New cohomogeneity one metrics with Spin(7) holonomy
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We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A8, is complete and non-singular on 8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S4, and are denoted by B8+, B8- and B8. The metrics on B8+ and B8- occur in families with a non-trivial parameter. The metric on B8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A8. The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP3. We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L2-normalisable harmonic 4-form for the A8 manifold, and two such 4-forms (of opposite dualities) for the B8 manifold. 2003 Elsevier B.V. All rights reserved.