A Liouville theorem for some asymptotically conical Calabi-Yau manifolds

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 Kähler manifold. We show that, if there exists a diffeomorphism $\mathrm{\Phi }:\mathcal{C}\setminus \overline{B_1(o)}\to M\setminus K$, for some compact $K\subset M$, such that ${\mathrm{\Phi }}^{\ast }J$ is asymptotic to $J_{\mathcal{C}}$ and $C^{-1}{\omega }_{\mathcal{C}}\le {\mathrm{\Phi }}^{\ast }\omega \le C{\omega }_{\mathcal{C}}$ for some $C\ge 1$, 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 Kähler metric on $T^{\ast }{S}^n$ which is quasi-isometric to the Stenzel metric must be equal to the Stenzel metric up to scaling and diffeomorphism. Similarly, any Ricci-flat Kähler metric on ${\mathcal{O}}_{{\mathbb{P}}^1}(-1{)}^{\oplus 2}$ 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.