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

TL;DR

This paper introduces a new parabolic flow on spin manifolds driven by the Dirac-Einstein functional, proving local well-posedness and laying groundwork for further studies on the Einstein-Dirac problem.

Contribution

It presents the first parabolic flow deforming Riemannian metrics via the Einstein-Dirac functional, establishing local existence and initiating a broader research program.

Findings

Proved local well-posedness of the flow.

Established a new approach to deforming metrics using Dirac-Einstein functional.

Laid foundation for future analysis of Einstein-Dirac equations.

Abstract

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.