(Log-)epiperimetric Inequality and Regularity over Smooth Cones for Almost Area-Minimizing Currents

Max Engelstein (1) , Luca Spolaor (1) , Bozhidar Velichkov (2)

1 MIT - Massachusetts Institute of Technology
2 CVGI - Calcul des Variations, Géométrie, Image

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing any given trace in the radial direction along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. [10], [15, 14], [16]), we need no a priori assumptions on the structure of the cone (e.g. integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new ε-regularity result for almost area-minimizing currents at singular points where at least one blow-up is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of L. Simon [12], but independent form it since almost minimizers do not satisfy any equation.

Mots clés: epiperimetric inequality; regularity of minimal surfaces; almost area-minimizing currents

Domaines: Mathématiques [math]

Max Engelstein, Luca Spolaor, Bozhidar Velichkov. (Log-)epiperimetric Inequality and Regularity over Smooth Cones for Almost Area-Minimizing Currents. Geometry and Topology, 2019, 23, pp.513-540. ⟨10.2140/gt.2019.23.513⟩. ⟨hal-01879179⟩