TY - JOUR
T1 - (Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents
AU - Engelstein, Max
AU - Spolaor, Luca
AU - Velichkov, Bozhidar
N1 - Publisher Copyright:
© 2019, Mathematical Sciences Publishers. All rights reserved.
PY - 2019
Y1 - 2019
N2 - 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 Leon Simon (1983), but independent from it since almost-minimizers do not satisfy any equation.
AB - 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 Leon Simon (1983), but independent from it since almost-minimizers do not satisfy any equation.
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U2 - 10.2140/gt.2019.23.513
DO - 10.2140/gt.2019.23.513
M3 - Article
AN - SCOPUS:85078172321
SN - 1465-3060
VL - 23
SP - 513
EP - 540
JO - Geometry and Topology
JF - Geometry and Topology
IS - 1
ER -