Surfaces without dominant morphism from affine plane

Let $X$ be a smooth contractible affine surface defined over the field of complex numbers $\mathbb{C}$. If $X$ is not isomorphic to the affine plane $\mathbb{C}^2$, then Kodaira logarithmic dimension of $X$ equals $1$ or $2$, which was proved by T. Fujita. It is interesting to know when $X$ admits dominant morphism from $\mathbb{C}^2$, since this surface $X$ have a lot in common with an affine plane.
The smooth contractible affine surfaces with Kodaira logarithmic dimension $1$ were described by T. Petrie and T. dom Dieck. In the talk we will prove that these surfaces do not admit a dominant morphism from an affine plane. Two different approaches to the proof will be shown.
The first approach is based on the fact that the images of any two of non-constant morphisms from a line to these surfaces coincide. The second approach gives us a way to prove that there is no dominant morphism from an affine space to a hypersurface, which is true for various types of hypersurfaces.
The talk is based on the papers of S. Kaliman and L. Makar-Limanov "On morphisms into contractible surfaces of Kodaira logarithmic dimension 1" and "Affine algebraic manifolds without dominant morphisms from Euclidean space".