(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents

TL;DR

This paper introduces a new logarithmic epiperimetric inequality for certain minimal cones, leading to improved regularity results for almost area-minimizing currents at singular points without requiring prior structural assumptions.

Contribution

The authors establish a novel logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularities, extending regularity theory for almost minimizers.

Findings

Proved a new logarithmic epiperimetric inequality for specific cones.

Derived an ε-regularity result for almost area-minimizing currents.

Achieved regularity results without assuming cone integrability or structure.

Abstract

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (e.g. integrability). Moreover, if the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new $\varepsilon$-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 Leon Simon, but independent from it since almost minimizers do not satisfy any equation.