Harmonic spinors on axisymmetric 3-manifolds with Melvin ends

TL;DR

This paper proves the existence of harmonic spinor fields in axisymmetric 3-manifolds with nonnegative scalar curvature that are asymptotic to Melvin's magnetic universe, aiding in the uniqueness proof of magnetized Schwarzschild solutions.

Contribution

It establishes the existence of harmonic spinors in a specific geometric setting, connecting geometric analysis with solutions in general relativity.

Findings

Harmonic spinors exist in axisymmetric manifolds with Melvin ends.

The result supports the uniqueness of magnetized Schwarzschild solutions.

Provides a new tool for analyzing magnetic universe models in relativity.

Abstract

We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.