Using $4D$ geometry, we can reduce the Laplacian flow in $G_2$ geometry to a flow in 3 or 4 dimensions. In this expository talk, I will describe Fine-Yao's recent work on a flow in 4 dimensions, which is related to a conjecture of Donaldson on hyperKaehler 4-manifolds. Time permitting, I will also describe a flow in 3 dimensions, which is connected to space-like submanifolds in Euclidean spaces with mixed signature.