Minkowski content estimates for generic area minimizing hypersurfaces

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