norm and the Levi-Civita connection associated to the metric $g = g(\tau)$.

4.  If $f$ is the function on $E_{{R}_0}\times (0,1]$ defined by $f(\tau)=\Psi_\tau^\ast \bar{f}$, then $\tau f$ extends to a smooth function on all of ${E}_{{R}_0}^1$ and there $g$ and $\tau f$ together satisfy $$\begin{aligned}
    \label{eq:fid0}
    &  \lim_{\tau\searrow 0} 4\tau f(x, \tau) =r_c^2(x),\quad r_c^2-\frac{N_0}{r_c^{2}} \le 4 \tau f \le r_c^2 + \frac{N_0}{r_c^{2}}, \end{aligned}$$ and $$\begin{aligned}
    \label{eq:fid1}
    & {{\frac{\partial }{\partial\tau}}}(\tau f) =\tau S,  \  \tau^2|\nabla f|^2 - \tau f = -\tau^2 S, \ \tau {\operatorname{Rc}}(g) + \tau \nabla\nabla f= \frac{g}{2}.\end{aligned}$$

Here $S$ denotes the scalar curvature of $g$.

We apply the above proposition to a shrinker $(M, \bar{g}, \bar{f})$ asymptotic to a Kähler cone $(E_0, g_c)$ along the end $V\subset M$ as in Theorem \[thm:kaehlerity\], and assume for the rest of the section that $g = g(\tau)$ is the smooth family of metrics on $E_{R_0}\times [0, 1]$ it provides. Let $J_c$ denote the complex structure associated to $(E_0, g_c)$, and let $J= J(\tau)$ denote the family of almost-complex structures obtained as solutions to the fiberwise-ODE . Then define ${\hat{{\operatorname{P}}}}$, ${\operatorname{A}}$, ${\operatorname{B}}$, ${\hat{{\operatorname{R}}}}$, and ${\hat{{\operatorname{S}}}}$ as above, in terms of the solution $g(\tau)$ and let $$\begin{aligned}
 {\mathbf{X}}(\tau) &= {\hat{{\operatorname{R}}}}(\tau)\oplus {\hat{{\operatorname{S}}}}(\tau) \in 