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

Abstract

The main theorem of the paper provides an existence criterion of holomorphic discs for higher $A_{\infty }$ 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 $\C\mathrm{P}^m$ with boundary on $\mathbb{R}{\mathrm{P}}^m$ are holomorphic.

19 pages, 1 figure