fixed $k_{0}$, there exists a sufficiently small constant $0<\mu_{1}(k_{0})<\mu(k_{0})$, such that the following estimate holds
 \begin{align*}
 \frac{\mathrm{d}}{\mathrm{d}\tau} I_{k_{0}}^{2}(\tau)\leq & -(\frac{1}{2}-C|1-a|)I_{k_{0}}^{2}(\tau)+CI_{k_{0}}^{3}(\tau)+C|1-a|I_{k_{0}}(\tau)\\
 & + C\nu C_{\omega}(\tau)\left(\|\hat{\omega}_{xxx}\|_{L^{\infty}(I)}^{2}+(1+\|\hat{\omega}_{xxx}\|_{L^{\infty}(I)})I_{k_{0}}^{2}(\tau)\right),
 \end{align*}
 where the energy is defined as
 \begin{align*}
 I_{k_{0}}^{2}(\tau)=\sum_{k=0}^{k_{0}}\mu_{1}^{k}(k_{0})E_{k}^{2}(\tau).
 \end{align*}
 Here, the constants depend on $k_{0}$ and $\mu,$ but once $k_{0}$ is fixed, with $\mu=\mu_{1}(k_{0})$, they become just constants. We will later make both $C_{\omega}(t)$ and $|1-a|$ sufficiently small to close the argument. By the Gagliardo-Nirenberg inequality, we have
 \begin{align*}
 \|\hat{\omega}_{xxx}\|_{L^{\infty}(I)}\lesssim \|\hat{\omega}^{(4)}\|_{L^{2}(I)}^{\frac{7}{8}} \|\hat{\omega}\|_{L^{2}(I)}^{\frac{1}{8}}.
 \end{align*}
 Consequently, we conclude that
 \begin{align*}
 \|\hat{\omega}_{xxx}\|_{L^{\infty}(I)}\lesssim I_{k_{0}}(\tau),
 \end{align*}
 for $k_{0}\geq 4$. For simplicity, we take $k_{0}=4$ as an example, which implies
 \begin{align*}
 \frac{\mathrm{d}}{\mathrm{d}\tau}I_{4}(\tau) \leq - (\frac{1}{2} - C|1-a|) I_{4}(\tau) + CI_{4}^{2}(\tau) + C|1-a| + C\nu C_{\omega}(\tau)(1+I_{4}(t))I_{4}(\tau).
 \end{align*}
 We now choose \(C_{\omega} (0) = |1-a|^{2} < \delta_{3}\) for a sufficiently small \(\delta_{3} > 0\) and use a bootstrap argument to obtain
 \begin{align*}
 I_{4} (\tau) \leq C|1-a|, \quad C_{\omega} (\tau) \leq 2|1-a|^2,
 \end{align*}
 where $C>0$ is some absolute constant.

\section{Appendix}\label{section appendix}
 In the Appendix, we provide the proof of Lemma \ref{le operator}. To extract the maximal damping effect, we expand the perturbed solution in a Fourier series and perform exact calculations.
 \begin{lemma}\label{le:linear operator}
 It holds that
 \begin{align*}
 \left\langle\mathcal{L}_{1} \hat{\omega},\hat{\omega} \right\rangle_{\mathcal{H}} \leq -\frac{1}{2}\|\hat{\omega}\|_{\mathcal{H}}^{2}.
 \end{align*}
 \end{lemma}
 \begin{proof}[Proof of Lemma \ref{le:linear operator}]
 Note that
 \begin{align}
 \mathcal{L}_{1}e_{k}^{(o)} = A_{k}e_{k+1}^{(o)} + B_{k}e_{k-1}^{(o)}, \quad k \geq 2,
 \end{align}
 where
 \begin{align*}
 e_{k}^{(o)} = \sin(kx), \quad k \geq 1
 \end{align*}
 and
 \begin{align}
 A_{k} = -\frac{(k+1)(k-1)^{2}}{2k^{2}}, \quad B_{k} = \frac{(k+1)^{2}(k-1)}{2k^{2}}, \quad k \geq 2.
 \end{align}
 For $k \geq 2,$ direct calculations give
 \begin{align*}
 \mathcal{L}_{1}\tilde{e}_{k}^{(o)} = & \frac{A_{k+1}}{k+1}e_{k+2}^{(o)} + \frac{B_{k+1}}{k+1}e_{k}^{(o)} - \frac{A_{k}}{k}e_{k+1}^{(o)} - \frac{B_{k}}{k}e_{k-1}^{(o)}\\
 = & -\frac{(k+2)k^{2}}{2(k+1)^{3}}e_{k+2}^{(o)} + \frac{(k+2)^{2}k}{2(k+1)^{3}}e_{k}^{(o)} + \frac{(k+1)(k-1)^{2}}{2k^{3}}e_{k+1}^{(o)} - \frac{(k+1)^{2}(k-1)}{2k^{3}}e_{k-1}^{(o)}\\
 = & -\frac{(k+2)^{2}k^{2}}{2(k+1)^{3}}\left(\frac{e_{k+2}^{(o)}}{k+2} - \frac{e_{k+1}^{(o)}}{k+1}\right) - \frac{(k+2)^{2}k^{2}}{2(k+1)^{3}}\frac{e_{k+1}^{(o)}}{k+1} + \frac{(k+2)^{2}k^{2}}{2(k+1)^3}\frac{e_{k}^{(o)}}{k}\\
 & + \frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\left(\frac{e_{k}^{(o)}}{k} - \frac{e_{k-1}^{(o)}}{k-1}\right) - \frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\frac{e_{k}^{(o)}}{k} + \frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\frac{e_{k-1}^{(o)}}{k-1}\\
 := & -d_{k+1}\tilde{e}_{k+1}^{(o)} + (-d_{k+1} + d_{k})\tilde{e}_{k}^{(o)} + d_{k}\tilde{e}_{k-1}^{(o)}.
 \end{align*}
 In addition, when $k=1$, we have
 \begin{align*}
 \mathcal{L}_{1} e_{1}^{(o)} = 0
 \end{align*}
 and
 \begin{align*}
 \mathcal{L}_{1}\tilde{e}_{1}^{(o)}=-d_{2}\tilde{e}_{2}^{(o)}+(d_{1}-d_{2})\tilde{e}_{1}^{(o)}.
 \end{align*}
 Thus we can write
 \begin{align*}
 \mathcal{L}_{1}\tilde{e}_{k}^{(o)} = -d_{k+1}\tilde{e}_{k+1}^{(o)} + (-d_{k+1} + d_{k})\tilde{e}_{k}^{(o)} + d_{k}\tilde{e}_{k-1}^{(o)},
 \end{align*}
 where $d_{k} = \frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}, k = 1, 2, \cdots.$
 If we expand \(\hat{\omega} (x,\tau)\) as
 \[\hat{\omega}(x, \tau) = \sum_{k \geq 1} \hat{\omega}^{(o)}_k(\tau)\, \tilde{e}^{(o)}_k,\]
 then the equation $\partial_{\tau} \hat{\omega} = \mathcal{L}_{1} \hat{\omega}$ reduces to the following infinite-dimensional system of ordinary differential equations
 \begin{align*}
 \partial_{\tau} \hat{\omega}_{k}^{(o)} (\tau) = - d_{k} \hat{\omega}_{k-1}^{(o)} (\tau) - (d_{k+1} - d_{k}) \hat{\omega}_{k}^{(o)} (\tau) + d_{k+1} \hat{\omega}_{k+1}^{(o)} (\tau), \quad k \geq 1,
 \end{align*}
 where $d_{1} \hat{\omega}_{0}^{(o)}$ is understood to be zero. Thus, we formally deduce that
 \begin{align}\label{sumomegak}
 \begin{aligned}
 \frac{1}{2} \partial_{\tau} \sum_{k\geq 1} \big(\hat{\omega}_{k}^{(o)}\big)^{2} & =
 \left\langle \mathcal{L}_{1}\hat{\omega},\hat{\omega} \right\rangle_{\mathcal{H}}\\
 & = \sum_{k\geq 1} -d_{k} \hat{\omega}_{k-1}^{(o)}\hat{\omega}_{k}^{(o)} + (-d_{k+1} + d_{k}) \big(\hat{\omega}_{k}^{(o)}\big)^{2} + d_{k+1} \hat{\omega}_{k}^{(o)}\hat{\omega}_{k+1}^{(o)}\\
 & = \sum_{k\geq 1} (-d_{k+1} + d_{k}) \big(\hat{\omega}_{k}^{(o)}\big)^{2}\\
 &\leq - \frac{1}{2} \big(\hat{\omega}_{k}^{(o)}\big)^{2}.
 \end{aligned}
 \end{align}
 Here, we have used the fact that for all $k\geq 1$,
 \begin{align*}
 d_{k+1} - d_{k} &= \frac{(k+2)^{2}k^{2}}{2(k+1)^{3}} - \frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\\
 &=\frac{k^{6} + 3k^{5} +5k^{4} + 5k^{3} - k^{2} -3k -1}{2k^{3}(k+1)^{3}}\\
 &=\frac{1}{2} + \frac{2k^{4} + 4k^{3} -k^{2} -3k -1}{2k^{3}(k+1)^{3}}\\
 & > \frac{1}{2}.
 \end{align*}
 While the formal computation in \ref{sumomegak} illustrates the dissipative structure of \(\mathcal{L}_1\), it is required to justify the validity of the summation, which may not converge in a straightforward manner. One rigorous approach is to invoke standard linear semigroup theory.
 To this end, we consider the real Hilbert space $Y$ formally spanned by the basis functions $\{\tilde{e}_{k}^{(o)}, k\geq 1\}$ in which this basis is orthonormal, namely
 \begin{align*}
 Y = \bigg\{\hat{\omega} (x,\tau) = \sum_{k\geq 1} \hat{\omega}_{k}^{(o)} (\tau) \tilde{e}_{k}^{(o)} \bigg| \{\hat{\omega}_{k}^{(o)}\}_{k\geq 1}\in l^{2}\bigg\}.
 \end{align*}
 The operator \(\mathcal{L}_1\) defines a densely defined, closed, unbounded linear operator on \(Y\). The estimate \eqref{sumomegak}, together with standard energy methods, implies via a direct application of the Hille--Yosida theorem that \(\mathcal{L}_1\) generates a strongly continuous semigroup satisfying
 \begin{align*}
 \|e^{\tau \mathcal{L}_{1}} \hat{\omega} (0)\|_{\mathcal{H}} \leq e^{-\frac{1}{2} \tau} \|\hat{\omega} (0)\|_{\mathcal{H}}.
 \end{align*}
 An alternative approach is to apply Galerkin's method to establish global well-posedness. In addition, the decay rate of the Fourier coefficients of the solution can be rigorously derived (see \cite{GJ2025} for more details).
 \end{proof}

