Perturbing away higher dimensional singularities from area minimizing hypersurfaces
Let \(C\subset \mathbb{R}^{n+1}\) denote a strictly minimizing hypercone with an isolated singularity and \(C_1\) its truncation to the unit ball. \(\Sigma \subset S^n\) denotes its boundary link. Consider a compact \(k\)-dimensional Riemannian manifold. Then \(T_0= C_1 \times M\) is minimizing in \(\mathbb{R}^{n+1} \times M\), has boundary \(\Sigma \times M\) and singular set \(\{0\} \times M\). The authors now introduce the notion of one sided perturbations of \(\Sigma \times M\) as certain classes of submanifolds \(Z\) of dimension \(n+k-1\) in \(S^n \times M\) which are \(C^{1,\alpha}\) close to \(\Sigma \times M\) and show smoothness of the minimizer spanning \(Z\).
scientific article; zbMATH DE number 820221