6 problems

Let $X$ be a Stein manifold diffeomorphic to ${\mathbb C}^n$ and having the density property, meaning that the closure of the Lie algebra generated by the complete holomorphic vect…

Volume-form preservation conjecture. The density property does not hold for $(\mathbb{C}^*)^n$; more precisely, every holomorphic automorphism of $(\mathbb{C}^*)^n$ preserves…

For $n\geq 2$, consider the complex algebraic torus $({\mathbb C}^*)^n$ and the density property, meaning that complete holomorphic vector fields generate the relevant holomorphic…

Let $X$ be a complex manifold with the density property, and suppose that $X$ is diffeomorphic to $\mathbb{C}^n$. Density-property conjecture. Then $X$ should be biholomorphic to…

Assume that $Y$ is a Stein manifold with the density property, and choose a complete Riemannian metric $\mathfrak{g}$ on $Y$. Complete dense embedding conjecture. If $\dim Y\geq 3$…

Varolin–Tóth conjecture. Then $M$ is biholomorphic to $\mathbb{C}^n$.