Our next result concerns the formation of finite time singularities in the generalized Proudman-Johnson model with H\"{o}lder continuous initial data.
	\begin{theorem}\label{the holder}
		{\bf (The  case $\nu=0$ and $a=1$)}  There exists a constant $\delta_{2}>0$, such that for all $\alpha\in(1-\delta_{2},1),$ the model \ref{PJE} with $\nu=0$ and $a=1$ develops a finite time singularity for some $C^{\alpha}$ initial data. Moreover, there exists a $C^{\alpha}$ self-similar profile, analogous to the setting in Theorem \ref{them profile}.
	\end{theorem}

	The final result concerns the finite time blow-up of model \ref{PJE} in the presence of viscosity. The dynamic rescaling formulation suggests that the viscous terms are asymptotically small. Building upon Theorem \ref{them profile} in the regime $1<a<1+\delta_{3},$ we establish the following result.

	\begin{theorem}\label{the viscous}
		{\bf (The  case $\nu>0$)} There exists a constant $\delta_{3}>0$ such that for all $a\in(1,1+\delta_{3}),$ the model \ref{PJE} with $\nu>0$ develops a singularity in finite time for some $C^{\infty}$ initial data.
	\end{theorem}
	We emphasize that, in contrast to the inviscid case, exact self-similar blow-up profiles do not exist due to the presence of viscosity.

	In our analysis, we employ the framework of dynamic rescaling to establish the formation of singularities. This formulation was first introduced by McLaughlin, Papanicolaou, and co-authors in their study
	of self-similar blow-up of the nonlinear Schr\"{o}dinger equation (see \cite{MP1986,LP1988}). It was later developed into a powerful modulation technique, and has been applied to various blow-up problems including the nonlinear Schr\"{o}dinger equation \cite{KM2006,MR2005}, the nonlinear heat equation \cite{MZ1997}, the generalized KdV equation \cite{MM2014}, and other
	dispersive problems. More recently, this approach has been successfully adapted to prove singularity formation in the gCLM models \cite{C2021, CHH2021}, the Euler equations \cite{CH2021, E2021}, and the Hou-Li model \cite{HW2024}.

	Our blow-up analysis for the inviscid model \ref{PJE} consists of several steps. In the first step, we reformulate the original equation \eqref{PJE} via dynamic rescaling, thereby transforming the singularity formation problem into the stability analysis of an approximate steady state in the rescaled variables. In the second step, we perform the stability analysis of this approximate steady state in the rescaled formulation, which primarily involves:
	\begin{itemize}
		\item the identification of an approximate steady state;
		\item energy estimates in a singularly weighted norm, providing both linear and nonlinear stability of this approximate profile.
	\end{itemize}
	By tracing back to the original variables, this yields finite time blow-up for the inviscid model \eqref{PJE}. In the third step, we establish convergence of the dynamically rescaled solution to the true self-similar profile. However, the blow-up analysis for model \ref{PJE} with viscosity is more challenging, as the viscous term does not provide damping and generates some bad terms when a singular weighted norm is employed. To address this, we establish the blow-up analysis for the viscous \ref{PJE} using an energy norm that combines a singular weighted energy norm with a sum of higher-order Sobolev norms, as employed in \cite{HW2024}.

	The paper is organized as follows. In Section \ref{section pre}, we present some preliminaries and useful lemmas. Section \ref{section COSSP} is devoted to the construction of a family of self-similar profiles for \ref{PJE} when $a$ is close to 1, together with their stability analysis and the proof of Theorem \ref{them profile}. Section \ref{section holder} contains the proof of Theorem \ref{the holder}, concerning finite-time blow-up from H\"{o}lder continuous initial data for the model \ref{PJE} in the case $a=1$. Finally, in Section \ref{section viscous}, we prove Theorem \ref{the viscous} through a special energy norm to estimate the viscous terms.

The introduction outlines the proof strategy for Theorem~\ref{them profile} (and related results) as follows.

\begin{itemize}
\item \emph{Dynamic rescaling reduction.} “In the first step, we reformulate the original equation \eqref{PJE} via dynamic rescaling, thereby transforming the singularity formation problem into the stability analysis of an approximate steady state in the rescaled variables.”

\item \emph{Stability analysis in rescaled variables.} “In the second step, we perform the stability analysis of this approximate steady state in the rescaled formulation,” which “primarily involves”
  \begin{itemize}
  \item “the identification of an approximate steady state;”
  \item “energy estimates in a singularly weighted norm, providing both linear and nonlinear stability of this approximate profile.”
  \end{itemize}
  “By tracing back to the original variables, this yields finite time blow-up for the inviscid model \eqref{PJE}.”

\item \emph{Convergence to the true self-similar profile.} “In the third step, we establish convergence of the dynamically rescaled solution to the true self-similar profile.”
\end{itemize}

The introduction also locates the proof of Theorem~\ref{them profile}: “Section \ref{section COSSP} is devoted to the construction of a family of self-similar profiles for \ref{PJE} when $a$ is close to 1, together with their stability analysis and the proof of Theorem \ref{them profile}.”

