Parallel spinor flows on three-dimensional Cauchy hypersurfaces

TL;DR

This paper studies the evolution of parallel spinors on three-dimensional hypersurfaces within Lorentzian four-manifolds, showing that the flow preserves key constraints and connecting it to Ricci-flat geometries and curvature flows.

Contribution

It introduces the first explicit solutions of parallel spinor flows, characterizes initial data for Ricci-flat spacetimes, and explores applications to curvature flows and geometric structures.

Findings

Flow preserves vacuum constraints despite non-Ricci-flat Lorentzian metrics.

Provides initial data characterization of Ricci-flat Lorentzian manifolds and pp-waves.

Constructs explicit solutions and applications to curvature flows and geometric structures.

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, which in turn yields the first initial data characterization of Ricci-flat pp-waves. 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 $\eta\,$-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 in Riemannian signature.