Uncollapsing highly collapsed G2 holonomy metrics. Structure of Collapsed Special Holonomy Spaces, Simons Collaboration Meeting.. Duke University. April 9, 2018 - April 13, 2018

International Meeting or Conference

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

  • April 9, 2018 - April 13, 2018

Service or Event Name

  • Structure of Collapsed Special Holonomy Spaces, Simons Collaboration Meeting.

Host Organization

  • Duke University

Location or Venue

  • Durham NC