some connected evenly covered neighborhood $U$ of $p$ contained in $E$. Then $J_{\tilde{U}} = (\sigma^{-1})^*J_E$ is a parallel complex structure on $\tilde{U} = \sigma(U)$ relative to which $\tilde{g}|_{\tilde{U}}$ is Hermitian. As Ricci solitons, both $(M, g)$ and $(\tilde{M}, \tilde{g})$ are real-analytic manifolds [@IveySolitons]. The simple-connectivity of $\tilde{M}$ then implies (as above) that $J_{\tilde{U}}$ extends to a parallel complex structure $\tilde{J}$ on $\tilde{M}$ relative to which $\tilde{g}$ is Hermitian [@Nomizu]. So $(\tilde{M}, \tilde{g}, \tilde{f})$ is a complete Kähler gradient Ricci soliton and $\tilde{M}$ must therefore be connected at infinity [@MunteanuWangKaehler]. But the fundamental group of a complete shrinking soliton is finite [@Wylie], so $\pi:\tilde{M}\to M$ is proper, and $M$ must also be connected at infinity. So we may assume that $E = M\setminus D$ for some compact $D\subset M$.

Now we claim that $\tilde{E} = \pi^{-1}(E)$ is connected in $\tilde{M}$. Since $\pi$ is proper, $\tilde{D} = \pi^{-1}(D)$ is compact. The set $\tilde{E} = \tilde{M}\setminus \tilde{D}$ therefore must have a unique unbounded connected component $\tilde{C}$. However, any two of these connected components must be isometric, so every component of $\tilde{E}$ would have to be unbounded as well. So $\tilde{E} =\tilde{C}$ is connected. This implies that $\pi_1(E, p)\hookrightarrow \pi_1(M, p)$ is surjective, and 