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Seminar

Symmetry in Geometry: Laplacian flow of homogeneous G2-structures and its solitons April 21, 2016 (01:30 PM PDT - 02:30 PM PDT)
Parent Program: --
Location: MSRI: Simons Auditorium
Speaker(s) Jorge Lauret (Universidad Nacional de Cordoba)
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We use the bracket flow/algebraic soliton approach to study the Laplacian flow of G2-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e. a G-invariant G2-structure on a homogeneous space G/K that flows by pull-back of automorphisms of G up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.

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