Polynomial convexity of compacts that lies in certain Levi-flat hypersurfaces in ${\mathbb{C}}^2$

In this paper, we first prove that the totally real discs lying in certain Levi flat hypersurfaces are polynomially convex. As applications we prove that the totally real discs lying in the boundary of certain polynomial polyhedra are polynomially convex. We also provide an if and only if condition for polynomial convexity of totally real discs lying in the boundary of Hartog’s triangle. We also provide sufficient conditions on general compact subsets lying on those hypersurfaces for polynomial convexity.