to the conical (Kähler) metric on some neighborhood of infinity. This transforms the essentially elliptic problem of unique continuation at infinity we are initially given into a parabolic problem of backward uniqueness on a finite time interval.

As in the introduction, let $(\Sigma, g_{\Sigma})$ denote a compact Riemannian manifold of dimension $n-1$, and let $g_c = dr^2 + r^2g_{\Sigma}$. Denote by $r_c: E_0 \to{\mathbb{R}}$ the radial distance $r_c(r, \sigma) = r$ relative to $g_c$.

\[prop:brf\] Suppose $(M, \bar{g}, \bar{f})$ is a shrinking Ricci soliton asymptotic to the regular cone $(E_0, g_c)$ along the end $V\subset M$. Then there exist $K_0$, $N_0$, and ${R}_0> 0$, and a smooth family of maps $\Psi_\tau: E_{{R}_0}\to V$ defined for $\tau\in (0,1]$ satisfying:

1.  For each $\tau\in (0, 1]$, $\Psi_{\tau}$ is a diffeomorphism onto its image and $\Psi_{\tau}(E_{{R}_0})$ is an end of $V$.

2.  The family of metrics $g(x, \tau) {\doteqdot}\tau\Psi_\tau^*\bar{g}(x)$ is a solution to the backwards Ricci flow for $\tau \in (0, 1]$, and converges smoothly as $\tau\searrow 0$ to $g(x, 0) \equiv g_c(x)$ on $E_{{R}_0}$.

3.  For all $m = 0, 1, 2, \ldots $, $$\begin{aligned}
    \label{eq:curvdecay}
    \sup_{E_{{R}_0}\times [0,1]} \left(r_c^{m+2}+1\right){\left\vert\nabla^{(m)}R\right\vert} & \le K_0.\end{aligned}$$ Here $|\cdot| = |\cdot|_{g(\tau)}$ and $\nabla = \nabla_{g(\tau)}$ denote the 