Area-minimizing submanifolds are not generically smooth

TL;DR

This paper proves that area-minimizing submanifolds generally have singularities, confirming White's conjecture, and provides a lower bound on the Hausdorff dimension of their singular sets depending on the submanifold's dimension and codimension.

Contribution

It establishes that area-minimizing submanifolds are not typically smooth and derives a lower bound on the Hausdorff dimension of their singular sets.

Findings

Area-minimizing submanifolds are not generically smooth.

Lower bound on singular set Hausdorff dimension: max{d-5, d-c}.

Confirms White's conjecture on generic smoothness.

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.