Abstract
For an area-minimizing flat chain modulo v with no boundary inside the unit ball, an absolute upper bound is given for the amount of area inside a shrunken ball. Such Harnack-type estimates lead to generalizations of area inside a shrunken ball. Such Harnack-type estimates lead to generalizations of Bernstein's Theorem. For example, for n⪇5, a 2-dimensional, area-minimizing locally flat chain modulo 2 without boundary in IRn which has at least 1 singularity must consist of 2 orthogonal planes.
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