Stability of Coassociative Conical Singularities

TL;DR

This paper introduces a new invariant called the stability index to analyze the stability of coassociative 4-folds with conical singularities under G_2 structure perturbations, and applies it to construct examples in compact manifolds.

Contribution

It defines the stability index based on the spectrum of the curl operator, providing a new tool to study coassociative singularities and constructing the first examples in compact G_2 manifolds.

Findings

Explicit calculation of stability index for cones on group orbits

Description of stability index for cones fibered by 2-planes over algebraic curves

Construction of the first known coassociative 4-folds with conical singularities in compact G_2 manifolds

Abstract

We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on its link. We explicitly calculate the stability index for cones on group orbits. We also describe the stability index for cones fibered by 2-planes over algebraic curves using the degree and genus of the curve and the spectrum of the Laplacian on the link. Finally we apply our results to construct the first known examples of coassociative 4-folds with conical singularities in compact manifolds with G_2 holonomy.