The introduction outlines the proof strategy for Theorem~\ref{them profile} (and related results) as follows.

\begin{itemize}
\item \emph{Dynamic rescaling reduction.} “In the first step, we reformulate the original equation
\begin{align}\label{PJE}
		\begin{cases}
			\omega_{t} + u\omega_{x} = a\omega u_{x}+\nu\omega_{xx}, & t >0, \ x \in \mathbb{T}, \\
			u_{xx} =\omega, & t >0, \ x \in \mathbb{T}.
		\end{cases}
	\end{align}
via dynamic rescaling, thereby transforming the singularity formation problem into the stability analysis of an approximate steady state in the rescaled variables.”

\item \emph{Stability analysis in rescaled variables.} “In the second step, we perform the stability analysis of this approximate steady state in the rescaled formulation,” which “primarily involves”
  \begin{itemize}
  \item “the identification of an approximate steady state;”
  \item “energy estimates in a singularly weighted norm, providing both linear and nonlinear stability of this approximate profile.”
  \end{itemize}
  “By tracing back to the original variables, this yields finite time blow-up for the inviscid model \eqref{PJE}.”

\item \emph{Convergence to the true self-similar profile.} “In the third step, we establish convergence of the dynamically rescaled solution to the true self-similar profile.”
\end{itemize}

The introduction also locates the proof of the main theorem: “Next we construct a family of self-similar profiles for \eqref{PJE} when $a$ is close to 1, together with their stability analysis and the proof of the main theorem.”

\label{them profile}
		{\bf (The  case $\nu=0$)} There exists an absolute constant $\delta_{1}>0,$  sufficiently small, such that the following statements hold:

		(1) For \(a\in (1, 1+\delta_{1})\) the model
		\begin{align}\label{PJE}
		\begin{cases}
			\omega_{t} + u\omega_{x} = a\omega u_{x}+\nu\omega_{xx}, & t >0, \\ x \in \mathbb{T}, \\
			u_{xx} =\omega, & t >0, \\ x \in \mathbb{T}.
		\end{cases}
	\end{align}
		with $\nu=0$ develops a finite time singularity for some \(C^{\infty}\) initial data;

		(2) For all $a\in(1-\delta_{1},1+\delta_{1}),$ the model above admits a self-similar solution of the form
		\begin{align*}
			\omega(x,t) = \frac{1}{1 + c_{\omega,a}t}\omega_{a}(x),
		\end{align*}
		where $\omega_{a}$ is an odd profile and $c_{\omega,a}$ is the scaling parameter,  satisfying
		\begin{align}\label{the estimates}
			\|\omega_{a} +\sin (x)\|_{\mathcal{W}} \lesssim |1-a|, \quad |c_{\omega,a} - (1-a)|\leq \min\{C|1-a|^{2}, |1-a|\},
		\end{align}
		for some absolute constant $C > 0$. More precisely:
		\begin{itemize}
			\item for $1 < a < 1+\delta_{1}$, the scaling parameter satisfies $c_{\omega,a} < 0$, and the corresponding solution $\omega(x,t)$ blows up in finite time $T = -\frac{1}{c_{\omega,a}}$;
			\item for $a = 1$, the explicit steady state $\omega_{1} = -\sin (x)$ solves the system with $c_{\omega,a} = 0$;
			\item for $1-\delta_{1}<a<1$, one has $c_{\omega,a} > 0$, and the solution $\omega(x,t)$ exists globally with $O(t^{-1})$ decay rate as $t\to\infty$.
		\end{itemize}

Question:
Consider the inviscid generalized Proudman--Johnson system on the torus \(\mathbb T=[-\pi,\pi]\):
\[
\begin{cases}
\omega_t+u\omega_x=a\,\omega u_x, & t>0,\ x\in\mathbb T,\\
u
u_{xx}=\omega, & t>0,\ x\in\mathbb T,
\end{cases}
\]
that is, the case \(\nu=0\). Which statement holds for parameters \(a\) near \(1\)?

There exists an absolute constant \(\delta_1>0\), sufficiently small, such that: (i) for every \(a\in(1,1+\delta_1)\), the inviscid system develops a finite-time singularity for some \(C^\infty\) initial data; and (ii) for every \(a\in(1-\delta_1,1+\delta_1)\), the system admits a self-similar solution
\[
\omega(x,t)=\frac{1}{1+c_{\omega,a}t}\,\omega_a(x),
\]
where \(\omega_a\) is an odd profile and \(c_{\omega,a}\) is a scaling parameter satisfying
\[
\|\omega_a+\sin x\|_{\mathcal W}\lesssim |1-a|,\qquad |c_{\omega,a}-(1-a)|\le \min\{C|1-a|^2,|1-a|\}
\]
for some absolute constant \(C>0\). More precisely: if \(1<a<1+\delta_1\), then \(c_{\omega,a}<0\) and the corresponding self-similar solution blows up at finite time \(T=-1/c_{\omega,a}\); if \(a=1\), then the explicit steady state \(\omega_1=-\sin x\) solves the system and \(c_{\omega,a}=0\); and if \(1-\delta_1<a<1\), then \(c_{\omega,a}>0\) and the self-similar solution exists globally with \(O(t^{-1})\) decay as \(t\to\infty\).

Choices:
B.
There exists an absolute constant \(\delta_1>0\), sufficiently small, such that: (i) for every \(a\in(1,1+\delta_1)\), every \(C^\infty\) solution of the inviscid system develops a finite-time singularity; and (ii) for every \(a\in(1-\delta_1,1+\delta_1)\), the system admits a self-similar solution
\[
\omega(x,t)=\frac{1}{1+c_{\omega,a}t}\,\omega_a(x),
\]
where \(\omega_a\) is an odd profile and \(c_{\omega,a}\) is a scaling parameter satisfying
\[
\|\omega_a+\sin x\|_{\mathcal W}\lesssim |1-a|,\qquad |c_{\omega,a}-(1-a)|\le \min\{C|1-a|^2,|1-a|\}
\]
for some absolute constant \(C>0\). More precisely: if \(1<a<1+\delta_1\), then \(c_{\omega,a}<0\) and the corresponding self-similar solution blows up at finite time \(T=-1/c_{\omega,a}\); if \(a=1\), then the explicit steady state \(\omega_1=-\sin x\) solves the system and \(c_{\omega,a}=0\); and if \(1-\delta_1<a<1\), then \(c_{\omega,a}>0\) and the self-similar solution exists globally with \(O(t^{-1})\) decay as \(t\to\infty\).
C.
There exists an absolute constant \(\delta_1>0\), sufficiently small, such that for every \(a\in(1-\delta_1,1+\delta_1)\), the inviscid system admits a self-similar solution
\[
\omega(x,t)=\frac{1}{1+c_{\omega,a}t}\,\omega_a(x),
\]
where \(\omega_a\) is an odd profile and \(c_{\omega,a}\) satisfies
\[
\|\omega_a+\sin x\|_{\mathcal W}\lesssim |1-a|,\qquad |c_{\omega,a}-(1-a)|\le \min\{C|1-a|^2,|1-a|\}
\]
for some absolute constant \(C>0\). In particular, \(a=1\) yields the steady state \(\omega_1=-\sin x\) with \(c_{\omega,a}=0\); for \(1<a<1+\delta_1\) one has \(c_{\omega,a}<0\), while for \(1-\delta_1<a<1\) one has \(c_{\omega,a}>0\).
D.
There exists an absolute constant \(\delta_1>0\), sufficiently small, such that: (i) for every \(a\in(1,1+\delta_1)\), the inviscid system develops a finite-time singularity for some \(C^\infty\) initial data; and (ii) for every \(a\in(1-\delta_1,1+\delta_1)\), the system admits a self-similar solution
\[
\omega(x,t)=\frac{1}{1+c_{\omega,a}t}\,\omega_a(x),
\]
where \(\omega_a\) is an odd profile and \(c_{\omega,a}\) is a scaling parameter satisfying
\[
\|\omega_a+\sin x\|_{\mathcal W}\lesssim |1-a|,\qquad |c_{\omega,a}-(1-a)|\le C|1-a|^2
\]
for some absolute constant \(C>0\). More precisely: if \(1<a<1+\delta_1\), then \(c_{\omega,a}<0\) and the corresponding self-similar solution blows up at finite time \(T=-1/c_{\omega,a}\); if \(a=1\), then the explicit steady state \(\omega_1=-\sin x\) solves the system and \(c_{\omega,a}=0\); and if \(1-\delta_1<a<1\), then \(c_{\omega,a}>0\) and the self-similar solution exists globally with \(O(t^{-1})\) decay as \(t\to\infty\).
E.
There exists an absolute constant \(\delta_1>0\), sufficiently small, such that: (i) for every \(a\in(1,1+\delta_1)\), the inviscid system develops a finite-time singularity for some \(C^\infty\) initial data; and (ii) for every \(a\in(1-\delta_1,1+\delta_1)\), the system admits a self-similar solution
\[
\omega(x,t)=\frac{1}{1+c_{\omega,a}t}\,\omega_a(x),
\]
where \(\omega_a\) is an odd profile and \(c_{\omega,a}\) is a scaling parameter satisfying
\[
\|\omega_a+\sin x\|_{\mathcal W}\lesssim |1-a|,\qquad |c_{\omega,a}-(1-a)|\le \min\{C|1-a|^2,|1-a|\}
\]
for some absolute constant \(C>0\). More precisely: if \(1<a<1+\delta_1\), then \(c_{\omega,a}>0\) and the corresponding self-similar solution exists globally with \(O(t^{-1})\) decay as \(t\to\infty\); if \(a=1\), then the explicit steady state \(\omega_1=-\sin x\) solves the system and \(c_{\omega,a}=0\); and if \(1-\delta_1<a<1\), then \(c_{\omega,a}<0\) and the self-similar solution blows up at finite time \(T=-1/c_{\omega,a}\).

