(Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents.

*(English)*Zbl 1409.53013Summary: 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 (eg work of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (eg integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new regularity result for almost area-minimizing currents at singular points where at least one blowup is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of L. Simon [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071)], but independent from it since almost-minimizers do not satisfy any equation.

##### MSC:

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

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\textit{M. Engelstein} et al., Geom. Topol. 23, No. 1, 513--540 (2019; Zbl 1409.53013)

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