A Liouville theorem for some asymptotically conical Calabi-Yau manifolds

Abstract

Let $(\mathcal{C}, J_{\mathcal{C}}, \omega_{\mathcal{C}}, g_{\mathcal{C}})$ be a Calabi-Yau cone and $(M, J, \omega, g)$ an open Ricci-flat K\"ahler manifold. We show that, if there exists a diffeomorphism $\Phi: \mathcal{C} \setminus \overline{B_1(o)} \rightarrow M \setminus K$, for some compact $K \subset M$, such that $\Phi^{*}J$ is asymptotic to $J_{\mathcal{C}}$ and $C^{-1} \omega_{\mathcal{C}} \leq \Phi^{*} \omega \leq C \omega_{\mathcal{C}}$ for some $C \geq 1$, then $(M, g)$ is asymptotically conical (AC) with tangent cone at infinity given by $(\mathcal{C}, d_{g_{\mathcal{C}}})$. As a consequence, we obtain that any Ricci-flat K\"ahler metric on $T^{*}S^n$ which is quasi-isometric to the Stenzel metric must be equal to the Stenzel metric up to scaling and diffeomorphism. Similarly, any Ricci-flat K\"ahler metric on $\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus2}$ which is quasi-isometric to the Candelas-De la Ossa metric must be equal to the Candelas-De la Ossa metric up to scaling and diffeomorphism. This provides new examples of complete Calabi-Yau manifolds for which a Liouville-type theroem holds.