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(Log-)epiperimetric
inequality and regularity over smooth cones for almost
area-minimizing currents
Max Engelstein, Luca Spolaor and Bozhidar Velichkov |
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Geometry & Topology 23 (2019) 513–540
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Abstract |
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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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Keywords
regularity of minimal surfaces, epiperimetric inequality,
almost area-minimizing currents
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Mathematical Subject Classification 2010
Primary: 53A10
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References |
Publication
Received: 3 May 2018
Accepted: 30 August 2018
Published: 5 March 2019
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, Gang Tian
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Authors |
| Max Engelstein | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
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| Luca Spolaor | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
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| Bozhidar Velichkov | |
| Laboratoire Jean Kuntzmann Université Grenoble Alpes Grenoble France |
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