Generic metrics and the mass endomorphism on spin three-manifolds

TL;DR

This paper investigates the mass endomorphism on spin three-manifolds, proving that for generic metrics it is nonzero, which helps prevent bubbling phenomena in conformal spin geometry.

Contribution

It establishes that the mass endomorphism is generically nonzero on three-dimensional spin manifolds, providing a new tool for conformal spin geometry analysis.

Findings

Mass endomorphism is nonzero for generic metrics.

Nonzero mass endomorphism implies a strict inequality.

Results help avoid bubbling-off phenomena in conformal geometry.

Abstract

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $p\in M$ is called the mass endomorphism in $p$ associated to the metric $g$ due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.