Uncollapsing highly collapsed $G_2$ holonomy metrics.. Oxford University Geometry seminar. Oxford University . October 22, 2018 - October 22, 2018

Invited Talk

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$ holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical (ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture. We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

Service Performed By

Date

  • October 22, 2018

Service or Event Name

  • Oxford University Geometry seminar

Host Organization

  • Oxford University

Location or Venue

  • Mathematical Institute, Oxford