Area-minimizing submanifolds are not generically smooth

Abstract

We prove that area-minimizing submanifolds are not generically smooth, settling a conjecture of White that asks the generic smoothness of area-minimizing submanifolds. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is $\max\{d-5,d-c\},$ where $d$ denotes the dimension of the submanifold and $c$ denotes the codimension.