An inhomogeneous nearly parallel G₂-structure on the Berger space and a sine cone desingularisation

Ragini Singhal - Münster

Resumo:

Nearly G2 manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, Spin(7). We study nearly parallel G2-structures invariant under a cohomogeneity-one action of SO(4), and construct the first inhomogeneous nearly parallel G₂-structure on the Berger space SO(5)/SO(3). The cone over this metric has full holonomy Spin(7). The existence of the solution is established by a rigorous computer-assisted shooting argument. We also establish a local desingularisation of a finite quotient of the sine cone over S³ x S³ by gluing in the asymptotically conical C₇ bubble constructed by Foscolo-Haskins-Nordström. We show that as the singular orbit of a family of nearly parallel G₂ manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free G₂-metric of C₇. The talk is based on a joint work with Simon Salamon.

© Ragini Singhal.