Parallel spinor flows on three-dimensional Cauchy hypersurfaces

Abstract

The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian metrics admitting parallel spinors are not necessarily Ricci flat, the parallel spinor flow preserves the vacuum momentum and Hamiltonian constraints and therefore the Einstein and parallel spinor flows coincide on common initial data. Using this result, we provide an initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds. Furthermore, we explicitly solve the left-invariant parallel spinor flow on simply connected Lie groups, obtaining along the way necessary and sufficient conditions for the flow to be immortal. These are, to the best of our knowledge, the first non-trivial examples of evolution flows of parallel spinors. Finally, we use some of these examples to construct families of η -Einstein cosymplectic structures and to produce solutions to the left-invariant Ricci flow in three dimensions. This suggests the intriguing possibility of using first-order hyperbolic spinorial flows to construct special solutions of curvature flows.

Spinor flows with flux, I: short-time existence and smoothing estimates

“…We end the introduction by noting that in an interesting different direction, parallel spinors on globally hyperbolic Lorentzian four-manifolds have recently been studied by Murcia-Shahbazi [29,30] building on work of Cortés-Lazaroiu-Shahbazi [9]. In particular, in [29] the authors show that the existence of a globally defined parallel spinor in a globally hyperbolic Lorentzian 4 manifold induces a solution of a hyperbolic flow called the "parallel spinor flow" on a Cauchy hypersurface, and vice versa.…”