A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometry

TL;DR

This paper studies a conformally invariant Dirac-type equation on compact spin manifolds, proving strict inequalities unless the manifold is conformal to a sphere, and establishing existence results for the conformal Dirac-Einstein problem in dimension four.

Contribution

It provides the first general existence result for the conformal Dirac-Einstein equations in four dimensions, extending the understanding of such equations on spin manifolds.

Findings

A strict Aubib-type inequality unless the manifold is conformal to the sphere.

Existence of a ground state for the conformal Dirac-Einstein problem in dimension four.

First general existence result beyond perturbative cases in this setting.

Abstract

Given a closed Riemannian Spin manifold $(M,g)$ of dimension greater or equal than four, we consider a generalized conformally invariant equation involving the Dirac operator with a non-linearity of convolution type. We show that the Aubib-type inequality corresponding to the problem is always strict, unless $(M,g)$ is conformal to the round sphere. In particular, this result provides an existence result for a ground state to the conformal Dirac-Einstein problem in dimension four. We point out that aside from some perturbative or special cases, this presents the first general existence result for the conformal Dirac-Einstein equations in dimension four.