Lagrangian caps

TL;DR

This paper proves an h-principle for exact Lagrangian embeddings with concave Legendrian boundary, demonstrating the existence of embedded Lagrangian discs attached to a standard symplectic ball in higher dimensions.

Contribution

It establishes an h-principle for a new class of Lagrangian embeddings with boundary conditions, expanding understanding of symplectic topology in higher dimensions.

Findings

Existence of embedded Lagrangian discs attached to a symplectic ball

Proof of an h-principle for Lagrangian embeddings with Legendrian boundary

Construction of embeddings in standard symplectic space

Abstract

We establish an $h$-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball $B$ in the standard symplectic $\R^{2n}, 2n\geq 6$, there exists an embedded Lagrangian $n$-disc transversely attached to $B$ along its Legendrian boundary.