Calculus of Variations and Geometric Measure Theory
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C. De Lellis - J. Hirsch - A. Marchese - S. Stuvard

Regularity of area minimizing currents mod $p$

created by stuvard on 10 Sep 2019
modified on 23 Sep 2019


Submitted Paper

Inserted: 10 sep 2019
Last Updated: 23 sep 2019

Year: 2019

ArXiv: 1909.05172 PDF


We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an $m$-dimensional area minimizing current $\mathrm{mod}(p)$ cannot be larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional measure.

Keywords: Rectifiability, Area minimizing currents mod(p), Regularity of solutions of variational problems, Hausdorff dimension estimate


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