Surveys in Differential Geometry

Volume 24 (2019)

Associative submanifolds and gradient cycles

Pages: 39 – 65

DOI:  https://dx.doi.org/10.4310/SDG.2019.v24.n1.a2

Authors

Simon Donaldson (Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York, U.S.A.; and Department of Mathematics, Imperial College London, United Kingdom)

Christopher Scaduto (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Abstract

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