It is a challenging task to verify condition i) in Theorem 1.1, i.e. to find the region of bounded curvature in a Ricci-flat Calabi-Yau manifold. If $(M,\omega,J,g,\Omega)$ is a K3-surface with Ricci-flat metric, it was shown in [8] that there are universal constants $C>0$, $\tau>0$, and a finite subset $\{p_{j}\}\subset M$, $1\leq j\leq\tau$, such that

$$\sup_{B_{g}(p,1)}|K_{g}|\leq C,$$

for any $p\in M\backslash\bigcup\limits_{1\leq j\leq\tau}B_{g}(p_{j},2)$. From the author’s knowledge, no such estimate for higher dimensional Calabi-Yau manifolds is known except some trivial cases, for example $K3\times T^{2}$.