Surveys in Differential Geometry
Volume 24 (2019)
Associative submanifolds and gradient cycles
Pages: 39 – 65
We discuss a model for associative submanifolds in $G_2$ manifolds with $K3$ fibrations, in the adiabatic limit. The model involves graphs in a $3$‑manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds. We propose conjectures on the existence of associative and special Lagrangian submanifolds in certain product spaces, corresponding to the vertices of the graphs.
Published 29 December 2021