Holomorphic disks with boundary on compact Lagrangian surface

TL;DR

This paper proves the existence of holomorphic disks with boundaries on compact Lagrangian surfaces in certain Kähler and almost Kähler manifolds, answering a longstanding question and establishing nonexistence results for exact embeddings.

Contribution

It demonstrates the existence of holomorphic disks bounded by Lagrangian surfaces in specific geometric settings and addresses a question posed by Bennequin.

Findings

Existence of holomorphic disks for certain fundamental group classes.

Nonexistence of exact Lagrangian embeddings in specific Kähler surfaces.

Results on minimizers of partial energies in cotangent bundles.

Abstract

Let $L$ be a compact oriented Lagrangian surface in a K\"ahler surface endowed with a complete Riemannian metric (compatible with the symplectic structure and the complex structure) with bounded sectional curvatures and a positive lower bound on injectivity radius. We show that for every nontrivial class $[\gamma]$ of the fundamental group $\pi_1(L)$ such that $\gamma$ bounds a topological disk in $M$, there exists a holomorphic disk whose boundary belongs to $L$ and is freely homotopic to $\gamma$ on $L$. This answers a question of Bennequin on existence of $J$-holomorphic disks. Nonexistence of exact Lagrangian embeddings of certain surfaces is established in such K\"ahler surface if the fundamental form is exact. In the almost K\"ahler setting, especially, the cotangent bundles of compact manifolds, results on nonexistence of $J$-holomorphic disks and existence of minimizers of the partial energies in the sense of A. Lichnerowicz are obtained.