# Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

## TL;DR

This paper develops a method to solve the constraint equations for initial data sets of Lorentzian manifolds with lightlike parallel spinors, linking Riemannian metrics with parallel spinors to solutions of these equations.

## Contribution

It introduces a new approach to solve the constraint equations using curves in the moduli space of Riemannian metrics with parallel spinors.

## Key findings

- Any curve in the moduli space yields a solution on M×(a,b)
- Closed curves in the moduli space produce solutions on M×S^1
- Provides a constructive method to generate initial data sets

## Abstract

Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on $M$ with a parallel spinor gives rise to a solution of the constraint equations on $M\times (a,b)$ (resp. $M\times S^1$).