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

Zhenhua Liu

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

Differential Geometry 2026-07-27 v1

Authors: Zhenhua Liu

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.

Keywords

area-minimizing submanifold mod 2 homology generic smoothness singular set hausdorff dimension veronese embedding

Cite

@article{arxiv.2607.24735,
  title  = {Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces},
  author = {Zhenhua Liu},
  journal= {arXiv preprint arXiv:2607.24735},
  year   = {2026}
}