Calculus of Variations and Geometric Measure Theory

C. De Lellis - J. Hirsch - A. Marchese - L. Spolaor - S. Stuvard

Excess decay for minimizing hypercurrents mod $2Q$

created by marchese on 16 Aug 2023
modified by stuvard on 09 Nov 2023


Submitted Paper

Inserted: 16 aug 2023
Last Updated: 9 nov 2023

Year: 2023


We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\Sigma$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in {\textrm{spt}} (T)\setminus {\textrm{spt}}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in [11] as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $\Sigma$. This technical improvement is in fact needed in [5] to prove that the singular set of $T$ can be decomposed into a $C^{1,\alpha}$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.