Flows of G2-structures
Sergey Grigorian (Edinburg, Texas, USA):
Abstract:
G2-structures on 7-dimensional manifolds play a very important role in both
geometry and physics. One of the ways of better understanding the relationships
between different types of G2-structures is to study their flows. In this talk, we
will consider Laplacian flows of either closed or co-closed G2-structures.
Since the Laplacian is itself determined by the underlying G2-structure,
these flows give rise to non-linear partial differential equations.
We will show that these flows share many similarities, such as the corresponding
flow of the associated metric being equal to the Ricci flow to the leading order,
but also some major differences. It turns out that unlike the flow of closed G2-structures,
the Laplacian flow of co-closed G2-structures is not even weakly parabolic.
We then show that this flow can be modified to make it weakly parabolic at least in certain
directions and prove short-time existence and uniqueness of solutions for this new flow.