Compactness of Dirac-Einstein spin manifolds and horizontal deformations

TL;DR

This paper investigates the compactness properties of solutions to the coupled Einstein-Dirac equations on 3- and 4-dimensional manifolds, analyzing stability and providing explicit examples.

Contribution

It establishes a compactness theorem for critical points of the Hilbert-Einstein-Dirac functional under certain geometric conditions and studies their stability via second variation analysis.

Findings

Proved a compactness result for solutions in dimensions three and four.

Characterized horizontal deformations with vanishing second variation.

Presented explicit examples of solutions.

Abstract

In this paper we consider the Hilbert-Einstein-Dirac functional, whose critical points are pairs, metrics-spinors, that satisfy a system coupling the Riemannian and the spinorial part. Under some assumptions, on the sign of the scalar curvature and the diameter, we prove a compactness result for this class of pairs, in dimension three and four. This can be seen as the equivalent of the study of compactness of sequences of Einstein manifolds as in \cite{And0,N}. Indeed, we study the compactness of sequences of critical points of the Hilbert-Einstein-Dirac functional which is an extension of the Hilbert-Einstein functional having Einstein manifolds as critical points. Moreover we will study the second variation of the energy, characterizing the horizontal deformations for which the second variation vanishes. Finally we will exhibit some explicit examples.