Center Manifold: a case study

TL;DR

This paper demonstrates the C^{3,} regularity of area-minimizing currents near points of density one, advancing understanding of Almgren's center manifold construction without relying on nonparametric methods.

Contribution

It provides a new proof of regularity for area-minimizing currents near density-one points, simplifying Almgren's original approach.

Findings

Proves C^{3,} regularity of area-minimizing currents

Eliminates the need for nonparametric theory in this context

Advances understanding of Almgren's center manifold construction

Abstract

Following Almgren's construction of the "center manifold" in his Big regularity paper, we show the C^{3,\alpha} regularity of area-minimizing currents in the neighborhood of points of density one without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren's construction in its full generality.