On the existence of mass minimizing rectifiable G chains in finite dimensional normed spaces

TL;DR

This paper introduces density contractors in finite dimensional normed spaces, enabling solutions to the Plateau problem for rectifiable G chains through new methods ensuring lower semicontinuity of Hausdorff mass.

Contribution

It defines density contractors in finite dimensional normed spaces and proves their existence, facilitating solutions to the Plateau problem for rectifiable G chains.

Findings

Existence of density contractors for various dimensions.

Lower semicontinuity of the m-dimensional Hausdorff mass.

Solution to the Plateau problem in this setting.

Abstract

We introduce the notion of density contractor of dimension m in a finite dimensional normed space X. If m+1=dim X this includes the area contracting projectors on hyperplanes whose existence was established by H. Busemann. If m=2, density contractors are an ersatz for such projectors and their existence, established here, follows from work by D. Burago and S. Ivanov. Once density contractors are available, the corresponding Plateau problem admits a solution among rectifiable G chains, regardless of the group of coefficients G. This is obtained as a consequence of the lower semicontinuity of the $m$ dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.