*Accepted Paper*

**Inserted:** 5 feb 2018

**Last Updated:** 16 oct 2018

**Journal:** Geom. Topol.

**Year:** 2018

**Abstract:**

In this paper we prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing radially any nearby trace along appropriately chosen directions in the sphere. In contrast to previous epiperimetric inequalities for minimal surfaces by Reifenberg (Ann. of Math. 1964), Taylor (Invent. Math. 1973, Ann. of Math. 1976) and White (Duke Math. J. 1983), 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 epsilon-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 (Ann. of Math. 1983), but independent from it since almost minimizers do not satisfy any equation.

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