A nonpolynomially convex isotropic two-torus with no attached discs

TL;DR

This paper provides a real-analytic example demonstrating that Gromov's theorem on attached holomorphic discs does not hold for isotropic (subcritical) Lagrangian manifolds, challenging previous assumptions about polynomial convexity.

Contribution

It constructs a specific example of an isotropic two-torus in b^3 that lacks attached holomorphic discs, showing the theorem's limitations in the subcritical case.

Findings

Gromov's theorem does not extend to isotropic subcritical manifolds

Existence of a nonpolynomially convex isotropic two-torus without attached discs

Counterexample in b^3 using real-analytic methods

Abstract

We show --- with the means of a real-analytic example in $\mathbb{C}^3$ --- that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the isotropic (subcritical) case, even in the absence of an obvious obstruction, i.e, polynomial convexity.