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arxiv: 2408.02783 · v2 · submitted 2024-08-05 · 🧮 math.DG

Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity

Christine Breiner , Ben K. Dees , Chikako Mese This is my paper

Pith reviewed 2026-05-23 22:35 UTC · model grok-4.3

classification 🧮 math.DG
keywords harmonic mapsEuclidean buildingssingular setsHausdorff codimensionsuperrigiditynon-Archimedean valuationpluriharmonic maps
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The pith

Harmonic maps into Euclidean buildings have singular sets of Hausdorff codimension 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that harmonic maps from Riemannian manifolds to Euclidean buildings possess singular sets whose Hausdorff dimension is at most two less than the domain dimension, and this holds even when the buildings are not locally finite. This regularity extends the earlier result of Gromov and Schoen that applied only to locally finite buildings. Using this, the authors establish superrigidity results for algebraic groups defined over fields equipped with non-Archimedean valuations. They also demonstrate the existence of pluriharmonic maps from Kähler manifolds into Euclidean buildings. A reader would care because these results connect the analysis of energy-minimizing maps with rigidity phenomena in a broader class of geometric targets.

Core claim

We prove that harmonic maps into Euclidean buildings, which are not necessarily locally finite, have singular sets of Hausdorff codimension 2, extending the locally finite regularity result of Gromov and Schoen. As an application, we prove superrigidity for algebraic groups over fields with non-Archimedean valuation, thereby generalizing the rank 1 p-adic superrigidity results of Gromov and Schoen and casting the Bader-Furman generalization of Margulis' higher rank superrigidity result in a geometric setting. We also prove an existence theorem for a pluriharmonic map from a Kähler manifold to a Euclidean building.

What carries the argument

Regularity theory for energy-minimizing maps into Euclidean buildings that controls the singular set without requiring local finiteness.

If this is right

  • Superrigidity holds for algebraic groups over fields with non-Archimedean valuation.
  • The result generalizes the rank-1 p-adic superrigidity of Gromov and Schoen.
  • Bader-Furman higher-rank superrigidity appears as a consequence of harmonic map geometry.
  • Pluriharmonic maps exist from Kähler manifolds to Euclidean buildings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same codimension-2 control may hold for harmonic maps into other non-locally finite CAT(0) spaces with comparable link structures.
  • The geometric approach could yield new proofs of rigidity for representations into groups over local fields.
  • Pluriharmonic maps into buildings might constrain the fundamental groups of Kähler manifolds in previously inaccessible cases.

Load-bearing premise

The maps minimize the Dirichlet energy on a Riemannian domain manifold and the targets satisfy the standard axioms for Euclidean buildings used in prior harmonic map work.

What would settle it

A single harmonic map from a Riemannian domain to a Euclidean building whose singular set has Hausdorff codimension one would disprove the claim.

Figures

Figures reproduced from arXiv: 2408.02783 by Ben K. Dees, Chikako Mese, Christine Breiner.

Figure 1
Figure 1. Figure 1: In the locally finite case, a homogeneous degree 1 map L is effectively contained in PF . This diagram depicts an example when PF is an apartment. The thin lines represent walls of an apartment and the thick line represent F = L(R n ). The map L is effectively contained in PF since the preimage of walls is locally a finite set of hyperplanes. Therefore, the set of points mapping close to the complement of … view at source ↗
Figure 2
Figure 2. Figure 2: The sequences and blow up limit in the proof of Lemma 3.4 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
read the original abstract

We prove that harmonic maps into Euclidean buildings, which are not necessarily locally finite, have singular sets of Hausdorff codimension 2, extending the locally finite regularity result of Gromov and Schoen. As an application, we prove superrigidity for algebraic groups over fields with non-Archimedean valuation, thereby generalizing the rank 1 $p$-adic superrigidity results of Gromov and Schoen and casting the Bader-Furman generalization of Margulis' higher rank superrigidity result in a geometric setting. We also prove an existence theorem for a pluriharmonic map from a K\"ahler manifold to a Euclidean building.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that harmonic maps (with respect to the standard Dirichlet energy) from Riemannian domains into Euclidean buildings, not necessarily locally finite, have singular sets of Hausdorff codimension 2. This extends the Gromov-Schoen regularity theorem. Applications include a superrigidity result for algebraic groups over fields with non-Archimedean valuations (generalizing Gromov-Schoen rank-1 p-adic results and providing a geometric setting for the Bader-Furman extension of Margulis higher-rank superrigidity) and an existence theorem for pluriharmonic maps from Kähler manifolds to Euclidean buildings.

Significance. If the results hold, the work is significant for extending harmonic map regularity to a broader class of targets by showing that local finiteness is not required for the Hausdorff codimension control via monotonicity formulas and tangent-cone analysis. This enables new applications to non-Archimedean superrigidity and casts prior algebraic results in geometric terms. The direct adaptation of the Gromov-Schoen framework without additional gaps in the dimension estimate is a strength.

minor comments (2)
  1. [Abstract] In the abstract and introduction, the precise axioms of Euclidean buildings used (e.g., the precise form of the distance function and apartments) should be cross-referenced to the cited literature to aid readers.
  2. [Introduction] A short paragraph explaining why the local finiteness assumption from Gromov-Schoen can be dropped in the Hausdorff dimension argument (without altering the monotonicity or tangent cone steps) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in extending Gromov-Schoen regularity to non-locally finite targets, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external Gromov-Schoen framework

full rationale

The central regularity result (singular sets of Hausdorff codimension 2 for harmonic maps into possibly non-locally-finite Euclidean buildings) is obtained by direct adaptation of the monotonicity formula and tangent-cone analysis from Gromov-Schoen, using only the standard building axioms and Dirichlet energy; local finiteness is not required in the dimension-control steps. The superrigidity and pluriharmonic existence applications follow immediately from this regularity without additional fitted parameters or self-referential definitions. All cited prior results (Gromov-Schoen, Bader-Furman, Margulis) are external and independent; no self-citation chain or ansatz smuggling is load-bearing. The derivation is therefore self-contained against the stated axioms and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from the stated claims and standard background in the field; no explicit free parameters, invented entities, or ad-hoc axioms are described.

axioms (2)
  • domain assumption Euclidean buildings satisfy the CAT(0) and other structural properties required for harmonic map theory
    Invoked implicitly to extend the Gromov-Schoen regularity result.
  • domain assumption Harmonic maps are critical points of the Dirichlet energy functional
    Standard definition underlying the singular-set analysis.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the Possible Orders of Harmonic Maps into Euclidean Buildings

    math.DG 2026-04 unverdicted novelty 7.0

    Harmonic maps from surfaces to Euclidean buildings have orders of the form m/k with k dividing the Weyl group order of the building.

Reference graph

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