Observing minimal submanifolds with isolated singularities on open sets of metrics

Abstract

We develop a perturbative theory for minimal submanifolds with only (strongly) isolated singularities, with respect to varying background Riemannian metrics. As an application, we obtain a general "sporadic persistence" criterion, thus naturally complementing the non-persistence theorem by the authors. In particular, such a criterion is satisfied by the spherical suspension in ${\mathbb{S}}^{3m+2}$ of the Veronese embedding of $\mathbb{K}{\mathbb{P}}^2$ for $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ (the normed division algebras) in the round spheres of dimension $3m+1=4,7,13,25$ respectively, thereby providing families of continuously varying $(2m+1)$-dimensional stationary varifolds with isolated singularities that are observable on an open set of metrics. By our previous results, these phenomena cannot possibly occur in the codimension one scenario.