M. Miyanishi [Osaka J. Math. 19, 901–921 (1982; Zbl 0534.14025)] proved that for every non trivial algebraic action of the additive group \(\mathbb C_+\) on the affine \(3\)-space \(\mathbb C^3\), the algebra of invariant functions is isomorphic to a polynomial ring in two variables. This means equivalently that the corresponding algebraic quotient \(\mathbb C^3/\!/\mathbb C_+\) is isomorphic to the affine plane \(\mathbb C^2\).
In this article, the authors generalize this result to additive group actions on arbitrary smooth contractible affine algebraic threefolds. They establish that for a nontrivial algebraic \(\mathbb C_+\)-action on such a threefold \(X\), the algebraic quotient \(S=X/\!/\mathbb C_+\) is a smooth contractible surface. All such surfaces are rational [R. V. Gurjar and A. R. Shastri, J. Math. Soc. Japan 41, No. 2, 175–212 (1989; Zbl 0687.14031)]. This implies in particular that \(X\) is rational as well since there exists an open subset of \(S\) over which the quotient morphism \(\rho:X\rightarrow S\) restricts to the structural morphism of a principal homogeneous \(\mathbb C_+\)-bundle. As a consequence of the smoothness of the quotient surface \(S\), the authors conclude that if a smooth contractible affine threefold \(X\) admits a free \(\mathbb C_+\)-action with algebraic quotient \(S=X/\!/\mathbb C_+\), then \(X\) is equivariantly isomorphic to \(S\times\mathbb C\) where \(\mathbb C_+\) acts by translations on the second factor. Indeed, this result has been previously established by the first author [Invent. Math. 156, No. 1, 163–173 (2004; Zbl 1058.14076)] under the additional assumption that the quotient \(S\) is smooth. Finally, it is shown in this paper that if a smooth contractible affine threefold \(X\) equipped with a nontrivial \(\mathbb C_+\)-action admits a dominant morphism from a threefold of the form \(C\times\mathbb C^2\), then the quotient surface \(S=X/\!/\mathbb C_+\) is isomorphic to \(\mathbb C^2\).