Constructions of Delaunay-type solutions for the spinorial Yamabe equation on spheres

TL;DR

This paper constructs new singular solutions to the critical Dirac equation on spheres, including Delaunay-type solutions with two point singularities and solutions with a great circle singular set, advancing the understanding of spinorial Yamabe problems.

Contribution

It introduces novel Delaunay-type and great circle singular solutions for the spinorial Yamabe equation on spheres, expanding the class of known solutions and their applications.

Findings

Constructed solutions with two point singularities.

Constructed solutions with a great circle singular set.

Provided building blocks for solutions on general Spin manifolds.

Abstract

In this paper we construct singular solutions to the critical Dirac equation on spheres. More precisely, first we construct solutions admitting two points singularities that we call Delaunay-type solutions because of their similarities with the Delaunay solutions constructed for the singular Yamabe problem in \cite{MP1 , Schoen1989}. Then we construct another kind of singular solutions admitting a great circle as a singular set. These solutions are the building blocks for singular solutions on a general Spin manifold.