Conformal deformation of a Riemannian metric via an Einstein-Dirac parabolic flow

We introduce a new parabolic flow deforming any Riemannian metric on a spin manifold by following a constrained gradient flow of the total scalar curvature. This flow is built out of the well-known Dirac-Einstein functional. We prove local well-posedness of smooth solutions. The present contribution is the first installment of more general program on the Einstein-Dirac problem.