The existence criterion of holomorphic discs for higher $A_{\mathrm{\infty }}$ operations via minimal discs

The main theorem of the paper provides an existence criterion of holomorphic discs for higher $A_{\mathrm{\infty }}$ operations. The key step is to show that if a minimal disc in a Kähler manifold with boundary in a sequence of Lagrangian submanifolds intersecting transversely such that its partial Maslov indices are either all no less than $1$ or all no larger than $-1$, then there is a holomorphic disc with the same image as this minimal disc. As a by-product, we show that all minimal discs in $\text{C}{\mathrm{P}}^m$ with boundary on $\mathbb{R}{\mathrm{P}}^m$ are holomorphic.