Solitons for the Closed G2 Laplacian Flow

Abstract

Geometric flows are partial differential equations on smooth manifolds which describe the time evolution of some geometric structure on the manifold, such as a Riemannian metric. In the setting of G2 geometry, which is specific to seven dimensions, a natural geometric flow to consider is the (closed) G2 Laplacian flow. Critical points of this flow correspond to torsion-free G2 structures, which satisfy a system of nonlinear partial differential equations. We are interested in such G2 structures since they give rise to Riemannian metrics with exceptional holonomy. Solitons for the Laplacian flow are self-similar solutions which we hope provide models for finite-time singularities of the flow. Since the soliton equation is a nonlinear system of PDEs, to get a feel for concrete solutions we consider a dimensional reduction of the equation by studying cohomogeneity-one solitons. We consider the cohomogeneity-one setting where the group G of isometries is either SU(3) or Sp(2). In these regimes, we show that the soliton equation is equivalent to a (nonlinear) first-order system of real-analytic ODEs. We then describe the space of G-invariant closed G2 cones, identifying it with a smooth surface/curve in R3. We then investigate the uniqueness of a complete shrinking Sp(2)-invariant soliton which possesses an asymptotically conical geometry, and conjecture that it is rigid to first order. We finally discuss ways to study the general closed Laplacian soliton equation, reviewing previous results by others in related directions, with an eye towards asymptotically conical shrinking solitons in particular.