The stem does not reveal the correct option or give decisive hints; it only asks which statement is valid near a = 1.

This is largely a theorem-statement recognition task: the correct option is essentially the full near-a=1 result, with distractors formed by small alterations in quantifiers, signs, or bounds.

A test-taker must compare subtle mathematical differences across choices, but the task is mostly precise recall/discrimination rather than generating a conclusion from the PDE itself.

Several distractors are plausible and target common errors (existential vs universal quantifiers, sign reversal, stronger estimate). However, choice C is a weaker true statement, so the item is not cleanly single-answer and the distractor set is compromised.

No answer leakage and some subtle option design, but the item mainly tests recall of a theorem statement rather than genuine reasoning. Its biggest flaw is that one distractor appears true but weaker, creating ambiguity in a single-answer MCQ.


\begin{aligned}
S_n + \sum_{m=n+1}^{\infty} \frac{(c_{n,m})^2}{d_{\alpha,m}} \int_{\mathbb{B}^{n}}  \bigg\langle {( \nabla \varphi_{z,w} (\tau) )}^m , \overline{ \bigg( \int_{\mathbb{B}^n} \bigg\langle (\nabla \varphi_{z',w'} (\tau') )^m, \overline{B_m (\tau, \tau')}  \bigg\rangle dV_g (\tau') \bigg) }  \bigg \rangle  dV_g(\tau)
\end{aligned}
\end{equation}
and the proof is completed.
\end{proof}

\section{Explicit formula for totally geodesic isometric embeddings}\label{isometric embeddings}
In this section, we derive an explicit formula for the map~$\Phi$ associated with the holomorphic $\mathbb{B}^N$-fiber bundle over $M$, where the compact complex manifold $M$ admits a totally geodesic, isometric, equivariant holomorphic embedding into $\mathbb{B}^N$. The structure of the argument follows the approach used in Section~\ref{proof of main theorem}.

For a complex manifold $\widetilde M$, a lattice $\Gamma$ in $\textup{Aut}(\widetilde M)$ and a homomorphism $\rho\colon\Gamma\to\textup{Aut}(\mathbb B^N)$, we say that a map $f\colon \widetilde M\to\mathbb B^N$ is $\rho$-equivariant if  $f(\gamma z) = \rho(\gamma)f(z)$ holds for any $\gamma\in\Gamma$ and $z\in \widetilde M$.

\begin{theorem}[\cite{LS2}]\label{totally geodesic imbedding}
Let $\widetilde{M}$ be a complex manifold, $\Gamma$ be a torsion-free cocompact lattice of $\textup{Aut}(\widetilde{M})$ and $\rho\colon\Gamma\to SU(N,1)$ be a representation.
Suppose that there exists a $\rho$-equivariant totally geodesic isometric holomorphic embedding $\imath\colon \widetilde M\to\mathbb B^N$. Let
$M:=\widetilde M/\Gamma$ and $\Sigma:=\mathbb B^N/\rho(\Gamma)$.
Let $\Omega_{\rho}:=M\times_\rho \mathbb B^N$ be a holomorphic $\mathbb B^N$-fiber bundle over $M$ where any $\gamma\in \Gamma$ acts on $\widetilde M\times \mathbb B^N$ by $(\zeta,w)\mapsto (\gamma \zeta, \rho(\gamma) w)$. Then there exists an injective linear map
\begin{equation}\nonumber
\Phi: \bigoplus_{m=0}^{\infty} H^0 (M, \imath^* (S^m T_{\Sigma}^*)) \rightarrow
\begin{cases}
\displaystyle\bigcap_{\alpha>-1} A^2_\alpha (\Omega_{\rho}) \subset \mathcal{O} (\Omega_{\rho}) & \text{ if } n=N,\\
\displaystyle\bigcap_{\alpha\geq -1} A^2_\alpha (\Omega_{\rho}) \subset \mathcal{O} (\Omega_{\rho})  & \text{ if } n< N,\\
\end{cases}
\end{equation}
which has a dense image in $\mathcal{O}(\Omega_{\rho})$ equipped with the compact open topology. In particular,
\[
\begin{cases}
\dim A^2_{\alpha}(\Omega_{\rho}) = \infty, & \text{if } \alpha > -1, \\[6pt]
A^2_{-1}(\Omega_{\rho}) = \displaystyle\bigcap_{\alpha \ge -1} A^2_{\alpha}(\Omega_{\rho}), \quad
\dim A^2_{-1}(\Omega_{\rho}) = \infty, & \text{if } N > n.
\end{cases}
\]

\end{theorem}

For the definition of $A^{2}{\alpha}(\Omega_{\rho})$, see \cite[pp.~20]{LS2}. Although, in \cite{LS2} the symbol $\Omega$ is used for $\Omega_{\rho}$, we will use the notation $\Omega_{\rho}$ in this section to avoid confusion with the notation used in the previous sections.

