A Real-analytic Nonpolynomially Convex Isotropic Torus with no Attached Discs

We showbymeans of an example in that Gromov’s theoremon the presence of attached holomorphic discs for compact Lagrangianmanifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e., polynomial convexity.

Polynomial hull of a torus fibered over the circle

Given a 2-sheeted torus over the circle with winding number 1, we prove that its polynomial hull is a union of 2-sheeted holomorphic discs. Moreover, when the hull is non-degenerate its boundary is a Levi-flat solid torus foliated by such discs.