Holomorphic disks with boundary on compact Lagrangian surface
Jingyi Chen
TL;DR
This work proves that in a Kähler surface with homogeneous regularity, every nontrivial loop class in a compact Lagrangian $L$ that bounds a disk in the ambient manifold is realized by the boundary of a nonconstant holomorphic disk with boundary on $L$, orthogonal to $L$ along the boundary and with nonzero boundary Maslov index. The key mechanism combines area-minimizing disks with free boundary and holomorphic variation theory (a Riemann–Roch type analysis for bundle pairs) to extend Gromov’s disk existence to general $(L,M)$ and to derive consequences for exact Lagrangian embeddings and almost Kähler/cotangent-bundle contexts. The authors then develop a Lichnerowicz-type energy homotopy framework for almost Kähler manifolds, showing that the partial energy difference is invariant under boundary-preserving homotopies and that, when the fundamental form is exact, the partial energies equilibrate, yielding nonexistence results for nonconstant $J$-holomorphic maps into $(M,L)$. This framework is applied to cotangent bundles $T^*M$, where a complete homogeneous-regular metric is constructed via a conformal change, enabling free minimal disks with boundary on exact Lagrangians and clarifying the role of the Sasakian almost complex structure; in the compact case the projection $L\to M$ is a simple homotopy equivalence, aligning with related Floer-theoretic results. Overall, the paper connects minimal surface techniques with $J$-holomorphic disk existence, providing new instances of Bennequin’s question, extending Gromov’s disk theorems, and offering tools for understanding exact Lagrangian embeddings in almost Kähler and cotangent-bundle settings.
Abstract
Let $L$ be a compact oriented Lagrangian surface in a Kähler 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 $[γ]$ of the fundamental group $π_1(L)$ such that $γ$ bounds a topological disk in $M$, there exists a holomorphic disk whose boundary belongs to $L$ and is freely homotopic to $γ$ 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 Kähler surface if the fundamental form is exact. In the almost Kähler 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.