Regularity of area minimizing currents II: center manifold

TL;DR

This paper constructs a center manifold and a multi-valued approximation for higher codimension area minimizing currents, advancing the understanding of their singularities and setting the stage for a new proof of Almgren's dimension bound.

Contribution

It introduces a center manifold and a Lipschitz multi-valued map as tools to analyze singularities in higher codimension currents, a key step in the regularity theory.

Findings

Construction of a center manifold for area minimizing currents

Development of a Lipschitz multi-valued approximation

Foundation for a new proof of Almgren's dimension bound

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 center manifold is complemented with a Lipschitz multi-valued map on its normal bundle, which approximates the current with a highe degree of accuracy. In the third and final paper these objects are used to conclude a new proof of Almgren's celebrated dimension bound on the singular set.