Let $N = \imath^* T_{\Sigma} / T_M$ be the holomorphic normal bundle of $M$. Since $M$ is totally geodesically embedded in $\Sigma$, the normal bundle $N$ is holomorphically isomorphic to the orthogonal complement of $T_M$ in $\imath^* T_{\Sigma}$ with respect to the induced metric from the normalized Bergman metric of $\mathbb{B}^N$. As a result, the pullback of the symmetric power of the cotangent bundle of $M$ admits the following holomorphic decomposition:
$$
H^0(M, \imath^* (S^m T_{\Sigma}^*)) \cong \bigoplus_{\ell=0}^{m} H^0 (M, S^\ell T_{M}^* \otimes S^{m-\ell} N^*).
$$
$$
$$

\begin{theorem}
For $\psi^{\ell}_m \in H^0(M, S^\ell T_{M}^* \otimes S^{m-\ell} N^*)$, we have
    \begin{equation}\nonumber
    \begin{aligned}
        \Phi ( \psi^\ell_m )(\zeta, w)
        &= \frac{c_{n,m}{(m+\ell-1)}}{2m-1}\int_{\imath(\widetilde M)}
        \left\langle
        \left.\left(
        \sum_{j,\ell} g^{j\bar \ell}(\tau)\frac{\partial}{\partial\overline\tau_\ell}\frac{K(\imath(\zeta),\tau)}{K(w,\tau)}\frac{\partial}{\partial\tau_j}\right)^m\right|_{\tau\in\imath(\widetilde M)},
        \psi^\ell_m \right\rangle ~dV_{\widetilde M}
   \end{aligned}

\begin{aligned}
            \Theta(\psi)(0,w)
            &=\frac{\pi^n(N-n-1)!(2N-2)!}{(N-1)!\,(N-2)! }\sum_{|I|=N} \sum_{k=0}^\infty\sum_{|K|=k}
        \frac{(N+k-1)!}{(2N+k-1)!K!}\frac{\partial^K\psi_I}{\partial\tau^K }(0)w^{I+K}.
    \end{aligned}
\end{equation}
\end{proof}

\section{Application of the explicit form of $\Phi$}
\subsection{Poincar\'e series}\label{Poincare series}
In \cite[for $n=1$]{O16} and \cite[pp. 1265--1266 for $n\geq 2$]{LS1}, it is proved that the Poincar\'e series defined by
\begin{equation}\nonumber
	\sum_{\gamma\in \Gamma}  (\gamma_j(z)-\gamma_j(w))^{N}
\end{equation}
is a $\Gamma$-invariant holomorphic function on $\mathbb B^n\times\mathbb B^n$ for any $N\geq n+1$
with respect to the diagonal action, i.e.
this series converges when $N\geq n+1$.
Now, we consider a holomorphic coordinate system $(\widetilde z, \widetilde w)$ on $\mathbb{B}^n \times \mathbb{B}^n$ which is given by $\widetilde z := z$ and $\widetilde w := w-z$.  By the Taylor expansion of $\gamma_j (\tilde z + \tilde w)$ at $\tilde w=0$, we have
\begin{equation*}
\begin{aligned}
&\sum_{\gamma \in \Gamma} (\gamma_j (z) - \gamma_j (w))^N
= \sum_{\gamma \in \Gamma} (\gamma_j (\tilde z ) - \gamma_j (\tilde z + \tilde w) )^N \\
&= (-1)^N \sum_{|I|=N}\frac{N!}{I!} \bigg(  \sum_{\gamma \in \Gamma} \bigg( \frac{\partial \gamma_j (z) } {\partial z_1} \bigg)^{i_1} \cdots \bigg( \frac{\partial \gamma_j (z) } {\partial z_n} \bigg)^{i_n} \bigg)  {\tilde w}^{i_1} \cdots {\tilde w}^{i_n} + O( {| \tilde{w} |}^{N+1} ).
\end{aligned}

\label{Bergman kernels}

In this section, we define the weighted Bergman space $A^2_{\alpha}(\Omega)$ on $\Omega$ and express the weighted Bergman kernel of $A^2_{\alpha}(\Omega)$ using Bergman kernel

Let $\mathbb B^n = \{ z \in \mathbb{C}^n : |z| < 1 \}$ be the unit ball in $\mathbb{C}^n$, and let
$\Gamma \subset \mathrm{Aut}(\mathbb B^n)$ be a cocompact discrete subgroup.
We denote by $\Sigma = \mathbb B^n / \Gamma$
the corresponding compact complex hyperbolic space form, and $\Omega = (\mathbb B^n \times \mathbb B^n)/\Gamma$
be the quotient under the diagonal action $\gamma \cdot (z,w) = (\gamma z, \gamma w)$.
Then $\Omega$ is a holomorphic $\mathbb B^n$-fiber bundle over $\Sigma$.
In previous work~\cite{LS1},
the authors studied the relation between holomorphic symmetric differentials on $\Sigma$ and weighted $L^2$ holomorphic functions on $\Omega$.
More precisely,
authors proved that there exists a natural injective correspondence between symmetric differentials on $\Sigma$ and weighted $L^2$ holomorphic functions on $\Omega$.
For the holomorphic cotangent bundle $T^*_\Sigma$ of $\Sigma$, and its $m$-th symmetric power $S^m T^*_\Sigma$, there exists an injective linear map
$$
\Phi:\ \bigoplus_{m=0}^\infty H^0(\Sigma, S^m T^*_\Sigma)
\;\longrightarrow\;
\bigcap_{\alpha > -1} A^2_\alpha(\Omega) \subset \mathcal{O}(\Omega),
$$
whose image is dense in $\mathcal{O}(\Omega)$ with respect to the compact--open topology.
For $n=1$ this goes back to a result of Adachi~\cite{A21},
who obtained an explicit description in terms of the cross ratio on the unit disc.

In the one-dimensional case, if $\Sigma$ is a compact hyperbolic Riemann surface and
$$\psi = \psi(\tau)(d\tau)^{\otimes N} \in H^0(\Sigma, K_\Sigma^{\otimes N}),$$
Adachi showed that the corresponding function $\Phi(\psi)$ on $(\mathbb B^1 \times \mathbb B^1)/\Gamma$ is given by
$$
\Phi(\psi)(z,w)
=
\begin{cases}
\displaystyle
\frac{1}{B(N,N)}
\int_{\tau \in [z,w]}
\!\!
\left(
\frac{w - z}{(w - \tau)(\tau - z)}
\right)^{N-1}
\psi(\tau)^N\, d\tau, & N \ge 1,\\[1.2em]
\text{the constant }\psi, & N=0,
\end{cases}
$$
where $B(N,N)$ denotes the Beta function.
For $n \ge 2$, however, the construction of $\Phi$ in~\cite{LS1}, which was inspired by Adachi's recursive approach in the one-dimensional case, relied on solving a recursive system of $\bar\partial$-equations
using the raising operator associated to the K\"ahler form of the Bergman metric, and did not yield a closed formula.
While the existence and density of $\Phi$ were established, the dependence of $\Phi(\psi)$ on the symmetric differential $\psi$ remained implicit.

The aim of the present paper is to give an explicit formula for $\Phi(\psi)$ in all dimensions $n \ge 1$.
Let $K(z,w)$ denote the normalized Bergman kernel of $\mathbb B^n$,
$$
K(z,w) = \frac{1}{(1 - z \cdot \overline{w})^{n+1}},
$$
and let $g$ be the normalized Bergman metric on $\mathbb B^n$ with its volume form $dV_g$.
For each $z,w \in \mathbb B^n$, we define
$$
\varphi_{z,w}(\tau)
= \frac{1}{n+1} \log \frac{K(w,\tau)}{K(z,\tau)}
= \log \frac{1 - z \cdot \tau}{1 - w \cdot \tau},
$$
and let $\nabla \varphi_{z,w}(\tau)$ denote its gradient with respect to $g$.
Our main result is the following.

In the one-dimensional case, if $\Sigma$ is a compact hyperbolic Riemann surface and
$$\psi = \psi(\tau)(d\tau)^{\otimes N} \in H^0(\Sigma, K_\Sigma^{\otimes N}),$$
Adachi showed that the corresponding function $\Phi(\psi)$ on $(\mathbb B^1 \times \mathbb B^1)/\Gamma$ is given by
$$
\Phi(\psi)(z,w)
=
\begin{cases}
\displaystyle
\frac{1}{B(N,N)}
\int_{\tau \in [z,w]}
\!\!
\left(
\frac{w - z}{(w - \tau)(\tau - z)}
\right)^{N-1}
\psi(\tau)^N\, d\tau, & N \ge 1,\\[1.2em]
\text{the constant }\psi, & N=0,
\end{cases}
$$
where $B(N,N)$ denotes the Beta function.
For $n \ge 2$, however, the construction of $\Phi$ in~\cite{LS1}, which was inspired by Adachi's recursive approach in the one-dimensional case, relied on solving a recursive system of $\bar\partial$-equations
using the raising operator associated to the K\"ahler form of the Bergman metric, and did not yield a closed formula.
While the existence and density of $\Phi$ were established, the dependence of $\Phi(\psi)$ on the symmetric differential $\psi$ remained implicit.

