Quantitative regularity for $p$-minimizing maps through a Reifenberg Theorem

TL;DR

This paper extends regularity results for $p$-energy minimizing maps between Riemannian manifolds, providing a quantitative stratification of singularities and establishing rectifiability and Minkowski bounds for the singular set.

Contribution

It generalizes known regularity results from the case $p=2$ to arbitrary $p$, introducing a quantitative stratification and applying a Reifenberg-type theorem to analyze singularities.

Findings

Singular set can be quantitatively stratified based on almost-symmetries.

The singular set has finite Minkowski content.

The singular set is $k$-rectifiable, with $k$ depending on $p$ and the domain dimension.

Abstract

In this article we extend to generic $p$-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done by Cheeger and Naber in 2013. Then, adapting the work of Naber and Valtorta, we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is $k$-rectifiable for a $k$ which only depends on $p$ and the dimension of the domain.