\label{vbint-}
Let $0<r\leq\infty$.
Let $M$ be a domain in ${\bf C}^n$ with $\partial M\in C^2$. Assume that   the Levi form of $\partial M$ on $T^{(1,0)}\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\zeta_0\in\partial M$.
Let $E$ be a complex vector bundle of class $Lip$ $($resp. $\Lambda^{r+1})$ over $\overline M$. Assume that a connection  $D\colon Lip(M,E)\to L_{(1)}^\infty(M,E)$  $($resp. $\Lambda^{r+1}(M,E)\to \Lambda^{r}_{(1)}(M,E))$ satisfies $\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\overline M$ and $\zeta_0\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\Lambda^{r+1})$ structure of $E|_U$.

\begin{thm}\label{vbint-}
Let $0<r\leq\infty$.
Let $M$ be a domain in $\cc^n$ with $\pd M\in C^2$. Assume that   the Levi form of $\pd M$ on $T^{(1,0)}\pd M$ has either $n-1$ positive or at least three negative eigenvalues at  $\zeta_0\in\pd M$.
Let $E$ be a complex vector bundle of class $Lip$ $($resp. $\Lambda^{r+1})$ over $\ov M$. Assume that a connection  $D\colon Lip(M,E)\to L_{(1)}^\infty(M,E)$  $($resp. $\Lambda^{r+1}(M,E)\to \Lambda^{r}_{(1)}(M,E))$ satisfies $\Om^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\ov M$ and $\zeta_0\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\Lambda^{r+1})$ structure of $E|_U$.
\end{thm}

\label{sec1}
 Let $M\subset{\bf C}^n$ be a relatively compact domain with boundary $\partial M\in C^2$. Let $E$ be a smooth (i.e. $C^\infty$) complex vector bundle over the closure $\overline M$. Denote by $C^\infty(\overline M, E), C_{(k)}^\infty(E,\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\overline M$, respectively.
A Koszul {\it connection} $D$ in $E$ of is a ${\bf C}$-linear map
from $
 C_{(k)}^\infty(\overline M,E)$ to $ C^\infty_{(k+1)}(\overline M,E)
$
satisfying, for any $f\in C_{(k)}^\infty(\overline M)$ and $s\in C^\infty(\overline M,E)$,
$$
D(fs)=df\otimes s +(-1)^kf\wedge Ds.
$$ The curvature $R$ of $D$ is defined as
$
R=D\circ D\colon C^\infty_{(k)}(\overline M,E)\to C^\infty_{(k+2)}(\overline M,E).
$
 Let $e=(e_1,\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then
$$
De_i=\sum \omega^j_i\otimes e_j, \ De=\omega e;\quad D^2e_i=\sum \Omega^j_i\otimes e_j, \ D^2e=\Omega e,
$$
where $\omega$ and $\Omega$ are the connection and curvature forms.
If $\hat e=Ae$ is another frame, then
\begin{equation}\label{hatomom}
\hat \omega =(dA)A^{-1} +A\omega A^{-1}, \quad \widehat\Omega=A\Omega A^{-1}.
\end{equation}
Decomposing into types, $\omega=\omega^{(1,0)}+\omega^{(0,1)}$ and
$\Omega=\Omega^{(2,0)}+\Omega^{(1,1)}+\Omega^{(0,2)}$, yields
\begin{equation}\label{om02}
 \Omega^{(0,2)}=\dbar\omega^{(0,1)}-\omega^{(0,1)}\wedge\omega^{(0,1)}.
\end{equation}

The {\it integrability problem} studied in this paper is to find a frame $\hat e=Ae$ for which $\hat\omega^{(0,1)}=0$, which is equivalent to solving
\begin{equation}\label{dbAA}
A^{-1}\dbar A+\omega^{(0,1)}=0.
\end{equation}
If such frames $\hat e_k$ exist on an open covering $\{U_k\}$ of $\overline M$, then on overlap $U_j\cap U_k$ their transition matrices satisfy $\dbar A_{kj}=0$. In this case we say that $(E,\overline M,D)$ carries a holomorphic vector bundle structure. By (\ref{hatomom})-(\ref{om02}), the solvability of (\ref{dbAA}) requires the {\it formal} integrability condition
\begin{equation}\label{Om02=0}
\dbar\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)},\quad \text{i.e.}\quad \Omega^{(0,2)}=0.
\end{equation}
Koszul and Malgrange~~\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.

Our main result establishes integrability and optimal boundary regularity for the H\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.

\begin{thm}[convex configuration estimate]\label{conv-est} Let $r>0$ and $1\leq q\leq n-1$.
Let $(D^1,D_{r_2}^2)$ be an $(n-q)$-convex configuration. Let $D^{12}_r=D^1\cap D^2_r$.
The homotopy operators $H_q, H_{q+1}$ in \rta{hf-c} satisfies
\aln{}\|H_s\var\|_{\Lambda^{r+1/2}(D^{1 2}_{(1-\theta)r_2})}&\leq \f{C_r( \nabla\rho^1,\nabla^2\rho^1)}{\theta^{3r+2r_0}}\|\var\|_{\Lambda^r(D^{1 2}_{r_2})},\quad \forall\theta\in(0,1).
\end{align*}
Moreover, $C_r(\nabla\rho^1,\nabla^2\rho^1)$ is stable under small $C^2$ perturbations of $\rho^1$.
\end{thm}
\begin{proof}Recall from \re{hq1}-\re{nhq2-} that $H_s =H^{(1)}_s + H_s^{(2)}$ with
\aln{}
 H^{(1)}_s f&:=R_{ D^{2}; s-1}^0 \cL E_{D^{12}} f+R_{U^1;s-1 }^{01}[\db,\cL E_{D^{12}}] f,\\
H^{(2)}_sf&=L_{1^+;{s-1}
}^{01} \cL E_{D^{12}} f +L_{2;s-1}^{02} f+L_{12;s-1}^{012}f.
\end{align*}
Take a cut-off function $\chi\in C^\infty_0(D^{2}_{(1-\theta/3)r_2})$ with $\chi=1$ on $D^{2}_{(1-\theta/2)r_2}$, and decompose
$
 H^{(1)}_s f= H^{(1)}_s(\chi f)+ H^{(1)}_s( (1-\chi)f).
$
Then the estimate in~\cite{gong-shi-nn} says that
$$
\|H_s^{(1)}(\chi \var)\|_{\Lambda^{r+1/2}(D^{1 2}_{r_2})}\leq C_r \|\chi\var\|_{\Lambda^r(D^{1 2}_{r_2})}.
$$
We can choose $\chi$ such that $\|\chi\|_{C^r}\leq {C_r}{\theta^{-r-1}}$. Thus,
$$
\|\chi\var\|_{\Lambda^r(D^{12}_{r_2})}\leq {C_r }{\theta^{-r-r_0}}\|\var\|_{\Lambda^r(D^{12}_{r_2})}.
$$
Since $(1-\chi)\var$ vanishes on $D_{(1-\theta/2)r}^{12}$ and
\ga{}\label{giest}
|g^i(z,\zeta)\cdot(\zeta-z)|\geq |\zeta-z|^2/C,\quad z\in D^{1}\cap D^2,\zeta\in D^2\setminus D^1, i=0,1,2,
\end{gather}
then we can obtain $$
\|H_s^{(1)}((1-\chi) \var)\|_{\Lambda^{r+1/2}(D^{12}_{(1-\theta)r_2})}\leq C_r\theta^{-2r-r_0} \|(1-\chi)\var\|_{\Lambda^r(D^{12}_{r_2})}.
$$
The $H_s^{(2)}$ is given by boundary integrals on $S^1_+, S^2$ and $ S^{12}$ respectively. The estimate \re{giest} also yields
$\|H_s^{(2)}\var\|_{\Lambda^{r+1/2}(D^{12}_{(1-\theta)r_2})}\leq C_r \theta^{-2r-r_0} \|\var\|_{\Lambda^r(D^{12}_{r_2})}.
$
We have obtained the desired estimate of $H_s$.
\end{proof}
We now derive estimates for the concave case. We still have $H_s=H_s^{(1)}+H_s^{(2)}$, where $H_s^{(1)}$ has the same form as in the convex case. However,
$$
H_{s}^{(2)}f=\sum_{i=1}^2L_{i3,s}^{0i3}f- L^{01}_{1^+,s-1}\cL Ef +L_{12;s-1}^{123}f-\hat T_{ D^{23}_{(1-\theta)r}, s}L^{23}_{12;s} f.
$$
By \re{g3theta}, we have $
\RE\{g^3(z,\zeta)\cdot(\zeta-z)\} \geq\theta r_3^2$ for $z\in D^{23}_{(1-\theta)r}$ and $\zeta\in S^{13}\cup S^{23}\cup S^{12}\subset\pd D^{123}_r$. Some kernels in $H^{(2)}_s$ involve first-order $z$-derivatives of $\nabla\rho^1(z)$.
As in the convex case, we can obtain
$$\|H^{(2)}_sf\|_{\Lambda^{r+1/2}(D^{123}_{(1-\theta)r})}\leq {C_r(\nabla\rho^1 )}{\theta^{-r-r_0}}\|\rho^1\|_{\Lambda^{3/2}}\|f\|_{\Lambda^\e}.
$$The kernels of $H^{(1)}f$ involve the second-order $z$-derivatives $ \nabla\rho^2(z)$.
Then by ~\cite{gong-shi-nn}*{Thm. 5.12}, we have
$$\|H^{(1)}_sf
\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}\leq {C_r }{\theta^{-r-r_0}}\left(\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\e}+ \|f\|_{\Lambda^r}\right).
$$Therefore, we have obtained part $(a)$ below.
\begin{thm}[concave configuration estimate]\label{concave-est} Let $\e>0,r>0$ and $1\leq q\leq n-3$.
Assume that $\rho^1\in C^{2+\e}$. The following hold.
\bpp
\item The operators $H_q, H_{q+1}$ in Theorem~$\ref{cchf}$ satisfy
\al{}
\label{hqfr12}
\|H_sf\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}&\leq \f{C_{r,\e}}{\theta^{3r+r_0}} (\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\epsilon({D_{r}^{123}})}+ \|f\|_{\Lambda^r(D^{123}_{r})})
\end{align}
for $0\leq\delta\leq1/2$ and $0<\theta<1$. Furthermore, $C_{r,\e}=C_{r,\e}(\nabla\rho^1,\nabla^2\rho^1)$ is stable under small $C^2$ perturbations of $\rho^1$;
 the same estimate holds for the $\db$ solution operator $H_q$ in \rta{cchf-closed}.
 \item
The homotopy formula \rea{tsqf+-cv} in Theorem~$\ref{cchf}$ holds for $f\in \Lambda^{\e}_{(0,q)}(D^{123})$ when $\db f\in \Lambda^{\e}_{(0,q)}(D^{123})$, and the $\db$-solution operator $H_q$ in \rta{cchf-closed} is valid for all $\db$-closed $f\in\Lambda^{\e}_{(0,q)}(D^{123})$ and satisfies \rea{hqfr12}.
\epp
\end{thm}
\begin{proof}We verify $(b)$ using $(a)$. We know that $f=\db H_qf+H_{q+1}\db f$ when $f\in C^{1+\e}$. Let us write $H_q$ as $H_{D^{123}_r}$ to indicate the dependent on the domain. Now assume $f\in \Lambda^r_{(0,q)}(D^{123})$ with $\db f\in\Lambda^r$. We find a sequence $f_j\in C^{\infty}_{(0,q)}(D^{123})$ such that $f_j $ and $\db f_j $ converge to $f,\db f$ in $\Lambda^{r'}(D^{123})$ as $j\to\infty$ for any $r'<r$. Furthermore, $$\|f_j\|_{\Lambda^{r}(D^{123})}\leq C_r\|f\|_{\Lambda^{r}(D^{123})}, \quad \|\db f_j\|_{\Lambda^{r}(D^{123})}\leq C_r\|\db f\|_{\Lambda^{r}(D^{123})}.$$
Then $H_qf=\lim H_qf_j$ and $H_{q+1}\db f=\lim H_{q+1}\db f_j$ gives us \rea{tsqf+-cv}.
Assume now $f$ is $\db$-closed. Set $L_t(z)=z+tie_{q+2}$ with $t>0$, where $e_{q+2}$ is the $(q+2)$-th unit vector of $\cc^n$. Fix $\theta\in(0,1/2)$. When $0<t<t_\theta$, we have $L_t(D^{123}_{(1-\theta)r})\subset D^{123}_r$. Using a mollifier, we find $\db$-closed $f_j\in C^\infty(D^{123}_{(1-\theta)r})$ such that $f_j$ converges to $f$ in $\Lambda^{r'}(D^{123}_{(1-\theta)r})$-norm for any $r'<r$, whereas $\|f_j\|_{\Lambda^{r}(D^{123}_{(1-\theta)r})}\leq C_r\|f\|_{\Lambda^{r}(D^{123}_{r})}$. We have $f_j=\db H_{D^{123}_{(1-\theta')r},q}f_j$ on $D^{123}_{(1-\theta)r}$ for $0<\theta'<\theta/2$.
Letting $j\to\infty$, we obtain $f=\db H_{D^{123}_{(1-\theta')r},q}f$ on $D^{123}_{(1-\theta)r}$ and
\eq{Hth''}
\|H_{D^{123}_{(1-\theta')r},q}f\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}\leq \f{C_{r,\e}}{\theta^{3r+r_0}} (\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\epsilon }+ \|f\|_{\Lambda^r }).
\eeq
By the Arzel\`{a}-Ascoli theorem, $H_{D^{123}_{(1-\theta'_j)r},q}f$ converges to $H_{D^{123}_{r},q}f$ in $\Lambda^{r'+\delta}(D^{123}_{(1-\theta)r})$ norm for a sequence $\theta'_j\to0$ for any $r'<r$. Then \re{Hth''} yield
\re{hqfr12} for $H_q$.
\end{proof}

\setcounter{thm}{0}\setcounter{equation}{0}
\section{Integrability of complex vector bundles}\label{sec:NM}
In Section 2, we have seen that \rt{vbint-} follows from the following result.
\begin{thm}\label{vbint}
Let $M$ be a domain in $\mathbb C^n$ with $\pd M\in C^2$. Assume that the Levi-form of $M$ has
either $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\zeta_0\in\pd M$. Let $U$ be an open set in $\ov M$ and $\zeta_0\in U$. Assume that $\om$ is a $k_0\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\db\om=\om\wedge\om$ holds on $U\setminus\pd M$ in the sense of distributions. Assume that $\om$ is in $L^\infty(U)$ $($resp. $\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$ of class $\Lambda^{1/2}$ $($resp. $\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\zeta_0$ in $\ov M$ such that
$
\db A+A\om=0
$
in the sense of distributions on $U'\setminus\pd M$.
\end{thm}
\begin{proof}The proof is based on a KAM-type iteration method. Such an approach was developed by Webster~\ci{MR999729} for an interior version of Newlander-Nirenberg theorem. It was also used in~\cites{MR2742034,MR2829316} to establish the integrability of CR vector bundles on strictly pseudoconvex hypersurfaces in $\cc^n$ for $n\geq4$.

Our theorem concerns the integrability of $(E,D)$ in the H\"older-Zygmund spaces $\Lambda^r$, which is the standard H\"older spaces $C^r$ when $r$ is not an integer; see Section 5 for definition. The theorem also treats  a Koszul connection $D$ that maps the space $ Lip(M,E)$ of Lipschitz sections
 into  $ L^\infty_{(1)}( M,E)$, the space  of $E$-valued $1$-forms of class $L^\infty$ on $M$. Such a connection extends trivially to a Koszul connection on $E$ over $\overline M$.   The theory of $L^\infty$ connections, which is of independent interest, is required  even when the original connections are $C^\infty$ for the (strictly) $3$-concave case. Theorem~\ref{vbint-}  is reduced to Theorem~\ref{vbint}, and the reduction requires a careful discussion in Section~\ref{sec:int-formal} of   $L^\infty$ connections on Lipschitz bundles  and of  {\it admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; see Proposition~\ref{d2=0}.

\medskip

We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear.
 We   apply homotopy formulas to the study of (\ref{dbAA}).
The use of integral representations to establish regularity of $\overline\partial$-solutions on strongly pseudoconvex domains in ${\bf C}^n$ has a long history. Sup-norm estimate for $\dbar$-solutions to $(0,1)$-forms was proved by  Grauert and Lieb~\cite{MR273057} and  Henkin~\cite{MR0249660}.   Kerzman~\cite{MR0281944} obtained  $L^p$ and $C^{\beta}$ estimates of $\dbar$-solutions for $(0,1)$-forms and all $\beta<1/2$.
  Lieb~~\cite{MR283235} obtained the $L^\infty$ and the $C^\beta$ estimates of $\dbar$-solutions for $(0,q)$-forms.
Henkin and Romanov~\cite{MR0293121}  proved the sharp $C^{1/2}$ estimate of
$\dbar$-solutions for continuous $(0,1)$-forms.

We recall several important developments on integral representation formulas for  $\overline\partial$   that lead to derivative estimates on strongly pseudoconvex domains $D\subset{\bf C}^n$. When $\partial D$ is of class $ C^{k+2}$,   Siu~\cite{MR330515} showed that the Henkin solution operator satisfies a $C^{k+1/2}$ estimate  for  $\dbar$-closed $(0,1)$ forms of class $C^k$, and for  $\dbar$-closed $(0,q)$-forms   with $q\geq1$,  Lieb and Range~\cite{MR597825} constructed a new  $\dbar$ solution operator   and established  the same estimate. When $\partial D$ is only $ C^2$,  a homotopy formula was constructed  in~\cite{MR3961327} using the Stein extension, producing   homotopy operators with $\Lambda^{r+1/2}$ estimate for $r>1$.    Shi and Yao~\cites{MR4688544,MR4861589} later developed   a homotopy formula   employing the Rychkov extension and obtained $\Lambda^{r+1/2}$ estimates for all $r>0$, as well as $H^{s+1/2,p}$ estimates  for $1<p<\infty$ and $s>1/p$ when $\partial D\in C^2$; they also showed that the  estimates hold  for all $s\in{\bf R}$ when $\partial D$ is sufficiently smooth.  Very recently, Yao~~\cite{yaoc2} further reduced the boundary smoothness assumption to $ C^2$ for all $s\in{\bf R}$. We also refer the reader to references cited in~\cites{MR3961327,MR4688544,MR4861589,yaoc2}.

We also note that   Range and Siu~~\cite{MR338450} proved the $C^\beta$ estimate for all $\beta<1/2$ for $\dbar$-solutions for continuous $(0,q)$-forms on the (real) transversal intersection of
  strictly pseudoconvex domains. Whether the optimal gain of $1/2$  derivative   estimates  holds for $\dbar$-solutions on such transversal intersections remains an open problem.
  Higher-order derivative estimates in this setting were obtained by Brinkmann~~\cite{Br84}, Michel~~\cite{MR928297}, and Michel-Perotti~~\cite{MR1038709}.
 Peters~~\cite{MR1135535}  constructed a homotopy formula for the weakly transversal intersection of strictly pseudoconvex domains and established higher-order estimates, though with a loss of derivatives.
 It would be interesting to determine whether the results mentioned above can be applied to the integrability problem for intersections of strictly pseudoconvex domains. We also remark that the integrability of almost CR vector bundles on strongly pseudoconvex hypersurfaces in $\mathbb C^n$ with $n\geq4$ was proved by Webster~~\cite{MR1128608}, and sharper regularity results in the finitely smooth case were obtained by Gong-Webster~\cites{MR2742034,MR2829316}.
It remains an open question whether Theorem~\ref{vbint-} holds if $\partial M$ has two negative Levi eigenvalues. This appears to be related to the unsettled embedding problem for strongly pseudoconvex local CR structures in ${\bf R}^5$, investigated by Webster~\cite{MR0995504} and Gong-Webster~~\cite{MR2868966}.

We organize the paper as follows. In Section 2, we examine the formal integrability condition in detail for connections on Lipschitz complex vector bundles and introduce the notion of admissible frame changes. Sections 3 and 4 reformulate the local homotopy formulas in~\cites{MR986248,MR4866351}
 for shrinking domains. In Section 5, we recall  estimates established in \cites{MR986248,gong-shi-nn} and derive  new estimates on shrinking domains. Finally, in Section 6, we complete the proof of Theorem~\ref{vbint-}  by establishing Theorem~\ref{vbint}.

\setcounter{thm}{0}\setcounter{equation}{0}

Theorem~\ref{vbint-} is reduced to Theorem~\ref{vbint}, and the reduction requires a careful discussion in Section~\ref{sec:int-formal} of $L^\infty$ connections on Lipschitz bundles and of \textit{admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; see Proposition~\ref{d2=0}.

“We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\ref{dbAA}).”

The proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and “Finally, in Section 6, we complete the proof of Theorem~\ref{vbint-} by establishing Theorem~\ref{vbint}.”

To prove the main theorem, we reduce it to the following theorem.

\begin{thm}\label{vbint}
Let $M$ be a domain in $\mathbb C^n$ with $\pd M\in C^2$. Assume that    the Levi-form of $M$ has
either $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\zeta_0\in\pd M$. Let $U$ be an open set in $\ov M$ and $\zeta_0\in U$.    Assume that $\om$ is a $k_0\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\db\om=\om\wedge\om$ holds on $U\setminus\pd M$ in the sense of distributions. Assume that $\om$ is in $L^\infty(U)$ $($resp. $\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\Lambda^{1/2}$ $($resp. $\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\zeta_0$ in $\ov M$ such that
$
\db A+A\om=0
$
in the sense of distributions on $U'\setminus\pd M$.
\end{thm}

The reduction requires a careful discussion of $L^\infty$ connections on Lipschitz bundles and of \textit{admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; for this we use the following proposition.

\begin{prop}\label{d2=0}
Let $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\om \otimes e$ and $\om\in L^2_{loc}(U)$ and $d\om\in L^1_{loc}(U)$. If $A$ is admissible, then $\hat\om=(dA)A^{-1}+A\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\Om A^{-1}=\hat\Om$.
\end{prop}

“We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\ref{dbAA}).”

The proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and finally we complete the proof of the main theorem by establishing the theorem above.

Question: Let $0<r\le \infty$. Let $M\subset \mathbb C^n$ be a domain with $\partial M\in C^2$, and let $\zeta_0\in \partial M$. Assume that the Levi form of $\partial M$ on $T^{(1,0)}\partial M$ has either $n-1$ positive eigenvalues or at least three negative eigenvalues at $\zeta_0$. Let $E$ be a complex vector bundle over $\overline M$ of class $Lip$ (respectively of Hölder-Zygmund class $\Lambda^{r+1}$). Suppose $D$ is a connection on $E$ of the corresponding regularity, namely $D\colon Lip(M,E)\to L^{\infty}_{(1)}(M,E)$ (respectively $D\colon \Lambda^{r+1}(M,E)\to \Lambda^r_{(1)}(M,E)$), and that its curvature has vanishing $(0,2)$-part on $M$ in the sense of distributions. In a local frame $e$, if $De=\omega e$ and $D^2e=\Omega e$, this means $\Omega^{(0,2)}=0$, equivalently $\bar\partial\omega^{(0,1)}=\omega^{(0,1)}\wedge \omega^{(0,1)}$. A holomorphic vector bundle structure on $E|_U$ means local frames on an open cover of $U$ whose transition matrices are holomorphic (equivalently, local frame changes can be made so that the $(0,1)$-part of the connection vanishes). Under these assumptions, which existence statement holds?

There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1/2}$.

Choices:
B. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1}$.
C. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure.
D. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure, provided the identity $\bar\partial\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)}$ holds on all of $U$, including across $\partial M$, in the sense of distributions.
E. For every neighborhood $U$ of $\zeta_0$ in $\overline M$, the restriction $E|_U$ admits a holomorphic vector bundle structure. Moreover, one can choose a local frame change $A$ of the same regularity as the original bundle structure, namely $A\in Lip(U)$ in the $Lip$ case and $A\in \Lambda^{r+1}(U)$ in the $\Lambda^{r+1}$ case, such that $\bar\partial A+A\omega^{(0,1)}=0$ on $U\setminus\partial M$.

equation required only on interior/distribution on $U\setminus\partial M$

local existence on some neighborhood and reduced regularity of frame change


\begin{theorem}\label{Thm for E}
    If $E$ is uniformly weakly RC-positive over a compact K\"ahler manifold $X$, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.
\end{theorem}

\begin{lemma}\label{lemma abc}
  Assume the curvature $\Theta$ of $h$ is positive on every fiber $\mathcal{X}_t$. For $1\leq k\leq m$, the following are equivalent.
\begin{enumerate}[label=\Alph*.]
    \item\label{A} The curvature $\Theta$ has at least $n+k$ positive eigenvalues at every point in $\mathcal{X}$.
    \item\label{B} The horizontal component $\Theta_\mathcal{H}$ has at least $k$ positive eigenvalues at every point in $\mathcal{X}$.
    \item\label{C}  There exists a positive $\beta\in C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))$ such that the sum of any $m-k+1$ eigenvalues of $\Theta_\mathcal{H}$ with respect to $\beta$ is positive.
\end{enumerate}
Moreover, when $k=1$, statement C can be rephrased as $\Theta_\mathcal{H}\wedge \beta^{m-1}>0$ or equivalently $\Theta^{n+1}\wedge \beta^{m-1}>0$.
\end{lemma}

In \cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness.  RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \cite{YangForum} which also can be used to prove the same conjecture of  Yau. For the semipositive case of Yau's conjecture, see \cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.

Let us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\Theta^H$, which is an $\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\in E_t$ and a tangent vector $v\in T_tX$ with $t\in X$, we define the expression $$H(\Theta^H u,u)(v,\bar{v})$$ to be  $\sum_{j,k} H(\Theta^H_{j\bar{k}}u,u)v_j \bar{v}_k$ locally where we write the curvature $\Theta^H=\sum_{j,k}\Theta^H_{j\bar{k}}dt_j\wedge d\bar{t}_k$ and  $v=\sum_j v_j\partial/\partial t_j$.

\begin{definition}

A Hermitian metric $H$ on a holomorphic vector bundle $E\to X$ is called RC-positive if for any $t\in X$ and any nonzero $u\in E_t$, there is a nonzero tangent vector $v\in T_tX$ such that $H(\Theta^Hu,u)(v,\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\in X$, there is a nonzero tangent vector $v\in T_tX$ such that for any nonzero $u\in E_t$, we have $H(\Theta^H u,u)(v,\bar v)>0$.

A holomorphic vector bundle $E\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.
\end{definition}
It is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$  (\cite[Definition 3.3]{YangCamb}).

Similarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\Theta$ of the induced metric $h$ satisfies
\begin{enumerate}[label=\alph*.]
    \item\label{a} $\Theta$ is positive on every fiber.
    \item $\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.
    \item For any point $t\in X$, there exists a nonzero tangent vector $v\in T_tX$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,[\zeta])}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,[\zeta])}P(E^*)$.
\end{enumerate}
Following Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\zeta])}P(E^*)$ for any point $[\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.
\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.
\end{definition}
In Yang's solution to Yau's conjecture, two main theorems \cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity,  hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\cite[Question 7.11]{YangCamb} and \cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.

In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.

In \cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \cite{YangForum} which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau's conjecture, see \cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.

Let us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\Theta^H$, which is an $\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\in E_t$ and a tangent vector $v\in T_tX$ with $t\in X$, we define the expression $$H(\Theta^H u,u)(v,\bar{v})$$ to be $\sum_{j,k} H(\Theta^H_{j\bar{k}}u,u)v_j \bar{v}_k$ locally where we write the curvature $\Theta^H=\sum_{j,k}\Theta^H_{j\bar{k}}dt_j\wedge d\bar{t}_k$ and $v=\sum_j v_j\partial/\partial t_j$.

A Hermitian metric $H$ on a holomorphic vector bundle $E\to X$ is called RC-positive if for any $t\in X$ and any nonzero $u\in E_t$, there is a nonzero tangent vector $v\in T_tX$ such that $H(\Theta^Hu,u)(v,\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\in X$, there is a nonzero tangent vector $v\in T_tX$ such that for any nonzero $u\in E_t$, we have $H(\Theta^H u,u)(v,\bar v)>0$.

A holomorphic vector bundle $E\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.
\end{definition}
It is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$ (\cite[Definition 3.3]{YangCamb}).

Similarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\Theta$ of the induced metric $h$ satisfies
\begin{enumerate}[label=\alph*.]
 \item\label{a} $\Theta$ is positive on every fiber.
 \item $\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.
 \item For any point $t\in X$, there exists a nonzero tangent vector $v\in T_tX$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,[\zeta])}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,[\zeta])}P(E^*)$.
\end{enumerate}
Following Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\zeta])}P(E^*)$ for any point $[\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.
\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.
\end{definition}
In Yang's solution to Yau's conjecture, two main theorems \cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\cite[Question 7.11]{YangCamb} and \cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.

In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.

In this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$.

In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.

Another motivation for establishing Theorem \ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \cite[Theorem 1.4]{YangCamb} and \cite[Theorem 1.3]{YangForum}, for a compact K\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.
\begin{enumerate}
 \item The holomorphic tangent bundle $TX$ is uniformly RC-positive.
 \item The exterior power $\wedge^p TX$ is RC-positive for $1\leq p\leq n$.
 \end{enumerate}
 One can ask if the converse is true (\cite[Problem 4.15]{YangForum}). A partial converse is proved in \cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\wedge^p TX}(-1)$ is RC-positive for $1\leq p \leq n$. So, Theorem \ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.

For the proof of Theorem \ref{Thm for E}, instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \ref{Thm for E}
as a special case. We consider a proper holomorphic surjection $p:\mathcal{X}^{n+m}\to Y^m$ between two complex manifolds with $\mathcal{X}$ K\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\mathcal{X}_t$ for $t\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\mathcal{X}$. Let $$V_t=H^0(\mathcal{X}_t, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}).$$ We assume that $\dim V_t$ is independent of $t\in Y$. So, the direct image of the sheaf of sections of $L\otimes K_{\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\in Y$,
\begin{equation}\label{metric}
 H(u,u):=\int_{\mathcal{X}_t}h(u,u).
\end{equation}
We extend the metric $h$ to act on sections $u$ of $L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\mathcal{X}_t$. In terms of local coordinates, if $u=u'\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\mathcal{X}_t}$, then $h(u,u)=c_n u' \wedge \overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show
\begin{theorem}\label{Thm 1}
 If the curvature $\Theta$ of $h$ is positive on every fiber, and for any point $t\in Y$, there exists a nonzero tangent vector $v\in T_tY$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.
\end{theorem}

Now, we consider the fibration $p:P(E^*)\to X$, and we assume $X$ is K\"ahler to make sure $P(E^*)$ is K\"ahler (see \cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \ref{Thm 1}, we have Theorem \ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \ref{Thm 1} is associated with the direct image of $L\otimes K_{\mathcal{X}/Y}$. In the present situation, the relative canonical bundle
$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\otimes p^*\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\otimes \det E$. On the other hand, if we choose $O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large).

We choose a coordinate system $(t_1,\ldots,t_m)$ around the point $t_0$ in $Y$ such that $v_0=\partial/\partial t_1$ at $t_0$. Consider a fixed $u_0\neq 0$ in $V_{t_0}$. A standard argument allows us to extend $u_0$ to a local holomorphic section $u$ of $V$ such that $D'u=0$ at $t_0$ and $u(t_0)=u_0\neq0$. A straightforward computation gives
\begin{equation}\label{standard} \partial \bar{\partial} H(u,u)=- H(\Theta^V u,u) \text{ at } t_0.
\end{equation}
On the other hand, if we let $\mathbf{u}$ be a representative of $u$ and write $\mathbf{u}=u'\otimes e$ with $u'$ an $(n,0)$-form and $e$ some local frame of $L$, then
\begin{equation}
 H(u,u)=p_*(c_n u'\wedge \overline{u'} e^{-\phi})
\end{equation}
where $e^{-\phi}=h(e,e)$ and $c_n=i^{n^2}$. According to \cite[Proposition 4.2]{Berndtsson09}, we can choose a representative $\mathbf{u}$ such that in $\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j$, the $\eta_j$ is primitive on $\mathcal{X}_{t_0}$. Moreover,
$\partial^\phi u'=0$ at $t_0$. After using such a representative, we obtain
\begin{equation}\label{4.4}
\partial\bar{\partial}H(u,u)
 =
 -c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})
 +
 (-1)^n c_np_* (\bar{\partial}u'\wedge \overline{\bar{\partial}u'}e^{-\phi}) \text{ at } t_0.
\end{equation}
We apply the above $(1,1)$-form to the tangent vector $v_0=\partial/\partial t_1$ and get
\begin{equation}\label{4.6}
\partial\bar{\partial}H(u,u)(v_0,\bar{v}_0)
 =
 -c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})(v_0,\bar{v}_0)
 +
 (-1)^n c_np_* (\bar{\partial}u'\wedge \overline{\bar{\partial}u'}e^{-\phi}) (v_0,\bar{v}_0)\text{ at } t_0.
\end{equation}
Because $\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j$ and $\mathbf{u}=u'\otimes e$, we see $\sum \eta_j\wedge dt_j=\bar{\partial}\mathbf{u}=\bar{\partial}u'\otimes e $. If we write $\eta_j=\eta_j'\otimes e$, then $\bar{\partial}u'=\sum \eta_j'\wedge dt_j$. So the last term in (\ref{4.6}) is equal to \begin{equation}\label{4.7}
 (-1)^n c_n\int_{\mathcal{X}_{t_0}}(-1)^{n}\sum \eta'_j\wedge
\overline{\eta'}_k \wedge dt_j\wedge d\bar{t}_k e^{-\phi}(v_0,\bar{v}_0)= c_n \int_{\mathcal{X}_{t_0}} \eta'_1\wedge
\overline{\eta'}_1 e^{-\phi}\leq 0;
\end{equation}
the last inequality is by the fact that the $\eta_1$ is primitive on $\mathcal{X}_{t_0}$.

The case we care about most in this paper is when $k=1$ in Lemma \ref{lemma abc} because it corresponds to weak RC-positivity. The difficulty in proving a theorem like Theorem \ref{Thm for E} or Theorem \ref{Thm 1} for weak RC-positivity is that the $\beta$ in Lemma \ref{lemma abc} is on $\mathcal{X}$, so it does not quite fit into Berndtsson's computation, especially formula (\ref{4.11}).
So, we raise the question:
\begin{question}
 Is it possible to choose $\beta$ in Lemma \ref{lemma abc} so that $\beta=p^*\alpha$ for some Hermitian metric $\alpha$ on $Y$?
\end{question}
This question is somewhat bold because if it is possible to choose $\beta=p^*\alpha$, then we can use \cite[Theorem 4]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $S^k\otimes \det E$ has positive mean curvature for $k\geq 0$. Moreover, it is even possible to use \cite[Theorem 5]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $E$ has positive mean curvature. Since positive mean curvature implies RC-positivity (\cite[Theorem 3.6]{YangCamb}), this would mean that RC-positivity, weak RC-positivity, and mean curvature positivity are all equivalent. Such an equivalence is conjectured for tangent bundle $TX$ in
\cite[Problems 4.15 and 4.17]{YangForum}.

Another motivation for establishing Theorem \ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \cite[Theorem 1.4]{YangCamb} and \cite[Theorem 1.3]{YangForum}, for a compact K\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.
\begin{enumerate}
     \item The holomorphic tangent bundle $TX$ is uniformly RC-positive.
     \item The exterior power $\wedge^p TX$ is RC-positive for $1\leq p\leq n$.
 \end{enumerate}
 One can ask if the converse is true (\cite[Problem 4.15]{YangForum}). A partial converse is proved in \cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\wedge^p TX}(-1)$ is RC-positive for $1\leq p \leq n$. So, Theorem \ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.

We also prove a lemma (Lemma \ref{lemma abc} in Section \ref{section ?}) and discuss how a variant of this lemma might lead to a solution to the original question of Yang, namely, weak RC-positivity of $E$ implying RC positivity of $E$.

For the proof of Theorem \ref{Thm for E}, instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \ref{Thm for E}
as a special case. We consider a proper holomorphic surjection $p:\mathcal{X}^{n+m}\to Y^m$ between two complex manifolds with $\mathcal{X}$ K\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\mathcal{X}_t$ for $t\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\mathcal{X}$. Let $$V_t=H^0(\mathcal{X}_t, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}).$$ We assume that $\dim V_t$ is independent of $t\in Y$. So, the direct image of the sheaf of sections of $L\otimes K_{\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\in Y$,
\begin{equation}\label{metric}
  H(u,u):=\int_{\mathcal{X}_t}h(u,u).
\end{equation}
We extend the metric $h$ to act on sections $u$ of $L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\mathcal{X}_t$. In terms of local coordinates, if $u=u'\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\mathcal{X}_t}$, then $h(u,u)=c_n u' \wedge \overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show
\begin{theorem}\label{Thm 1}
    If the curvature $\Theta$ of $h$ is positive on every fiber, and for any point $t\in Y$, there exists a nonzero tangent vector $v\in T_tY$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.
\end{theorem}

Actually, the precise statement we prove in Theorem \ref{Thm 1} is: For a fixed point $t_0 \in Y$, if the curvature $\Theta$ of $h$ is positive on the fiber $\mathcal{X}_{t_0}$ , and  there exists a nonzero tangent vector $v\in T_{t_0}Y$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t_0,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t_0,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive at $t_0$.

Now, we consider the fibration $p:P(E^*)\to X$, and we assume $X$ is K\"ahler to make sure $P(E^*)$ is K\"ahler (see \cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \ref{Thm 1}, we have Theorem \ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \ref{Thm 1} is associated with the direct image of $L\otimes K_{\mathcal{X}/Y}$. In the present situation, the relative canonical bundle
$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\otimes p^*\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\otimes \det E$. On the other hand, if we choose $O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large).

The proof of Theorem \ref{Thm 1} is an adaptation of \cite[Section 3]{wu2025mean}, but we still include the details for completeness  (the original argument is due to Berndtsson in \cite[Section 4]{Berndtsson09} and \cite[Section 2]{BoMathz}. See also \cite{CampanaCaoMihai}).

This paper is organized as follows. In Section \ref{section prelim}, we give a local expression for the assumption (uniform weak RC-positivity) in Theorem \ref{Thm 1} which will be used in the proof of the main theorem. In Section \ref{section proof}, we prove Theorem \ref{Thm 1}. In Section \ref{section ?}, we discuss
a characterization of weak RC-positivity and its possible application.

I would like to thank Shin-ichi Matsumura for bringing to my attention the question of RC-positivity and weak RC-positivity. I am grateful to L\'aszl\'o Lempert, Siarhei Finski, and Xiaokui Yang for their interest in the paper. Thanks are also due to the Erd\H{o}s Center, Budapest for the support.

To prove Theorem~\ref{Thm for E}, the text says that “instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem~\ref{Thm for E}$ as a special case.” Concretely, one considers a proper holomorphic submersion $p:\mathcal{X}^{n+m}\to Y^m$ with $\mathcal{X}$ K\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\mathcal{X}$, and the direct-image bundle $V$ with fibers
\[
V_t=H^0(\mathcal{X}_t,\, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}),
\]
endowed with the “naturally defined Hermitian metric”
\[
H(u,u):=\int_{\mathcal{X}_t} h(u,u).\tag{\ref{metric}}
\]
Under the hypothesis (in Theorem~\ref{Thm 1}) that “the curvature $\Theta$ of $h$ is positive on every fiber” and that for each $t\in Y$ there is a nonzero $v\in T_tY$ with “$\Theta(\widetilde v,\overline{\widetilde v})|_{(t,z)}>0$ for any lift $\widetilde v$ of $v$,” one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”

Then Theorem~\ref{Thm for E} is obtained by applying Theorem~\ref{Thm 1} to $p:P(E^*)\to X$ (with $X$ K\"ahler so $P(E^*)$ is K\"ahler). In this specialization, one uses that
\[
K_{P(E^*)/X}\simeq O_{P(E^*)}(-r)\otimes p^*\det E.
\]
Choosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\otimes\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”

Finally, the proof of Theorem~\ref{Thm 1} is described as “an adaptation of \cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).

To prove the main theorem, the text says that “instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains the main theorem as a special case.” Concretely, one considers a proper holomorphic submersion $p:\mathcal{X}^{n+m}\to Y^m$ with $\mathcal{X}$ K\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\mathcal{X}$, and the direct-image bundle $V$ with fibers
\[
V_t=H^0(\mathcal{X}_t,\, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}),
\]
endowed with the “naturally defined Hermitian metric”
\[
H(u,u):=\int_{\mathcal{X}_t} h(u,u).\tag{\ref{metric}}
\]
We first prove the following theorem.
\begin{theorem}\label{Thm 1}
    If the curvature $\Theta$ of $h$ is positive on every fiber, and for any point $t\in Y$, there exists a nonzero tangent vector $v\in T_tY$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.
\end{theorem}
Under these hypotheses, one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”

Then the main theorem is obtained by applying the preceding theorem to $p:P(E^*)\to X$ (with $X$ K\"ahler so $P(E^*)$ is K\"ahler). In this specialization, one uses that
\[
K_{P(E^*)/X}\simeq O_{P(E^*)}(-r)\otimes p^*\det E.
\]
Choosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\otimes\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”

Finally, the proof of the preceding theorem is described as “an adaptation of \cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).

Question: Let $E\to X$ be a holomorphic vector bundle over a compact K\"ahler manifold $X$. Assume that $E$ is uniformly weakly RC-positive in the following sense: there exists a Hermitian metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ such that, if $\Theta$ denotes the curvature of $h$, then (i) $\Theta$ is positive on every fiber of $P(E^*)\to X$, and (ii) for every point $t\in X$ there exists a nonzero tangent vector $v\in T_tX$ such that for every point $[\zeta]\in P(E_t^*)$ and every lift $\widetilde v\in T_{(t,[\zeta])}P(E^*)$ of $v$, one has $\Theta(\widetilde v,\overline{\widetilde v})>0$ at $(t,[\zeta])$. Here $S^kE$ denotes the $k$th symmetric power of $E$, and $\det E$ its determinant line bundle. A holomorphic vector bundle $F\to X$ is called uniformly RC-positive if it admits a Hermitian metric $H$ such that for every $t\in X$ there exists a nonzero vector $v\in T_tX$ with $H(\Theta^H u,u)(v,\bar v)>0$ for every nonzero $u\in F_t$, where $\Theta^H$ is the Chern curvature of $H$. Under these assumptions, which statement holds?

For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is uniformly RC-positive, and the bundle $S^kE$ is uniformly RC-positive for all sufficiently large $k$.

Choices:
B. For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is uniformly RC-positive, and the bundle $S^kE$ is uniformly RC-positive for every integer $k\ge 0$.
C. For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is uniformly RC-positive.
D. For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is RC-positive, and the bundle $S^kE$ is RC-positive for all sufficiently large $k$.
E. There exists an integer $k_0\ge 0$, depending only on $\operatorname{rk}(E)$, such that for every integer $k\ge k_0$, both $S^kE\otimes \det E$ and $S^kE$ are uniformly RC-positive.

large-k requirement for absorbing the metric on $K^{-1}_{P(E^*)/X}$

the additional conclusion about $S^kE$ for sufficiently large $k$

uniformity conclusion of the direct-image theorem

non-uniform dependence of the threshold for large $k$


\begin{thm} \label{thm:decaymap}
For all H\"older observables $v,w:\Sigma\to\R$, there is a constant
$C>0$ such that
$$
\left|\int_{\Sigma} v \cdot (w\circ g^n)\,d\mu_{\Sigma}-\int_{\Sigma}v\,d\mu_{\Sigma}\int_{\Sigma}w\,d\mu_{\Sigma}\right|
\leq C n^{-1}
\quad\text{for all $n\ge1$}.
$$
Moreover, for all H\"older observables $v,w:\Sigma\to\R$  supported in $\Sigma_0$,
\[
\int_{\Sigma} v \cdot (w\circ g^n)\,d\mu_{\Sigma}-\int_{\Sigma}v\,d\mu_{\Sigma}\int_{\Sigma}w\,d\mu_{\Sigma}\sim \bar\tau\sigma_R^2
n^{-1} \int_\Sigma v\,d\mu_\Sigma\, \int_\Sigma w\,d\mu_\Sigma \quad\text{for all $n\ge1$}.
\]
\end{thm}

\label{app:C}

For convenience, we list the Chernov axioms as stated in~\cite[Section 3]{LMM24}. A more in-depth discussion of the axioms, and where they differ from treatments in~\cite{Balint-Toth,Ch

\label{sec-intro}

It is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.
Ergodicity with respect to volume was proved for surfaces in~\cite{Hopf39} and for general dimension in~\cite{Anosov67}. The Bernoulli property was shown in~\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\cite{DenkerPhilipp84,Ratner73}.
In major breakthroughs, exponential decay of correlations was established in~\cite{Dolgopyat98} for surfaces and~\cite{Liverani04} in general dimension.

In contrast, there are few results on statistical properties beyond the Bernoulli property for
geodesic flows on nonpositively curved manifolds when there exist points of zero curvature.
In~\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.
In particular, we considered a class of closed surfaces with one flat geodesic
and proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP
(with standard normalisation).
In this paper, we provide examples of geodesic flows on nonpositively curved surfaces
that satisfy a CLT and WIP with \emph{nonstandard normalisation}.

Our approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\cite{ChernovMarkarian}.
General results on statistical limit laws and decay of correlations seem infeasible,
but it is possible to analyse interesting classes of examples.
Moreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.
On the other hand, the prerequisites for the study of dispersing billiards are firmly established.
For geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.
A case in point is the smoothness of the foliations induced by the Green bundles and the necessity
to extend existing results such as those in~\cite{GerberNitica99,GerberWilkinson99}, see Section~\ref{sec:GW}.

\vspace{1ex}
The class of geodesic flows that we study in this paper is as follows.
Let $r\in[5,\infty)$ and fix $0<L<\ve_0$.
Define ${\mathcal R}$ to be the surface of revolution
with profile~$\xi$ given by
$\xi(s)=1$ for $|s|\leq L$ and $\xi(s)=1+(|s|-L)^r$ for $L\leq |s|\leq \ve_0$.
We form
a closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\beta_\pm$ bounding ${\mathcal R}$,
see Figure~\ref{fig:surface-intro}.
We call $S$ a \emph{surface with flat cylinder ${\mathcal C}=[-L,L]\times{\mathbb S}^1$}, where
${\mathbb S}^1={\mathbb R}/(2\pi{\mathbb Z})$.
Away from~${\mathcal C}$ the surface $S$ has negative curvature.

\begin{figure}[hbt!]
\centering
\def\svgwidth{11cm}
   \input{nonpositive_curvature-intro.pdf_tex}
\caption{Surface
with flat cylinder ${\mathcal C}$.
The region ${\mathcal R}$ between the two curves $\beta_\pm$ is a surface of revolution
with profile $\xi$.}
\label{fig:surface-intro}
\end{figure}

Let $g_t:M\to M$ denote the geodesic flow on the unit tangent bundle
$M=T^1S$, and let $\mu$ denote the normalised Riemannian volume on the
three-dimensional manifold $M$. Then $\mu$ is an ergodic $g_t$-invariant
probability measure \cite{Pesin77b}.

The flat cylinder ${\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.
Let $v:M\to{\mathbb R}$ be H\"older with $\int_M v\,d\mu=0$. We define $I_v=\alpha_0^2+\alpha_\pi^2$ where
$\alpha_0$ and $\alpha_\pi$ are the averages of $v$ over these two families of periodic orbits.
(The notation derives from the explicit formulas~\eqref{eq:alpha} given in Section~\ref{sec:WIP}.)
For $t>0$, define $v_t=\int_0^t v\circ g_s\,ds:M\to{\mathbb R}$.
Our first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.

It is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.
Ergodicity with respect to volume was proved for surfaces in~\cite{Hopf39} and for general dimension in~\cite{Anosov67}. The Bernoulli property was shown in~\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\cite{DenkerPhilipp84,Ratner73}.
In major breakthroughs, exponential decay of correlations was established in~\cite{Dolgopyat98} for surfaces and~\cite{Liverani04} in general dimension.

In contrast, there are few results on statistical properties beyond the Bernoulli property for
geodesic flows on nonpositively curved manifolds when there exist points of zero curvature.
In~\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.
In particular, we considered a class of closed surfaces with one flat geodesic
and proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP
(with standard normalisation).
In this paper, we provide examples of geodesic flows on nonpositively curved surfaces
that satisfy a CLT and WIP with \emph{nonstandard normalisation}.

Our approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\cite{ChernovMarkarian}.
General results on statistical limit laws and decay of correlations seem infeasible,
but it is possible to analyse interesting classes of examples.
Moreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.
On the other hand, the prerequisites for the study of dispersing billiards are firmly established.
For geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.
A case in point is the smoothness of the foliations induced by the Green bundles and the necessity
to extend existing results such as those in~\cite{GerberNitica99,GerberWilkinson99}, see Section~\ref{sec:GW}.

\vspace{1ex}
The class of geodesic flows that we study in this paper is as follows.
Let $r\in[5,\infty)$ and fix $0<L<\ve_0$.
Define ${\mathcal R}$ to be the surface of revolution
with profile~$\xi$ given by
$\xi(s)=1$ for $|s|\leq L$ and $\xi(s)=1+(|s|-L)^r$ for $L\leq |s|\leq \ve_0$.
We form
a closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\beta_\pm$ bounding ${\mathcal R}$,
see Figure~\ref{fig:surface-intro}.
We call $S$ a \emph{surface with flat cylinder ${\mathcal C}=[-L,L]\times{\mathbb S}^1$}, where
${\mathbb S}^1={\mathbb R}/(2\pi{\mathbb Z})$.
Away from~${\mathcal C}$ the surface $S$ has negative curvature.

Let $g_t:M\to M$ denote the geodesic flow on the unit tangent bundle
$M=T^1S$, and let $\mu$ denote the normalised Riemannian volume on the
three-dimensional manifold $M$. Then $\mu$ is an ergodic $g_t$-invariant
probability measure \cite{Pesin77b}.

The flat cylinder ${\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.
Let $v:M\to{\mathbb R}$ be H\"older with $\int_M v\,d\mu=0$. We define $I_v=\alpha_0^2+\alpha_\pi^2$ where
$\alpha_0$ and $\alpha_\pi$ are the averages of $v$ over these two families of periodic orbits.
(The notation derives from the explicit formulas~\eqref{eq:alpha} given in Section~\ref{sec:WIP}.)
For $t>0$, define $v_t=\int_0^t v\circ g_s\,ds:M\to{\mathbb R}$.
Our first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.

\begin{equation} \label{eq:alpha}
\alpha_0=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,0)\,ds\,d\theta, \qquad
\alpha_\pi=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,\pi)\,ds\,d\theta.
\end{equation}

\begin{figure}[hbt!]
\centering
\def\svgwidth{11cm}
   \input{nonpositive_curvature-intro.pdf_tex}
\caption{Surface
with flat cylinder $\cC$.
The region $\cR$ between the two curves $\beta_\pm$ is a surface of revolution
with profile $\xi$.}
\label{fig:surface-intro}
\end{figure}

The flat cylinder $\cC$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.
Let $v:M\to\R$ be H\"older with $\int_M v\,d\mu=0$. We define $I_v=\alpha_0^2+\alpha_\pi^2$ where
$\alpha_0$ and $\alpha_\pi$ are the averages of $v$ over these two families of periodic orbits.
(The notation derives from the explicit formulas~\eqref{eq:alpha} given in Section~\ref{sec:WIP}.)
For $t>0$, define $v_t=\int_0^t v\circ g_s\,ds:M\to\R$.
Our first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.

\begin{rmk} As in~\cite[Remark~4.6]{LMMsub}, the word ``typically'' is interpreted here in the very strong sense that $\sigma^2=0$ only within a closed subspace of infinite codimension amongst H\"older observables $v$ with $\int_M v\,d\mu=I_v=0$. See for example the discussion in~\cite[End of Section~4]{HM07}.

\begin{ltheorem}[Nonstandard WIP] \label{thm:WIP}
If $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,
$W_n \to_w W$ in $C[0,1]$ as $n\to\infty$,
where $W$ is a Brownian motion with variance $\sigma_v^2$.
\end{ltheorem}

\begin{rmk} \label{rmk:lower}
 The upper bound in Theorem~\ref{thm:decay} is sharp in the sense that if $v$ is H\"older with $\int_Mv\,d\mu=0$ and $I_v\neq0$, then
\[
t\int_M v \cdot (v\circ g_t)\,d\mu\not\to0
\quad\text{as $t\to\infty$}.
\]
This is a consequence of the nonstandard limit law in Theorem~\ref{thm:CLT}.
See~\cite[Corollary~1.3]{BalintGouezel06} or~\cite[Proposition~9.14]{BBM19}.
\end{rmk}

\begin{lemma}\label{lem:hyperbolicity}
Let $r\ge 5$, $\omega\in(1,r/2)$. The following are true:
\begin{description}
\item[{\rm (1)}] Lebesgue almost every $x\in\Sigma_0$ has an LSM/LUM $W^{s/u}_x$ for $f$.
\item[{\rm (2)}]
For all $x,\overline{x}\in \Sigma_{\rm in}$,
\begin{enumerate}
\item[{\rm (a)}] There is a continuous function $a:\Sigma_{\rm in}\to\R$ such that
$E^u_x$ is spanned by $\begin{bmatrix}a(x) \\ 1\end{bmatrix}$.
Moreover,
$|a(x)-a(\bar x)|\ll |\log d(x,\overline{x})|^{-\omega}$.
\item[{\rm (b)}] There is a $C^{1+{\rm Lip}}$ function $\Theta$ such that
$W^u_x$ is locally the graph $\{(\Theta(\psi),\psi)\}$ of $\Theta$.
\end{enumerate}
\item[{\rm (3)} Growth bounds:] {\color{white} a}
\begin{enumerate}
\item[{\rm (a)}] If $x\in\mfD_n^>$, then $\|df|_{E^u_x}\|\approx n^{3-\frac{2}{r}}$.
\item[{\rm (b)}] If $x\in\mfD_n^<$, then $\|df|_{E^u_x}\|\approx n^3$.
\end{enumerate}
\item[{\rm (4)} Distortion bounds:] If $x,\overline{x}\in\mfD_n^>$ or $x,\overline{x}\in\mfD_n^<$,
with $\overline{x}\in W^u_x$, then
$$
\big|\log \|df|_{E^u_x}\|-\log \|df|_{E^u_{\overline x}}\|\big| \ll
d(fx, f \overline{x})^\frac13.
$$
\item[{\rm (5)} Jacobian of holonomies:]
If $x,\overline{x}\in\mfD_n^>$ or $x,\overline{x}\in\mfD_n^<$,
with $\overline{x}\in W^s_x$, then
\[
\big|\log \|df|_{E^u_x}\|-\log \|df|_{E^u_{\overline{x}}}\|\big|\ll |\log d(x,\overline{x})|^{-\omega}.
\]
\end{description}
\end{lemma}

\begin{thm} \label{thm:JR}
Let $\{\alpha_i:i\in I\}\subset\R$ be a list of the values attained by $J$.
Let $\sigma_i\ge0$ be constants, $i\in I$, such that
$\sum_{i\in I}\sigma_i^2\in(0,\infty)$.
Assume that there are constants $C>0$, $\eps\in(0,1)$ such that
\begin{equation} \label{eq:JR1}
 \sum_{i\in I}|\mu_\Delta(R_0=n,\,J=\alpha_i)-2\sigma_i^2 n^{-3}|\le Cn^{-(3+\eps)}
\quad\text{for all $n\ge1$},
\end{equation}
and
\begin{equation} \label{eq:JR2}
\mu_\Delta(R_0=k,\,R_0\circ f_\Delta^n=\ell)\le Ck^{-(2+\eps)}\ell^{-(2+\eps)}
\quad\text{for all $k,\ell,n\ge1$}.
\end{equation}
Then $JR_0$ satisfies a nonstandard CLT with variance
$\sigma_J^2=\sum_{i\in I}\alpha_i^2\sigma_i^2\ge0$. That is,
$(n\log n)^{-1/2}\sum_{j=0}^{n-1}(JR_0)\circ f_\Delta^j\to_d N(0,\sigma_J^2)$.
\end{thm}

\begin{cor} \label{cor:RCLT}
$R:\Delta\to\Z^+$ satisfies a nonstandard CLT
on $(\Delta,f_\Delta,\mu_\Delta)$. That is,
$(n\log n)^{-1/2}(\sum_{j=0}^{n-1}R\circ f_\Delta^j-n\bar R)\to_d N(0,\sigma_R^2)$ where
$\bar R=\int_\Delta R\,d\mu_\Delta$.
\end{cor}

In this section, we consider decay of correlations for the geodesic flow $g_t:M\to M$ and the global Poincar\'e map $g:\Sigma\to\Sigma$.
The general setup is the same as in~\cite[Section~7]{LMM24}.
We recall the basics here.\footnote{We caution that for example $\tau$ here was called $\sigma$ in~\cite{LMM24}.}
\begin{enumerate}[$\bullet$]
\item The flow $g_t$ and map $g$ are related by the identity
$g=g_h$ where $h:\Sigma\to\R^+$ is a H\"older roof function with $0<\inf h<\sup h<\infty$.
\item
The first return maps $g:\Sigma\to\Sigma$ and $f:\Sigma_0\to\Sigma_0$ are related by $f=g^R$ where $R:\Sigma_0\to\Z^+$ is a discrete first return time.
\item
The uniformly hyperbolic first return map $f:\Sigma_0\to\Sigma_0$ is modelled by a Young tower with exponential tails~\cite{Young98}: there is a subset $Y\subset\Sigma_0$ with product structure and bounded distortion, a probability measure $\mu_Y$ on $Y$, and a ``good'' return time $\tau:Y\to\Z^+$ constant along stable leaves with $\mu_Y(\tau>n)=O(e^{-cn})$ for some $c>0$, such that $f^\tau:Y\to Y$ is measure-preserving. Moreover, quotienting along stable leaves yields a full-branch Gibbs-Markov map $F:Z\to Z$ as in Section~\ref{sec:CLT}, and $\tau$ is constant on the associated partition elements.
\item
The global Poincar\'e map $g:\Sigma\to\Sigma$ is modelled by a subexponential Young tower over $g^{\varphi_*}=(g^R)^\tau=f^\tau:Y\to Y$ where the return time $\varphi_*:Y\to\Z^+$ is given by $\varphi_*=\sum_{\ell=0}^{\tau-1}R\circ f^\ell$.
\item
We can regard $f^\tau:Y\to Y$ as a (non-first return) Poincar\'e map for the flow $g_t$ with roof function $\varphi=h_{\varphi_*}=\sum_{\ell=0}^{\varphi_*-1}h\circ g^\ell$.
\end{enumerate}

\begin{equation} \label{eq:alpha}
\alpha_0=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,0)\,ds\,d\theta, \qquad
\alpha_\pi=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,\pi)\,ds\,d\theta.
\end{equation}

\begin{ltheorem}[Nonstandard/Standard CLT] \label{thm:CLT}
If $I_v>0$, then $v$ satisfies a nonstandard CLT. That is,
$(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2)$ as $t\to\infty$,
where $\sigma_v^2=b_SI_v>0$ and $b_S$ is a positive constant depending only on the surface $S$.

If $I_v=0$, then there exists $\sigma^2\ge0$, typically nonzero, such that
$v$ satisfies a standard CLT. That is,
$t^{-1/2}v_t \to_d N(0,\sigma^2)$ as $t\to\infty$.
\end{ltheorem}

\begin{ltheorem} \label{thm:decay}
 Let $v,\,w:M\to\R$ be sufficiently smooth observables. Then there is a constant
$C>0$ such that
$$
\left|\int_M v \cdot (w\circ g_t)\,d\mu-\int_Mv\,d\mu\int_Mw\,d\mu\right|
\leq C t^{-1}
\quad\text{for all $t>0$}.
$$
\end{ltheorem}

\begin{rmk} As in~\cite[Remark~4.6]{LMMsub}, the word ``typically'' is interpreted here in the very strong sense that $\sigma^2=0$ only within a closed subspace of infinite codimension amongst H\"older observables $v$ with $\int_M v\,d\mu=I_v=0$. See for example the discussion in~\cite[End of Section~4]{HM07}.

We note that the vanishing of $\int_Mv\,d\mu$ is ``without loss of generality'' since one can always centre the observable $v$ by considering $v-\int_Mv\,d\mu$.
The vanishing of $I_v$ is ``codimension two'' since $\alpha_0=\alpha_\pi=0$
may occur in generic two-parameter families. In contrast, the degenerate case $\sigma^2=0$
requires infinitely many coincidences beyond the constraint $I_v=0$.
\end{rmk}

In the case $I_v>0$, we also prove a nonstandard WIP (which implies the CLT).
Define the sequence of continuous functions
\[
W_n\in C[0,1], \qquad W_n(t)=(n\log n)^{-1/2}v_{nt},\quad n\ge1.
\]
Since each function $W_n$ depends on the initial condition in the probability space $(M,\mu)$, we
can regard $W_n$ as a random element in the metric space $(C[0,1],\|\;\|_\infty)$.

\begin{ltheorem}[Nonstandard WIP] \label{thm:WIP}
If $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,
$W_n \to_w W$ in $C[0,1]$ as $n\to\infty$,
where $W$ is a Brownian motion with variance $\sigma_v^2$.
\end{ltheorem}

There are only a few situations in deterministic dynamical systems where
the nonstandard CLT/WIP is known to hold.
The first example was for intermittent maps $f$ satisfying $f(x)=x+bx^{3/2}$ at a neutral fixed point $x=0$; see~\cite{Gouezel04} for the nonstandard CLT and~\cite{DedeckerMerlevede09} for the nonstandard WIP.
For various billiard examples, namely the infinite horizon Lorentz gas, Bunimovich stadium and billiards with cusps, we refer to~\cite{BalintChernovDolgopyat11,BalintGouezel06,LMMsub,SzaszVarju07}.

Our proofs of Theorems~\ref{thm:CLT} and~\ref{thm:WIP} closely follows abstract arguments in~\cite{BalintChernovDolgopyat11} and~\cite{LMMsub}, using the latter to reduce to a nonstandard CLT for a simpler class of observables, and using the former to establish this simplified nonstandard CLT.

Our final main result concerns the rate of decay of correlations for the flow $g_t$.

\begin{ltheorem} \label{thm:decay}
 Let $v,\,w:M\to{\mathbb R}$ be sufficiently smooth observables. Then there is a constant
$C>0$ such that
$$
\left|\int_M v \cdot (w\circ g_t)\,d\mu-\int_Mv\,d\mu\int_Mw\,d\mu\right|
\leq C t^{-1}
\quad\text{for all $t>0$}.
$$
\end{ltheorem}

\begin{rmk} An observable is ``sufficiently smooth'' if it is $C^k$ in the flow direction for some $k\ge0$ that depends only on the surface $S$. In
particular, if we choose the surface $S$ to be $C^\infty$ away from ${\mathcal C}$, then any observable that is $C^\infty$ on $M$ and flat near $\{s=\pm L\}$ is sufficiently smooth.

The analogous result for the infinite horizon Lorentz gas and Bunimovich stadia  was obtained in~\cite{BBM19}.
\end{rmk}

\begin{rmk} \label{rmk:lower}
 The upper bound in Theorem~\ref{thm:decay} is sharp in the sense that if $v$ is H\"older with $\int_Mv\,d\mu=0$ and $I_v\neq0$,  then
\[
t\int_M v \cdot (v\circ g_t)\,d\mu\not\to0
\quad\text{as $t\to\infty$}.
\]
This is a consequence of the nonstandard limit law in Theorem~\ref{thm:CLT}.
See~\cite[Corollary~1.3]{BalintGouezel06} or~\cite[Proposition~9.14]{BBM19}.
\end{rmk}

The remainder of this paper is organised as follows.
In Section \ref{sec:surfaces}, we review known facts about the geometry and dynamics of
geodesic flows on nonpositively curved surfaces, with special attention to surfaces with flat cylinder.
In Section \ref{sec:R-dynamics}, we make a systematic study of the dynamics of the geodesic flow on the surface of revolution ${\mathcal R}$.
In Section~\ref{sec:GW}, we prove sufficient regularity of the stable/unstable foliations
induced by the Green bundles, partly extending~\cite{GerberWilkinson99}.
In Section~\ref{sec:f}, we construct a uniformly hyperbolic first return map $f:\Sigma_0\to\Sigma_0$ satisfying the Chernov axioms~\cite{Chernov-1999}, showing that $f$ is modelled by a Young tower with exponential tails~\cite{Young98}.

In Section~\ref{sec:CLT}, we recall and slightly extend an abstract result on the nonstandard CLT due to~\cite{BalintChernovDolgopyat11}.
In Section~\ref{sec:WIP}, we prove Theorems~\ref{thm:CLT} and~\ref{thm:WIP}.
Finally, in Section~\ref{sec:decay} we prove Theorem~\ref{thm:decay} as well as a related result, Theorem~\ref{thm:decaymap}, for a suitable global Poincar\'e map $g$.
The Chernov axioms are stated for convenience in Appendix~\ref{app:C}.

\vspace{-2ex}
\paragraph{Notation}
We use ``big O'' and $\ll$ notation interchangeably, writing $a_n=O(b_n)$ or $a_n\ll b_n$
if there are constants $C>0$, $n_0\ge1$ such that
$a_n\le Cb_n$ for all $n\ge n_0$.
We write $a_n\approx b_n$ if $a_n\ll b_n$ and $b_n\ll a_n$, and
$a_n\sim b_n$ if $a_n/b_n\to1$.

Our proofs of Theorems~\ref{thm:CLT} and~\ref{thm:WIP} \emph{closely follows abstract arguments} in~\cite{BalintChernovDolgopyat11} and~\cite{LMMsub}: using~\cite{LMMsub} to \emph{reduce to a nonstandard CLT for a simpler class of observables}, and using~\cite{BalintChernovDolgopyat11} to \emph{establish this simplified nonstandard CLT}. In the case $I_v>0$, they also prove a nonstandard WIP (which implies the CLT), formulated via $W_n(t)=(n\log n)^{-1/2}v_{nt}$ and convergence $W_n\to_w W$ in $C[0,1]$ to Brownian motion with variance $\sigma_v^2$.

Our proofs of To prove the main theorem,  and
\begin{ltheorem}[Nonstandard WIP] \label{thm:WIP}
If $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,
$W_n \to_w W$ in $C[0,1]$ as $n\to\infty$,
where $W$ is a Brownian motion with variance $\sigma_v^2$.
\end{ltheorem}
\emph{closely follows abstract arguments} in~\cite{BalintChernovDolgopyat11} and~\cite{LMMsub}: using~\cite{LMMsub} to \emph{reduce to a nonstandard CLT for a simpler class of observables}, and using~\cite{BalintChernovDolgopyat11} to \emph{establish this simplified nonstandard CLT}. In the case $I_v>0$, they also prove the preceding theorem (which implies the CLT), formulated via $W_n(t)=(n\log n)^{-1/2}v_{nt}$ and convergence $W_n\to_w W$ in $C[0,1]$ to Brownian motion with variance $\sigma_v^2$.

Question:
Let $S$ be the closed $C^4$ nonpositively curved surface with a flat cylinder ${\mathcal C}=[-L,L]\times \mathbb S^1$ described as follows: ${\mathcal C}$ is foliated by two families of periodic geodesics with orientations corresponding to angle $0$ and $\pi$, and $S$ has negative curvature away from ${\mathcal C}$. Let $M=T^1S$, let $g_t:M\to M$ be the geodesic flow, and let $\mu$ be the normalised Riemannian volume on $M$. For a H\"older observable $v:M\to\mathbb R$ with $\int_M v\,d\mu=0$, define
\[
v_t=\int_0^t v\circ g_s\,ds,
\]
and
\[
\alpha_0=\frac{1}{2L}\int_{\mathbb S^1}\int_{-L}^L v(s,\theta,0)\,ds\,d\theta,
\qquad
\alpha_\pi=\frac{1}{2L}\int_{\mathbb S^1}\int_{-L}^L v(s,\theta,\pi)\,ds\,d\theta,
\]
with
\[
I_v=\alpha_0^2+\alpha_\pi^2.
\]
As $t\to\infty$, which limiting statement holds for $v_t$?

There is a dichotomy. If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2),
\]
where $\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then there exists $\sigma^2\ge 0$, typically nonzero, such that $v$ satisfies a standard central limit theorem:
\[
t^{-1/2}v_t \to_d N(0,\sigma^2).
\]

Choices:
B.
There is a dichotomy. If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2),
\]
where $\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the observable $v$. If $I_v=0$, then there exists $\sigma^2\ge 0$, typically nonzero, such that $v$ satisfies a standard central limit theorem:
\[
t^{-1/2}v_t \to_d N(0,\sigma^2).
\]
C.
If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2)
\]
as $t\to\infty$, for some $\sigma_v^2>0$.
D.
There is a dichotomy. If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
t^{-1/2}v_t \to_d N(0,\sigma_v^2),
\]
where $\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then there exists $\sigma^2\ge 0$, typically nonzero, such that $v$ satisfies a standard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma^2).
\]
E.
There is a dichotomy. If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2),
\]
where $\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then $v$ does not satisfy any central limit theorem with Gaussian limit under the normalisation $t^{-1/2}$.

dependence of the variance constant $b_S$

dropped the explicit formula $\sigma_v^2=b_S I_v$ and the $I_v=0$ branch

which branch carries the nonstandard $(t\log t)^{1/2}$ normalisation

existence of a standard CLT in the degenerate case $I_v=0$


In recent years the pluriclosed flow \cite{PCF, PCFReg} and generalized K\"ahler-Ricci flow \cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\"ahler, manifolds \cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \cite{inoue1974surfaces}. In \cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.

