Minkowski content estimates for generic area minimizing hypersurfaces

Let \(\Gamma \) $\mathrm{\Gamma }$ be a smooth, closed, oriented, \((n-1)\) $(n-1)$ -dimensional submanifold of \(\mathbb {R}^{n+1}\) ${\mathbb{R}}^{n+1}$ . It was shown by Chodosh–Mantoulidis–Schulze [6] that one can perturb \(\Gamma \) $\mathrm{\Gamma }$ to a nearby \(\Gamma '\) ${\mathrm{\Gamma }}^{\prime }$ such that all minimizing currents with boundary \(\Gamma '\) ${\mathrm{\Gamma }}^{\prime }$ are smooth away from a set with Hausdorff dimension less than \(n-9\) $n-9$ . We prove that the perturbation can be made such that the singular set of the minimizing current with boundary \(\Gamma '\) ${\mathrm{\Gamma }}^{\prime }$ has Minkowski dimension less than \(n-9\) $n-9$ .