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Interior regularity of area minimizing currents within a $C^{2,α}$-submanifold (2404.17407v2)

Published 26 Apr 2024 in math.AP and math.DG

Abstract: Given an area-minimizing integral $m$-current in $\Sigma$, we prove that the Hausdorff dimension of the interior singular set of $T$ cannot exceed $m-2$, provided that $\Sigma$ is an embedded $(m+\bar{n})$-submanifold of $\mathbb{R}{m+n}$ of class $C{2,\alpha}$, where $\alpha>0$. 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 $C{2,\alpha}$.

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