Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces

Abstract

We prove that area-minimizing submanifolds in mod $2$ homology are not generically smooth, except in the case of geodesics, minimal surfaces and minimal hypersurfaces. This settles a conjecture of White that asks the generic smoothness of area-minimizing submanifolds in mod $2$ homology. 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 $(d-3),$ where $d$ denotes the dimension of the submanifold. As a crucial step, we prove that the cone over the Veronese minimal embedding of $\rpt$ is mod $2$ area-minimizing, settling another long-standing open problem.

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