Rectifiability of the singular strata for harmonic maps to Euclidean buildings

TL;DR

This paper introduces a new framework for analyzing the structure of singularities in harmonic maps into Euclidean buildings, proving their rectifiability and providing bounds on their Minkowski content.

Contribution

It defines the singular strata for harmonic maps into $F$-connected complexes and proves their rectifiability, extending previous results on the full singular set.

Findings

Singular strata are rectifiable.

Bounds on Minkowski content for certain strata.

Extension of rectifiability results to more general targets.

Abstract

We define a natural notion of the singular strata for harmonic maps into $F$-connected complexes (which include locally finite Euclidean buildings), and prove the rectifiability of these strata. We additionally establish bounds on the Minkowski content for certain quantitative strata, following the rectifiable Reifenberg program of [NV17]. This builds on a result of the second author [D], which showed that the full singular set is $(n-2)$-rectifiable.