both $\operatorname{Hol}^0_p(M, g)$ and $\operatorname{Hol}_p(E, g)$ leave $\hat{J}$ invariant. Therefore $P_{\beta} \hat{J} P_{\beta}^{-1} = P_{\alpha} \hat{J} P_{\alpha}^{-1}= \hat{J}$, and so $P_{\gamma}\hat{J}P_{\gamma}^{-1} = P_{\sigma}\hat{J}P_{\sigma}^{-1}$ as desired. Of course, the restriction of $J_E$ to any path contained in $E$ must coincide with the parallel translation of $\hat{J}$ along that path, so the connectedness of $E$ implies that $J|_E = J_E$.

We now give an ad hoc argument to show that the Kähler structure extends to the entire manifold. The key ingredient is the result of Munteanu-Wang [@MunteanuWangKaehler] which guarantees that any complete Kähler shrinking gradient Ricci soliton is connected at infinity. We apply this to the universal cover of $M$ to determine that the preimage of any end in $M$ is connected in the universal cover.

\[lem:globallyk\] Suppose $(M, g, f)$ is a complete noncompact gradient Ricci soliton and $g|_E$ is Kähler on some end $E\subset M$ with complex structure $J_E$. Then $J_E$ extends uniquely to a complex structure $J$ on $M$ relative to which $g$ is Kähler.

The soliton structure $(M, g, f)$ lifts to a complete soliton structure $(\tilde{M}, \tilde{g}, \tilde{f})$ on the universal cover $\pi:\tilde{M}\to M$. Fix $p\in E$ and a local section $\sigma: U\to \tilde{M}$ of $\pi$ over 