The Euclidean section of the classical Lorentzian black-hole solutions has been used in approximating the functional integral in the Euclidean path-integral approach to quantum gravity. In this paper the claim that classical black-hole uniqueness theorems apply to the Euclidean section is disproved. In particular, it is shown that although a Euclidean version of Israel's theorem does provide a type of uniqueness theorem for the Euclidean Schwarzschild solution, a Euclidean version of Robinson's theorem does not allow one to form conclusions about the uniqueness of the Euclidean Kerr solution. Despite the failure of uniqueness theorems, "no-hair" theorems are shown to exist. Implications are discussed. A precise mathematical statement of the Euclidean black-hole uniqueness conjecture is made and the proof left as an unsolved problem in Riemannian geometry.

• Received 29 May 1980

DOI:https://doi.org/10.1103/PhysRevD.22.1837