Deformations of asymptotically conical Spin(7)-manifolds

We consider the moduli space $\mathcal{M}_{\nu}$ of torsion-free, asymptotically conical (AC) Spin(7)-structures which are defined on the same manifold and asymptotic to the same Spin(7)-cone with decay rate $\nu<0$. We show that $\mathcal{M}_{\nu}$ is an orbifold if $\nu$ is a generic rate in the non-$L^2$ regime $(-4,0)$. Infinitesimal deformations are given by topological data and solutions to a non-elliptic first-order PDE system on the compact link of the asymptotic cone. As an application, we show that the classical Bryant-Salamon metric on the bundle of positive spinors on $S^4$ has no continuous deformations as an AC Spin(7)-metric.