considered here (and in fact are sections of different bundles), the structural assumptions in Theorem \[thm:bu\] are the only properties of the system used in the application of the Carleman inequalities. But for the labeling of some constants, the argument which follows equations (6.1) and (6.2) in [@KotschwarWangConical] can be used here without change.

Once we know that ${\hat{{\operatorname{R}}}}$ and $\nabla {\hat{{\operatorname{P}}}}$ vanish on $E_{R_2}\times[0, \tau_0]$ for some $R_2$ and $\tau_0 > 0$, we can apply Lemma \[lem:jext\] to conclude that $g(\tau)$ is Kähler with respect to $J_c$ on the same set. Fix $s\in (0, \tau_0]$ and let $W = \Psi_s(E_{R_2})$, and $\Phi = \Psi^{-1}_s: W\to E_{R_2}$, where $\Psi_s: E_{R_2}\to V$ is the map from Proposition \[prop:brf\]. Then $\bar{g}|_W = s^{-1}\Phi^*(g(s))$ is Kähler relative to $J = \Phi^*J_c$, and $\Phi$ is the desired biholomorphism.

Global Kählerity in the complete case.
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To finish the proof of Theorem \[thm:kaehlerity\], we now argue that the Kähler structure defined on the end $W\subset M$ must extend to all of $M$ when $(M, g)$ is complete. The real-analyticity of Ricci solitons provides almost all that we need.

\[lem:genext\] Suppose $(M, g)$ is a complete connected real-analytic manifold, $E\subset M$ is open and connected, and 