Stability of coassociative conical singularities

We study the stability of coassociative 4-folds withconical singularities under perturbations of the ambient$\GG_2$ structure by defining an integer invariant of acoassociative cone which we call the stability index. Thestability index of a coassociative cone is determined bythe spectrum of the curl operator acting on its link. Weexplicitly calculate the stability index for cones on grouporbits. We also describe the stability index for conesfibred by 2-planes over algebraic curves using the degreeand genus of the curve and the spectrum of the Laplacian onthe link. Finally, we apply our results to construct thefirst known examples of coassociative 4-folds with conicalsingularities in compact manifolds with $\GG_2$ holonomy.

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Published 11 September 2012