Regularity of area minimizing currents II: center manifold

Abstract

This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely, the construction of a center manifold, i.e., an approximate average of the sheets of an almost flat area minimizing current. Such a center manifold is accompanied by a Lipschitz multivalued map on its normal bundle, which approximates the current with a high degree of accuracy. In the third and final paper these objects are used to conclude the proof of Almgren’s celebrated dimension bound on the singular set.