Invariant Calabi-Yau structures on punctured complexified symmetric spaces

In this paper, we show that $G$-invariant Calabi-Yau structures on the complexification $G^{\mathbb{C}}\mathrm{/}{K}^{\mathbb{C}}$ of a symmetric space $G\mathrm{/}K$ of compact type are constructed from solutions of a Monge-Amp$\grave{\mathrm{e}}$re type equation. Also, we give an explicit descriptions of the Monge-Amp$\grave{\mathrm{e}}$re type equation in the case where the rank of $G\mathrm{/}K$ is equal to one or two. Furthermore, we prove the existence of solutions of the Monge-Amp$\grave{\mathrm{e}}$re type equation.