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

TL;DR

This paper establishes an existence criterion for holomorphic discs related to higher $A_$ operations in Ke4hler manifolds, linking minimal discs with holomorphic ones under specific boundary and index conditions.

Contribution

It provides a new criterion connecting minimal discs and holomorphic discs in Ke4hler manifolds, especially with boundary on Lagrangian submanifolds.

Findings

Minimal discs with boundary on Lagrangian submanifolds under certain Maslov index conditions are holomorphic.

All minimal discs in m ext{P}^m with boundary on m ext{P}^mm ext{P}^m are holomorphic.

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 K\"ahler 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 $\R\mathrm{P}^m$ are holomorphic.