Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

Abstract

We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighborhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed by Eschenburg and Wang. As a by-product of these methods, we find many new examples of Einstein metrics of cohomogeneity one.

46 pages

References in corpus (3)

• Special cohomogeneity one metrics with Q^111 or M^110 as principal orbit
• The $Spin(7)$-structures on complex line bundles and explicit Riemannian metrics with SU(4)-holonomy
• Eight-dimensional SU(3)-manifolds of cohomogeneity one

Cited by in corpus (9)

• Asymptotically conical Calabi-Yau manifolds, I
• Einstein Metrics from Symmetry and Bundle Constructions: A Sequel
• Half-flat structures on S^3xS^3
• Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds
• Einstein metrics of cohomogeneity one with $S^{4m+3}$ as principal orbit
• SU(4)-holonomy via the left-invariant hypo and Hitchin flow
• Linear perturbations of metrics with holonomy Spin(7)
• About left-invariant geometry and homogeneous pseudo-Riemannian Einstein structures on the Lie group SU(3)
• Exceptional holonomy on vector bundles with two-dimensional fibers