Calculus of Variations and Geometric Measure Theory

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 04 Dec 2020


Published Paper

Inserted: 10 sep 2019
Last Updated: 4 dec 2020

Journal: Geom. Funct. Anal.
Volume: 30
Number: 5
Pages: 1224-1336
Year: 2020
Doi: 10.1007/s00039-020-00546-0

ArXiv: 1909.05172 PDF
Links: GAFA


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