Area minimizing currents mod $2Q$: linear regularity theory

TL;DR

This paper develops a theory of Q-valued functions minimizing a generalized Dirichlet integral, laying groundwork for approximating area minimizing currents mod 2Q and proving partial regularity results.

Contribution

It introduces a new regularity theory for Q-valued functions minimizing a generalized Dirichlet integral, enabling advances in understanding area minimizing currents mod 2Q.

Findings

Foundation for approximating area minimizing currents mod 2Q

Establishment of a partial regularity theorem for all p

Development of linear regularity theory for area minimizing currents

Abstract

We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension.