uld include another suitable criteria in general case which is still an active area of research.

The a priori estimates
----------------------

We are going to derive the a priori estimates for the geometric quantities. We fix a small constant $\delta>0$ and a constant $u_1\in(u_0,0)$ and denote $$\begin{aligned}
\mathscr{F}=\mathscr{F}(u_0,u_1)=\max\{1,\sup_{u_0\le u\le u_1}|\varphi(u)|\}.\end{aligned}$$ Without loss of generality, we also assume that $\Omega(u_0)\le1$. By the monotonicity of $\Omega_0$, we have $\Omega_0(u)\le1$ for all $u\in[u_0,0)$. Then we are going to prove

\[estimate\] There exists a universal large constant $C_0\ge1$ such that the following statement is true. Suppose that $$\begin{aligned}
\mathcal{A}=\mathcal{A}(\delta,u_0,u_1)=\max\{1,\sup_{0\le\ub\le\delta}\mathscr{F}^{-1}(|rL\phi(\ub,u_0)|+|u_0||\omega(\ub,u_0)|)\}<+\infty,\end{aligned}$$ and for some $C\ge C_0$ we have $$\begin{aligned}
\label{smallness}
C^2\delta|u_1|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\le1,\ \text{where}\ \mathscr{W}=\mathscr{W}(u_0,u_1)=\max\left\{1,\left|\log\frac{\Omega_0(u_1)}{\Omega_0(u_0)}\right|\right\}.\end{aligned}$$ Then we have the following estimates for $0\le\ub\le\delta$, $u_0\le u\le u_1$:[^3] $$\begin{aligned}
\label{geometryestimate}\frac{1}{2}\Omega_0\le\Omega\le 2\Omega_0,&\ \frac{1}{2}|u|\le r\le 2|u|,\\
\label{estimate-Lphi}|rL\phi|\lesssim&\mathscr{F}\mathcal{A},\\
\label{estimate-Lbphi}|r\Lb\phi-\psi|\lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A},\\
\label{estimate-h}|h-h_0|\lesssim&\delta|u|^{-1}\Omega_0^{-2}\mathscr{F}^2\mathcal{A}^2,\\
\label{estimate-hb}|{\underline{h}}+1|\lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A},\\
\label{estimate-omega}|u||\omega|\lesssim&\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{aligned}$$ Moreover, we have the improved estimate $$\begin{aligned}
\label{estimate-Lphiimp}
|rL\phi(\ub,u)-\varphi(u)|\lesssim|rL\phi(\ub,u_0)-\varphi(u_0)|+\delta|u|^{-1}\mathscr{F}^2\mathscr{W}^{\frac{1}{2}}\mathcal{A}^2.\end{aligned}$$

We begin the proof by