Interior regularity of area minimizing currents within a 𝐶2,𝛼-submanifold

Given an area-minimizing integral m m -current in Σ Σ , we prove that the Hausdorff dimension of the interior singular set of T T cannot exceed m − 2 m-2 , provided that Σ Σ is an embedded ( m + n ¯ ) (m+n̄) -submanifold of R m + n Rᵐ⁺ⁿ of class C 2 , α C2,α , where α > 0 α >0 . This result establishes the complete counterpart, in the arbitrary codimension setting, of the interior regularity theory for area-minimizing integral hypercurrents within a Riemannian manifold of class C 2 , α C2,α .