Polynomial convexity and non-polynomial convexity of totally real discs in $\mathbb{C}^2$

Abstract

We study the polynomial convexity of smooth totally real discs contained in the boundaries of strongly pseudoconvex domains in $\mathbb{C}^2$. Using Wermer's classical example, we construct a bounded domain with smooth strongly pseudoconvex boundary containing a non-polynomially convex totally real disc. We also construct a nonsingular Levi-flat hypersurface in a non-Runge domain containing such a disc. This provides a partial negative answer, to a conjecture proposed in an earlier work. Furthermore, we prove that the Wermer graph over the closed disc of radius $R$ is polynomially convex for every $R\in (0,1)$. Finally, we establish a sufficient condition for the polynomial convexity of totally real discs contained in the smoothly bounded strongly pseudoconvex domains in $\mathbb{C}^2$.