12 oct 2022 -- 14:00
Aula Seminari, DISMA, Politecnico di Torino
Differential Geometry Seminar Torino
Abstract.
G2 structures arise naturally when a 7-dimensional Riemannian manifold M carries a distinguished positive differential 3-form φ. A G2-structure endows the manifold with an orientation and Riemannian metric g. (Thus, the pointwise automorphism group of φ is isomorphic to a compact subgroup of SO(7).) A G2-structure is torsion-free if φ is parallel with respect to the Levi-Civita connection of g, in which case (M,φ) is called a G2-manifold. Non-trivial G2 manifolds, which have reduced holonomy, are difficult to construct. Inspired by the Ricci flow, several authors have studied evolution equations for G2 structures, with the hope that these limit to closed or co-closed structures.
This talk concerns a flow on isometric G2-structures which is the negative gradient flow for total torsion. (Two G2-structures are said to be isometric if they induce the same metric g.) The flow was recently introduced by Dwivedi, Giannotis and Karigiannis, who proved long-time existence and convergence to a structure with divergence-free torsion or to a shrinking soliton. In this talk, we construct soliton structures isometric to a fixed torsion-free background such as the canonical flat G2-structure on R7 or the product of a Calabi-Yau 3-fold with a circle. The restriction of the flow to structures like these, which have a high degree of symmetry, provides important test cases for convergence. This is joint work with Spiro Karigiannis.
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If you would like to attend the talk remotely, please send an e-mail to dgseminar.torino@gmail.com