Solitons for the closed G2 Laplacian flow in the cohomogeneity-one setting

Introduction Geometric flows, such as the Ricci flow, are partial differential equations on smooth manifolds which describe the time evolution of some geometric structure on the manifold, such as a Riemannian metric. Many geometric flows are heuristically similar to the heat equation as they are often parabolic in some suitable interpretation, and hence can be thought of as deforming an initial geometric structure to one that is “nicer” in some sense. Unlike strictly parabolic linear equations like the heat equation, the nonlinearity of many geometric flows frequently leads to the formation of singularities in finite time. To understand these singularities, we study special, self-similar solutions called solitons, with the expectation that developing singularities of the flow are modeled on these soliton solutions.