The aim of the present paper is to give an explicit formula for $\Phi(\psi)$ in all dimensions $n \ge 1$.
Let $K(z,w)$ denote the normalized Bergman kernel of $\mathbb B^n$,
$$
K(z,w) = \frac{1}{(1 - z \cdot \overline{w})^{n+1}},
$$
and let $g$ be the normalized Bergman metric on $\mathbb B^n$ with its volume form $dV_g$.
For each $z,w \in \mathbb B^n$, we define
$$
\varphi_{z,w}(\tau)
= \frac{1}{n+1} \log \frac{K(w,\tau)}{K(z,\tau)}
= \log \frac{1 - z \cdot \tau}{1 - w \cdot \tau},
$$
and let $\nabla \varphi_{z,w}(\tau)$ denote its gradient with respect to $g$.
Our main result is the following.

The integrand in Theorem~1.3 is invariant under the diagonal action of $\mathrm{Aut}(\mathbb B^n)$,
and hence $\Phi(\psi)$ descends to a well-defined holomorphic function on $\Omega$.
From this point of view, the gradient $\nabla \varphi_{z,w}$ plays the role of a vector-valued cross ratio on the unit ball:
it is built from the Bergman kernel and transforms equivariantly under automorphisms,
compensating for the absence of a scalar cross-ratio invariant when $n \ge 2$.

\section{Infinitesimal expression of $\Phi(\psi)$}
\subsection{Formal adjoint operator of $\bar\partial$}
Let $(M,g)$ be an $n$-dimensional hermitian manifold with a volume form $dV_g$ induced by its hermitian metric $g$.
For a local coordinate $(z_1,\ldots, z_n)$ of $M$, the metric is expressed by $\sum g_{i\bar j} dz_i\otimes d\overline z_j$ and its fundamental form $\omega$ is given by $\omega = \sqrt{-1} \sum g_{i\bar j} dz_i\wedge d\bar z_j$. The volume form $dV_g = \frac{\omega^n}{n!}$ has a local expression
$$
dV_g = \sqrt{-1}^n\det (g_{i\bar j}) dz_1\wedge d \bar z_1\wedge\cdots\wedge dz_n\wedge d\bar z_n.
$$
Let $E$ be a hermitian vector bundle over $M$. Define the Hodge $\star_{E}$-operator by
$$
s \wedge \star_{E} t = \left\langle s, t \right\rangle dV_\omega , \quad s, t \in C^{\infty}(\Lambda^{(p,q)} T^*_M \otimes E)
$$
where the wedge product $\wedge$ is combined with the canonical pairing $E \times E^* \rightarrow \mathbb{C}$ and $\left\langle \; , \; \right\rangle$ denotes the inner product induced by the metrics on $E$ and $M$.
Then the formal adjoint operator $\bar \partial^*_{E}$ of $\bar \partial$ is given by
\begin{equation}\label{adjoint}
\bar \partial^*_{E} = - \star_{E^*} \circ \bar \partial \circ \star_{E}.
\end{equation}
See \cite{D} for a general discussion.
For simplicity, we will omit the lower subscript $E$ of $\bar \partial^*_{E}$ if there is no danger of confusion.

\medskip
\begin{proof}[Proof of Theorem~\ref{explicit form}]
For $\gamma \in \mathrm{Aut}(\mathbb{B}^n)$, let $\Gamma' = \gamma^{-1}\Gamma\gamma$ denote the conjugate subgroup of $\Gamma$. Then, for any $\psi \in H^0(\Sigma, S^N T^*_{\Sigma})$, the pullback $\gamma^*\psi$ defines a symmetric differential on $\Sigma' = \mathbb{B}^n / \Gamma'$. Let $\Theta_{\Gamma'}$ and $I_{\Gamma'}$ denote the maps corresponding to $\Theta$ and $I$, respectively, associated with the subgroup $\Gamma'$.
Consider the recursive $\bar\partial$-equation \eqref{d-bar} with base step $\varphi_N = \gamma^*\psi$. Since the Bergman metric is invariant under $\mathrm{Aut}(\mathbb{B}^n)$, we have
$$
\bar\partial(\gamma^*\varphi_{N+m})
= \gamma^*(\bar\partial\varphi_{N+m})
= -(N+m-1)\gamma^*(\mathcal{R}_G \varphi_{N+m-1})
= -(N+m-1)\mathcal{R}_G(\gamma^*\varphi_{N+m-1}),
$$
and hence $\{\gamma^*\varphi_{N+m}\}_{m=0}^{\infty}$ is the solution of the recursive system. This implies that
$$
I_{\Gamma'}(\gamma^*\psi)(z,w)
= I_{\Gamma}(\psi)(\gamma(z), \gamma(w)).
$$
On the other hand,
\begin{equation}\nonumber
\begin{aligned}
\Theta_{\Gamma'}(\gamma^*\psi)(z,w)
&= \int_{\mathbb{B}^n} \left\langle (\varphi_{z,w})^N,\, \gamma^*\psi \right\rangle \, dV_g
= \int_{\mathbb{B}^n} \left\langle \big( d\gamma(\varphi_{z,w}) \big)^N,\, \psi \right\rangle \, dV_g \\
&= \int_{\mathbb{B}^n} \left\langle \big( \varphi_{\gamma(z), \gamma(w)} \big)^N,\, \psi \right\rangle \, dV_g
= \Theta_{\Gamma}(\psi)(\gamma(z), \gamma(w)).
\end{aligned}
\end{equation}
As a result, it suffices to prove the theorem in the case $z = 0$.

\begin{theorem}\label{integral formula of Bergman Kernel}
Let $B_m$ denote the Bergman kernel of $H^0(\Sigma, S^m T^*_\Sigma)$,
and let $\{\psi_{m,j}\}_{j=1}^{d_m}$ be an orthonormal basis of this space.
For $\alpha>-1$, the weighted Bergman kernel $K_{\alpha}$ of $A^2_{\alpha}(\Omega)$ is given by
\begin{equation}\nonumber
\begin{aligned}
&K_{\alpha}\big((z,w),(z',w')\big) \\
&= S_n + \sum_{m=n+1}^{\infty} \frac{(c_{n,m})^2}{d_{\alpha,m}} \int_{\mathbb{B}^{n}} \bigg\langle {( \nabla \varphi_{z,w} (\tau) )}^m , \overline{\bigg( \int_{\mathbb{B}^n} \left\langle (\nabla \varphi_{z',w'} (\tau') )^m, \overline{B_m (\tau, \tau')} \right \rangle dV_g (\tau') \bigg) } \bigg \rangle dV_g(\tau)
\end{aligned}
\end{equation}
where
$$
S_n := \frac{n!}{\pi^n 2^{2n} (\mathrm{Vol} (\Sigma) )} + \sum_{m=1}^{n} \sum_{j=1}^{d_m} \frac{ \Phi(\psi_{m,j})(z,w) \overline{\Phi(\psi_{m,j})(z',w')} }{ d_{\alpha,m} }
$$
and
$$
d_{\alpha,m}:= \frac{ \pi^n 2^{2n}}{n!} \frac{ \Gamma(n+\alpha+1) \Gamma(m+1) }{ \Gamma (m+n + \alpha +1) } \sum_{\ell=0}^{\infty} \frac{(m+1)_{\ell}}{(n+m+\alpha+1)_{\ell}} \frac{(m)_{\ell} (m)_{\ell}}{(2m)_{\ell}} \frac{1}{\ell !} \prod_{j=1}^{\ell} \bigg(1+ \frac{n-1}{m+j} \bigg)
$$
where $(m)_{\ell}:= m (m+1)\cdots(m+\ell-1)$.

On the other hand, by the proof of \cite[Lemma 4.14, Corollary 4.15, and Lemma 4.17]{LS1},
$$
\bigcup_{m=0}^{\infty} \bigg \{ \frac { \Phi(\psi_{m,j}) } { \| \Phi(\psi_{m,j} ) \|_{\alpha} } \bigg\}_{j=1}^{d_m}
$$
is a complete orthonormal basis on $A^2_{\alpha}(\Omega)$ when $\alpha>-1$. Therefore, by Theorem \ref{explicit form} and \eqref{intergral of bergman kernel}, $K_{\alpha}((z,w),(z',w'))$ is equal to
\begin{equation}\nonumber
\begin{aligned}
S_n + \sum_{m=n+1}^{\infty} \frac{(c_{n,m})^2}{d_{\alpha,m}} \int_{\mathbb{B}^{n}} \bigg\langle {( \nabla \varphi_{z,w} (\tau) )}^m , \overline{ \bigg( \int_{\mathbb{B}^n} \bigg\langle (\nabla \varphi_{z',w'} (\tau') )^m, \overline{B_m (\tau, \tau')} \bigg\rangle dV_g (\tau') \bigg) } \bigg \rangle dV_g(\tau)
\end{aligned}
\end{equation}
and the proof is completed.
\end{proof}

The integrand in Theorem~1.3 is invariant under the diagonal action of $\mathrm{Aut}(\mathbb  B^n)$,
and hence $\Phi(\psi)$ descends to a well-defined holomorphic function on $\Omega$.
From this point of view, the gradient $\nabla \varphi_{z,w}$ plays the role of a vector-valued cross ratio on the unit ball:
it is built from the Bergman kernel and transforms equivariantly under automorphisms,
compensating for the absence of a scalar cross-ratio invariant when $n \ge 2$.

The explicit formula for $\Phi$ has several consequences.
It provides integral representations for Poincar\'e series associated with symmetric differentials (Section~\ref{Poincare series}),
gives concrete descriptions of weighted Bergman kernels on $\Omega$ using the Bergman kernel of $S^NT^*_\Sigma$ (Section~\ref{Bergman kernels}),
and extends naturally to the case of totally geodesic isometric holomorphic embeddings into higher-dimensional balls (Section~\ref{isometric embeddings}).

\medskip

The paper is organized as follows.
In Section~2 we recall the construction of $\Phi$ from~\cite{A21, LS1} together with basic properties of the Bergman kernel and its gradient.
In Section~3 we compute the infinitesimal behavior of $\Phi(\psi)$ and obtain a formal jet expansion.
In Section~4 we identify this expansion with the integral expression in Theorem~1.3.
Section~5 discusses applications to Poincar\'e series and weighted Bergman kernels on $\Omega$,
and Section~6 extends the formula to totally geodesic isometric embeddings into higher-dimensional balls.

\medskip
{\bf Acknowledgement}
The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government(MSIT) (No. RS-2024-00339854). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2025-00561084).

Sections~2--4 outline the argument leading to Theorem~\ref{explicit form}: in Section~2 the authors "recall the construction of $\Phi$" (from~\cite{A21, LS1}) and the needed "basic properties of the Bergman kernel and its gradient"; in Section~3 they "compute the infinitesimal behavior of $\Phi(\psi)$" and obtain "a formal jet expansion"; in Section~4 they "identify this expansion with the integral expression in Theorem~1.3."

Sections~2--4 outline the argument leading to the main theorem: next the authors recall the construction of $\Phi$ (from~\cite{A21, LS1}) and the needed "basic properties of the Bergman kernel and its gradient"; then they "compute the infinitesimal behavior of $\Phi(\psi)$" and obtain "a formal jet expansion"; finally they "identify this expansion with the integral expression in Theorem~1.3."

\label{explicit form}
Let $\Sigma = \mathbb B^n/\Gamma$ be a compact complex hyperbolic space form and
$\Omega = (\mathbb B^n \times\mathbb
B^n)/\Gamma$ the associated ball bundle.
For each $N \ge n + 1$ and each $\psi \in H^0(\Sigma, S^N T^*_\Sigma)$, we have
$$
\Phi(\psi)(z,w)
= c_{n,N}
\int_{\mathbb B^n}
\big\langle
(\nabla \varphi_{z,w}(\tau))^{\otimes N}, \psi(\tau)
\big\rangle
\, dV_g(\tau),
$$
for all $(z,w) \in \mathbb B^n \times \mathbb B^n$, where
$$
c_{n,N}
= \frac{(-1)^N (2N - 1)(N - 2)!}{\pi^n (N - n - 1)!}.
$$,

Question:
Let \(\mathbb B^n=\{z\in\mathbb C^n:|z|<1\}\), let \(\Gamma\subset \mathrm{Aut}(\mathbb B^n)\) be a cocompact discrete subgroup, and set \(\Sigma=\mathbb B^n/\Gamma\) and \(\Omega=(\mathbb B^n\times \mathbb B^n)/\Gamma\) for the diagonal action. Let \(g\) be the normalized Bergman metric on \(\mathbb B^n\) with volume form \(dV_g\), and let
\[
K(z,w)=\frac{1}{(1-z\cdot \overline w)^{n+1}}
\]
be the normalized Bergman kernel. For \(z,w,\tau\in\mathbb B^n\), define
\[
\varphi_{z,w}(\tau)=\frac{1}{n+1}\log\frac{K(w,\tau)}{K(z,\tau)},
\]
and let \(\nabla\varphi_{z,w}(\tau)\) be its gradient with respect to \(g\). If \(N\ge n+1\) and \(\psi\in H^0(\Sigma,S^N T^*_{\Sigma})\), where \(\Phi\) is the natural map from symmetric differentials on \(\Sigma\) to holomorphic functions on \(\Omega\), which explicit formula holds for \(\Phi(\psi)(z,w)\) for all \((z,w)\in\mathbb B^n\times\mathbb B^n\)?

\(\Phi(\psi)(z,w)\) is given by
\[
\Phi(\psi)(z,w)=c_{n,N}\int_{\mathbb B^n}\big\langle (\nabla\varphi_{z,w}(\tau))^{\otimes N},\psi(\tau)\big\rangle\,dV_g(\tau)
\]
for every \((z,w)\in\mathbb B^n\times\mathbb B^n\), where \(\langle\cdot,\cdot\rangle\) is the natural pairing and
\[
c_{n,N}=\frac{(-1)^N(2N-1)(N-2)!}{\pi^n(N-n-1)!}.
\]

Choices:
B.
\(\Phi(\psi)(z,w)\) is given by
\[
\Phi(\psi)(z,w)=c_{n,N}\int_{\mathbb B^n}\big\langle (\nabla\varphi_{z,w}(\tau))^{\otimes N},\psi(\tau)\big\rangle\,dV_g(\tau)
\]
for every \((z,w)\in\mathbb B^n\times\mathbb B^n\) and every \(N\ge 1\), where
\[
c_{n,N}=\frac{(-1)^N(2N-1)(N-2)!}{\pi^n(N-n-1)!}.
\]
C.
For each \(N\ge n+1\) and \(\psi\in H^0(\Sigma,S^N T^*_{\Sigma})\), there exists a constant \(c_{n,N}\) such that
\[
\Phi(\psi)(z,w)=c_{n,N}\int_{\mathbb B^n}\big\langle (\nabla\varphi_{z,w}(\tau))^{\otimes N},\psi(\tau)\big\rangle\,dV_g(\tau)
\]
for all \((z,w)\in\mathbb B^n\times\mathbb B^n\).
D.
\(\Phi(\psi)(z,w)\) is given by
\[
\Phi(\psi)(z,w)=c_{n,N}\big\langle \Big(\int_{\mathbb B^n}\nabla\varphi_{z,w}(\tau)\,dV_g(\tau)\Big)^{\otimes N},\psi\big\rangle
\]
for every \((z,w)\in\mathbb B^n\times\mathbb B^n\), where \(\langle\cdot,\cdot\rangle\) is the natural pairing and
\[
c_{n,N}=\frac{(-1)^N(2N-1)(N-2)!}{\pi^n(N-n-1)!}.
\]
E.
\(\Phi(\psi)(z,w)\) is given by
\[
\Phi(\psi)(z,w)=c_{n,N}\int_{\mathbb B^n}\big\langle (\nabla\varphi_{z,w}(\tau))^{\otimes N},\psi(\tau)\big\rangle\,dV_g(\tau)
\]
for every \((z,w)\in\mathbb B^n\times\mathbb B^n\), where \(\langle\cdot,\cdot\rangle\) is the natural pairing and
\[
c_{n,N}=\frac{(-1)^N(2N+1)(N-1)!}{\pi^n(N-n)!}.
\]

The stem does not state the formula or the constant explicitly; it only sets up the notation and asks for the resulting representation formula. There is little direct answer leakage beyond signaling that an explicit integral formula exists.

