Complete Hyperk Ahler 4n-manifolds with a Local Tri-hamiltonian R N -action

We classify those manifolds mentioned in the title which have nite topological type. Namely we show that any such connected M 4n is isomorphic to a hyperkk ahler quotient of a at quaternionic vector space H d by an abelian group. We also show that a compact connected and simply connected 3-Sasakian manifold of dimension 4n ? 1 whose isometry group has rank n + 1 is isometric to a 3-Sasakian quotient of a sphere by a torus. As a corollary, a compact connected quaternion-KK ahler 4n-manifold with positive scalar curvature and isometry group of rank n + 1 is isometric to H P n or Gr 2 (C n+2). Hyperkk ahler metrics in dimension 4n with n commuting tri-holomorphic Killing vector elds form a large class of explicitly known Ricci-at metrics. The at H n , the Taub-NUT, and the Eguchi-Hanson metric are of this form. So are the multi-Eguchi-Hanson metrics (a.k.a. ALE-spaces of type A k) and their Taub-NUT-like deformations, both due to Gibbons and Hawking. Higher dimensional examples include the Calabi metrics on T C P n and the asymptotic metrics on moduli spaces of magnetic monopoles. A powerful method of constructing such metrics was given by Lindstrr om and Ro cek 17] (see also 14, 18]). It associates a hyperkk ahler metric having a local tri-Hamiltonian (meaning Hamiltonian for all three KK ahler forms) action of R n to every real-valued function on an open subset of R 3 R n which is polyharmonic, i.e. harmonic on any aane 3-dimensional subspace of the form a+R 3 Rv, a 2 R 3 R n , v 2 R n. This construction gives locally all hyperkk ahler metric with a local eeective tri-Hamiltonian (hence isometric) action of R n. On the other hand, the global properties of the resulting metrics-in particular completeness-have not been investigated. There is another construction-that of hyperkk ahler quotient-which, while in this case more restrictive, has the advantage that the global properties of the resulting manifold are accessible. In particular, a hyperkk ahler quotient of a complete manifold by a compact Lie group is complete. A basic example is the hyperkk ahler quotient of H (S 1 R 3) by the diagonal circle action which yields the Taub-NUT metric. More generally, we obtain complete hyperkk ahler 4n-manifolds with a tri-Hamiltonian (hence isometric) T n-action by taking hyperkk ahler quotients of quaternionic vector spaces by …