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Rectifiability of the singular strata for harmonic maps to Euclidean buildings

Christine Breiner, Ben K. Dees

TL;DR

This work addresses the regularity of harmonic maps $u:\Omega\to X$ where $X$ is an $F$-connected complex (e.g., Euclidean buildings) by establishing rectifiability of the singular strata. Building on the Reifenberg-type framework of NV17 and $Q$-valued techniques, the authors define strata that correctly detect homogeneity in the presence of target splittings, and prove that the $k$-th singular stratum $\mathcal{S}^k(u)$ is countably $k$-rectifiable, with quantitative strata obeying Minkowski-type bounds. Central to the argument are notions of almost $k$-homogeneity, cone splitting, and mean flatness (Jones $\beta$-numbers), which together yield the required rectifiability via a covering argument and quantitative stratification. The results extend prior $(n-2)$-rectifiability of the full singular set to a finer stratification, with global consequences obtained through standard localization and covering techniques. The findings sharpen the regularity theory for harmonic maps into buildings and related CAT(0) targets, with potential implications for rigidity phenomena in geometric group theory and algebraic geometry.

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.

Rectifiability of the singular strata for harmonic maps to Euclidean buildings

TL;DR

This work addresses the regularity of harmonic maps where is an -connected complex (e.g., Euclidean buildings) by establishing rectifiability of the singular strata. Building on the Reifenberg-type framework of NV17 and -valued techniques, the authors define strata that correctly detect homogeneity in the presence of target splittings, and prove that the -th singular stratum is countably -rectifiable, with quantitative strata obeying Minkowski-type bounds. Central to the argument are notions of almost -homogeneity, cone splitting, and mean flatness (Jones -numbers), which together yield the required rectifiability via a covering argument and quantitative stratification. The results extend prior -rectifiability of the full singular set to a finer stratification, with global consequences obtained through standard localization and covering techniques. The findings sharpen the regularity theory for harmonic maps into buildings and related CAT(0) targets, with potential implications for rigidity phenomena in geometric group theory and algebraic geometry.

Abstract

We define a natural notion of the singular strata for harmonic maps into -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 -rectifiable.

Paper Structure

This paper contains 23 sections, 24 theorems, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Preliminaries
  4. CAT(0) spaces
  5. $F$-connected complexes
  6. Harmonic maps
  7. Order and smoothed order
  8. Tangent maps and homogeneity
  9. The Singular Strata
  10. Motivating Examples
  11. Defining the Strata
  12. Almost Homogeneity and Quantitative Strata
  13. Local and Global Results
  14. Local Results
  15. Global Results
  16. ...and 8 more sections

Key Result

Theorem 1.1

For $X$ an $F$-connected complex, $\Omega\subset\mathbb{R}^n$ an open domain, and $u:\Omega\to X$ a harmonic map, the singular set $\mathcal{S}^k(u)$ is countably $k$-rectifiable.

Figures (2)

  • Figure 1: The domain of $f_Y$ is shown on the left as a disc; the target tripod on the right. Four points on the tripod are marked---one on each leg, and the common point $0_Y$. Their respective preimages are labeled in the domain.
  • Figure 2: The domain of $u$ is sketched on the left, and the target complex on the right. The domain is separated into three "sectors," corresponding to the preimages of the three half-planes which comprise $Y\times I$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 61 more