The first main result is to establish the global existence of the flow:

\begin{conj} \label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\omega_0$, the normalized pluriclosed flow \ref{e:PCF_normalized} with initial metric $\omega_0$ exists on $[0, \infty)$, and converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{conj}

The first main result is to establish the global existence of the flow:

The expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is
\begin{align}\label{e:normalized equation}
 \omega^{a,b}_h(t) = \sum_{i=1}^s \sqrt{-1}((1-e^{-t})\frac{3}{4}+e^{-t}a_i)\frac{1}{(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i + \sqrt{-1}e^{-t}b_i\operatorname{Im}w_i dz_i\wedge d\bar{z}_i.
\end{align}
Later, we shall denote the time-dependent normalized model metric starting with $\omega^{a,b}_h$ as $\omega^{a,b}_h(t)$. We shall write $\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\leq i\leq s$.
Observe that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\leq C$. Moreover, it follows that
\begin{align*}
 \frac{\omega_h^{a,b}(t)}{t+1}\to \sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i,
\end{align*}
which can be considered as a degenerate metric on $X(K, U)$. As explained in \cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:
\begin{thm} \label{t:homogOT} (\cite{fusi2024pluriclosed}) Let $\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\omega_0$ converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{thm}

\noindent It is reasonable to conjecture that this behavior holds in the general case:
\begin{conj} \label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\omega_0$, the normalized pluriclosed flow \ref{e:PCF_normalized} with initial metric $\omega_0$ exists on $[0, \infty)$, and converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{conj}

We shall prove the first part of the conjecture in Section \ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.
\begin{prop}\label{p:GH convergence argument}
 Let $\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that
 \begin{itemize}
 \item $C^{-1}\leq \operatorname{tr}_{\omega(t)}\omega_h(t)\leq C$,
 \item $\lim\limits_{t\to\infty}\operatorname{tr}_{\omega(t)}\omega_h(t) = 1$ for all $x\in X(K,U)$.
 \end{itemize}
 Then, we have
 \begin{align*}
 (X(K,U),\omega(t))\to (\mathbb{T}^s,d(K, U)),
\end{align*}
to the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.
\end{prop}
\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \cite{fusi2024pluriclosed}, \cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:
 \begin{align*}
 F:X(K,U)\to \mathbb{T}^s
 \end{align*}
 where the fiber is diffeomorphic to $\mathbb{T}^{3s}$. In particular, $E_\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\{z\}\times \mathbb{C}^s$ is dense in the $\mathbb{T}^{3s}$ fiber (See Section 2 of \cite{VerbiOT}), the degenerate metric $\sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i$ will induce a metric on the base $\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\mathbb{T}^s$ induced from $\sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i$, which is flat.

Secondly, when $s = 1$, then $E_\mathbb{C}$ and $E_\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\dot{\phi}$.
\begin{lemma}\label{l:potential derivative lower bound}
 On Inoue surface $S_M$, for the potential $\phi$, there exists a constant $C>0$, such that:
 \begin{align*}
 \dot{\phi}\geq -C.
 \end{align*}
\end{lemma}
\begin{proof}
 Note $s=1$. Choose a large constant $\Lambda>0$, consider the quantity $\dot{\phi} + \phi+\frac{1}{\Lambda}\phi$. Then
 \begin{align*}
 (\frac{\partial}{\partial t} - \Delta)(\dot{\phi}+\phi+\frac{1}{\Lambda}\phi) = 1 +\frac{1}{\Lambda}\dot{\phi}+ \frac{1}{\Lambda}e^{-t}\operatorname{tr}_{g_\mathbb{C}}g^0_\mathbb{C} - \frac{1}{\Lambda}e^{-t}\operatorname{tr}_{g_\mathbb{H}}g^0_\mathbb{H} - (\frac{1}{\Lambda}(1-e^{-t})+1)\frac{3}{4}\operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}.
 \end{align*}
 Now, for $A>0$, a very large constant, such that $\dot{\phi}+\phi +\frac{1}{\Lambda}\phi < -A$. By \eqref{e:potential eq}, notice that in this case $s=1$, we know:
 \begin{align*}
 \dot{\phi} +\phi = \log\frac{\det g_\mathbb{C}/c_1}{\det e^{-t}h_\mathbb{C}} - \log\frac{\det g_\mathbb{H}}{\det \frac{3}{4}h_\mathbb{H}} = -\log(c_1e^{-t}\operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C}) + \log (\frac{3}{4}\operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}).
 \end{align*}
 By Corollary \ref{c:Inoue C0}, in particular $|\phi|\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\dot{\phi}\leq -A'$, and
 \begin{align*}
 \operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}\leq \frac{4}{3}e^{\frac{C_0}{\Lambda}-A}e^{-t}c_1 \operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C}\leq \frac{4}{3}e^{\frac{C_0}{\Lambda}-A}c_0e^{-t}\operatorname{tr}_{g_\mathbb{C}}g^0_\mathbb{C},
 \end{align*}
 for some constant $c_0$, only depends on the initial metric.
 By choosing $\Lambda$ large, such that $1-\frac{C_0}{\Lambda}>\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\frac{C_0}{\Lambda}-A} \leq \frac{1}{\Lambda}$. Thus, choose an $A = \frac{\Lambda}{2}$ large enough, such that $\dot{\phi}(t)\geq -\frac{1}{2}A$, for $t\in[0,T]$, where $T$ is large enough. Then for the first time $\dot{\phi}+\phi +\frac{1}{\Lambda}\phi = -A$, we have:
 \begin{align*}
 (\frac{\partial}{\partial t}-\Delta)(\dot{\phi} + \phi +\frac{1}{\Lambda}\phi)&\geq (\frac{1}{\Lambda } - \frac{C_1e^{-t}}{\Lambda}-c_0e^{\frac{C_0}{\Lambda}-A})e^{-t}\operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C} + 1+ \frac{1}{\Lambda}\dot{\phi}\\
 &>0.
 \end{align*}
 By the maximum principle, the result follows.
\end{proof}
Combine with the potential derivative upper bound (Lemma \ref{l:potential derivative upper bound}), we have the following metric lower bound.
\begin{thm}\label{t:Inoue metric lower bound}
 On an Inoue surface $S_M$, for generalized K\"ahler metric $\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\omega(t)$with initial data $\gw_0$ will satisfy
 \begin{align*}
 \omega(t)\geq C\omega_h(t),
 \end{align*}
 where $\omega_h(t)$ is the model flow.
\end{thm}
\begin{proof}
 By Lemma \ref{l:potential derivative upper bound} and Lemma \ref{l:potential derivative lower bound}, we have that:
 \begin{align*}
 -C\leq\dot{\phi} +\phi \leq C
 \end{align*}
 for some constant $C$. Now, from the equation \eqref{e:potential eq} and $s=1$, we have:
 \begin{align*}
 -C\leq \operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}/\operatorname{tr}_{g_\mathbb{C}}e^{-t}h_\mathbb{C}\leq C.
 \end{align*}
 Note that in the normalized pluriclosed flow model case, \eqref{e:normalized equation}, the $E_\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\mathbb{H}$ part will be equivalent to $h_\mathbb{H}$. Thus, by Corollary \ref{c:lower z}, we have the desired lower bound for $E_\mathbb{C}$.
\end{proof}
\begin{proof} [Proof of Theorem \ref{t:mainthm2}]
Combine Lemma \ref{l:C0 bound}, Corollary \ref{c:imporved C0}, Lemma \ref{l:potential derivative upper bound}, Corollary \ref{c:Inoue C0}, Lemma \ref{l:potential derivative lower bound}, and Theorem \ref{t:Inoue metric lower bound}, the results follow.
\end{proof}

\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\"ahler-Ricci flow (GKRF) \cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\`ere type \cite{StreetsSTB}.

\begin{thm} \label{t:mainthm2} For generalized K\"ahler-Ricci flow on an Oeljeklaus-Toma manifold
\begin{enumerate}
\item The scalar potential $\phi$ satisfies
\begin{align*}
    - C \leq \phi \leq C e^{-t}(1 + t).
\end{align*}
\item Assuming there exists a Tricerri-type metric in $[\gw_0]$, we have
\begin{align*}
    - C e^{-t}(1 + t) \leq \phi \leq C e^{-t}(1 + t).
\end{align*}
\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:
\begin{align*}
    -C\leq \dot{\phi}\leq C.
\end{align*}
\item On Inoue surfaces of type $S_M$, we have:
\begin{align*}
        \omega(t)\geq C\omega_h(t),
\end{align*}
where $\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.
\end{enumerate}
\end{thm}

The proof of Theorem~\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”

The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”

\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \infty)$.,

Question: Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\omega_0$ satisfies $\partial\bar\partial\omega_0=0$)?

The pluriclosed flow with initial metric $g_0$ exists for all forward time; equivalently, its solution is defined on the entire interval $[0,\infty)$.

Choices:
B. The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\to\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$.
C. The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$.
D. There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$.
E. For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure.

adds asymptotic Gromov--Hausdorff convergence not asserted in the theorem

dropped the global-in-time conclusion $[0,\infty)$ to mere short-time existence on a maximal interval

replaces infinite-time existence by a uniform manifold-dependent finite existence time

confuses use of homogeneous background metrics in the proof with the actual initial-data/general-flow statement


Question: Let $(M^{2n},J)$ be an OeljeklausToma (OT) manifold, and let $g_0$ be a pluriclosed Hermitian metric on $M$ (equivalently, if $\omega_0$ is its associated $(1,1)$-form, then $\partial\bar\partial\omega_0=0$). Which existence statement holds for the pluriclosed flow on $M$ starting from $g_0$?

There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and this solution is defined for all times $t\in[0,\infty)$.

Choices:
B. There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and there is a maximal time $T<\infty$ such that the solution is defined precisely for $t\in[0,T)$.
C. There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and this solution is defined on some interval $t\in[0,T)$ for some $T>0$.
D. For every pluriclosed Hermitian metric $g_0$ on $M$, there exists a constant $T=T(M)>0$ such that the pluriclosed flow with initial condition $g(0)=g_0$ is defined for all times $t\in[0,T]$.
E. There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and this solution is unique and converges as $t\to\infty$ to a canonical homogeneous background metric on $M$.

infinite-time interval replaced by finite maximal existence time

dropped global-in-time conclusion, retaining only existence on some positive time interval

uniform dependence of existence time only on the manifold rather than global existence for each initial metric

imports convergence to a canonical homogeneous background metric from the proof strategy instead of the theorem statement