Abundance of affine algebraic structures on stabilized cotangent bundles of surfaces

We show that for any $g\ge 2$, there exist uncountably many non-isomorphic smooth affine threefolds that are Stein deformation equivalent to $T^{\ast }{\mathrm{\Sigma }}_g\times \mathbb{C}$, the product of the cotangent bundle of a genus $g$ oriented surface with the complex plane, answering a question of Ivan Smith. In fact, we prove that our family of smooth affine threefolds is parametrized by a $(6g-7)$-dimensional complex orbifold.