This is essentially a direct recall of a theorem/proposition: the question asks for the exact explicit formula that holds under the stated hypotheses. It is close to a restatement rather than a nontrivial application.

Some reasoning is needed to distinguish the exact threshold on N, the correct placement of the tensor power inside the integral, and the normalization constant. However, success depends mainly on theorem recall/recognition rather than generating a conclusion from first principles.

The distractors are plausible and mathematically targeted: one weakens the statement by hiding the explicit constant, one alters the range of N, one introduces a structurally incorrect tensor/integral interchange, and one perturbs the normalization constant. These reflect realistic failure modes.

A mathematically careful but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is largely a direct restatement of a known formula rather than a genuinely generative reasoning task.


\begin{theorem}\label{thm:main1}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.
\end{theorem}

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} $T(X) \subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

\begin{abstract}
Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.
\end{abstract}

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

\begin{theorem}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ to a free abelian group of countably infinite rank.
\end{theorem}
\begin{proof}
Let $\mathbb{Z}[\mathcal{S}(X_n)] = \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n)} \mathbb{Z}$. Elements of $\mathbb{Z}[\mathcal{S}(X_n)]$ can be regarded as functions $\mathcal{S}(X_n) \to \mathbb{Z}$ with finite support. Define a homomorphism $\Phi' : T(X_n) \to \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n) } \mathbb{Z}$ by taking $(\Phi'(f))(\mathfrak{s}) = SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$. The function $\mathfrak{s} \mapsto SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.

The image $im(\Phi')$ of $\Phi'$ is a subgroup of the free abelian group $\mathbb{Z}[\mathcal{S}(X_n)]$ and hence is free abelian. Let $\Phi : T(X_n) \to im(\Phi')$ be the homomorphism obtained by factoring $\Phi'$ through its image. The theorem will follow if we can show that $im(\Phi')$ has infinite rank (since $\mathbb{Z}[\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\Phi')$ can be at most countably infinite). For $n=10$, Lemma \ref{lem:torelli} immediately implies that $im(\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \# \overline{\mathbb{CP}^2}$. Let $f \in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \# id_{\overline{\mathbb{CP}^2}} \in T(X_n)$. Let $\kappa$ be a spin$^c$-structure on $\overline{\mathbb{CP}^2}$ with $c(\kappa)^2 = -1$. Let $\mathfrak{s} \in \mathcal{S}(X_{n-1})$. Then $\mathfrak{s} \# \kappa \in \mathcal{S}(X_n)$. The blowup formula gives
\[
SW^c_{X_n , \mathfrak{s} \# \kappa , \mathbb{Z}}(f \# id_{\overline{\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \mathfrak{s} , \mathbb{Z}}(f).
\]
Hence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \mathfrak{s} , \mathbb{Z}} : T(X_n) \to \mathbb{Z}$ is non-zero and hence $\Phi'( T(X_n) )$ has infinite rank.
\end{proof}

\begin{theorem}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}
\begin{proof}
Define $\Phi' : M_+(X) \to \mathbb{Z}^\infty$ to be given by
\[
\Phi' = \bigoplus_{n=1}^{\infty} SW_{X , \mathcal{O}_{2n-1}} : M_+(X) \to \bigoplus_{n=1}^{\infty} \mathbb{Z}.
\]
To prove the result, it suffices to show that the image of $\Phi'$ is not finitely generated. Then we let $\Phi$ be the homomorphism obtained by replacing the codomain of $\Phi'$ with the image of $\Phi'$. The argument is almost identical to the proof of \cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.
\end{proof}

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

Our second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

Note that by Schreier's lemma \cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.

The first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \cite{bar} and Konno \cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ where $n \ge 3$ is odd and also for $X = n (S^2 \times S^2) \# nK3$, $n \ge 1$. Note that these are precisely the manifolds of the form $E(m) \# (S^2 \times S^2)$, $m \ge 2$. Theorem \ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \# (S^2 \times S^2) = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$.

\subsection{Outline of the proofs of the main results}
To each diffeomorphism $f \in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \times X$ by identifying the ends via $f$. If $\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:
\begin{itemize}
\item[(1)]{{\bf The constant chamber:} assume that $f \in T(X)$. Then $f$ acts trivially on $H^2(X ; \mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\mathcal{H}^+ \to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\mathcal{H}^+$ under the above trivialisation.}
\item[(2)]{{\bf The zero chamber:} assume that $c(\mathfrak{s})^2 \ge 0$ and $c(\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}
\end{itemize}

These two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \mathfrak{s}}(f)$ and $SW^0_{X , \mathfrak{s}}(f)$ depending only on $(X , \mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \mathfrak{s}} : T(X) \to \mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \in T(X)$, the invariants $SW^c_{X , \mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism
\[
\Phi : T(X) \to \bigoplus_{\mathfrak{s}} \mathbb{Z}
\]
where the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \ref{thm:main1} follows by showing that the image of $\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\{ \mathfrak{s}_n\}$ and diffeomorphisms $\{ t_n \}$ for which $SW^c_{X , \mathfrak{s}_n}( t_n ) \neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \cite{rub1}, \cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \# (S^2 \times S^2) \cong E(1) \# (S^2 \times S^2)$.

\noindent{\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.

To prove Theorem~\ref{thm:main1}, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of Theorem~\ref{thm:main1} then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.

To prove the main theorem, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.,

Question: For each integer $n\ge 10$, let $X_n=2\mathbb{CP}^2\# n\overline{\mathbb{CP}^2}$. Here the mapping class group $M(X_n)$ is the group of smooth isotopy classes of diffeomorphisms of $X_n$, and the Torelli group $T(X_n)\subseteq M(X_n)$ consists of those smooth isotopy classes represented by diffeomorphisms that are continuously isotopic to the identity. Which statement holds for $T(X_n)$?

For every $n\ge 10$, there exists a surjective homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$, where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank.

Choices:
B. For every $n\ge 10$, there exists a homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$ whose image has finite index in $\mathbb{Z}^\infty$.
C. For every $n\ge 10$, there exists a homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$ with infinite image.
D. There exists a single surjective homomorphism $\Phi$ such that for every $n\ge 10$, one has $\Phi:T(X_n)\to \mathbb{Z}^\infty$.
E. For every $n\ge 9$, there exists a surjective homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$, where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank.

surjectivity-vs-large-image via infinite-rank image inside direct sum

dropped surjectivity, retaining only existence of an infinite-image homomorphism

dependence of the homomorphism on the manifold parameter $n$

sharp lower bound $n\ge 10$ coming from the base case and induction

The stem defines the objects involved but does not reveal the conclusion. The correct statement is not encoded in the wording beyond standard setup.

This is close to theorem recognition: the correct option appears to restate a known result almost verbatim. However, the alternatives vary quantifiers, range bounds, and strength of conclusion, so it is not a pure tautology.

Some reasoning is needed to distinguish surjectivity from weaker infinite-image claims, dependence on n, and the sharp threshold n>=10. But the item mainly tests recall of the exact theorem statement rather than deeper derivation.

The distractors are mathematically plausible and target realistic failure modes: weakening surjectivity, confusing large image with surjective image, mishandling quantifier dependence, and shifting the lower bound from 10 to 9.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall than genuinely generative reasoning.