Amanda Petcu, Department of Pure Mathematics, University of Waterloo
"Partial progress on a conjecture of Donaldson by Fine and Yao (Part 2)"
Given a compact hypersymplectic manifold $X^4$, Donaldson conjectured that the hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperk\"{a}hler structure. Fine and Yao consider a manifold with closed $G_2$-structure that is set up as $\mathbb{T}^3 \times X^4$. They examine the $G_2$-Laplacian flow under in this setting and give a flow of hypersymplectic structures which evolve according to the equation
\[\partial_t \underline{\omega} = d(Q d^*(Q^{-1} \underline{\omega}))\]
where $\underline{\omega}$ is the triple that gives the hypersymplectic structure and $Q$ is a $3 \times 3$ symmetric matrix that relates the symplectic forms $\omega_i$ to one another. Lotay-Wei have established long time existence of the $G_2$-Laplacian flow provided the velocity of the flow remains bounded. Fine-Yao use this extension theorem in their setup and manage to improve it by proving long time existence of the hypersymplectic flow provided the torsion tensor $T$ remains bounded. Furthermore, one can relate the scalar curvature and torsion tensor of manifold with closed $G_2$-structure and thus they conclude long time existence for the hypersymplectic flow provided the scalar curvature remains bounded. In this talk we will go over some details from this paper by Fine-Yao.
MC 5403