A rescaling principle for quasiregular curves with applications to hyperbolicity

Jonathan Pim; Toni Ikonen

A Miniowitz–Zalcman rescaling principle for quasiregular curves into calibrated manifolds equates Brody hyperbolicity with normality and, for conformal curves, with Kobayashi hyperbolicity.

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2026-07-13 03:38 UTC pith:RXI4WOAS

load-bearing objection Solid GMT-based Miniowitz–Zalcman for quasiregular curves that cleanly settles the Broder–Iliashenko–Madnick hyperbolicity question and supplies usable examples.

arxiv 2607.09353 v1 pith:RXI4WOAS submitted 2026-07-10 math.DG math.CV

A rescaling principle for quasiregular curves with applications to hyperbolicity

Toni Ikonen , Jonathan Pim This is my paper

classification math.DG math.CV MSC 30C6553C3853C6546E3649Q15

keywords quasiregular curvescalibrated manifoldsMiniowitz–Zalcman rescalingBrody hyperbolicityKobayashi hyperbolicityconformal curvesSpecial Lagrangianassociative submanifolds

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes a rescaling principle for quasiregular curves into calibrated manifolds: if a family of such curves fails to be normal at a point, then a non-constant entire quasiregular curve exists as a locally uniform (and uniformly Hölder) limit of rescaled maps. This immediately yields Brody’s theorem in the calibrated setting: absence of non-constant entire curves is equivalent to normality of the family of quasiregular curves from the unit ball (or any oriented n-manifold). When normality holds, the same geometric assumptions give a quantitative local modulus of continuity controlled by injectivity radius and sectional curvature of the target. For the narrower class of conformal curves into closed calibrated manifolds the authors further prove that Brody hyperbolicity is equivalent to Kobayashi (and Rω) hyperbolicity, answering a question of Broder–Iliashenko–Madnick; an intermediate Marty-type bound on derivatives of conformal curves is the bridge. Finally they produce new entire non-constant quasiregular curves that factor through Special Lagrangian, associative and related calibrated submanifolds of closed special-holonomy manifolds, thereby obstructing Brody hyperbolicity in those geometries.

Core claim

If a family of K-quasiregular curves into a closed n-calibrated m-manifold is not normal at a point, then a non-constant entire K-quasiregular curve exists and arises as a locally uniform, uniformly Hölder limit of suitably rescaled maps. The same principle characterises Brody K-hyperbolicity by normality of the family from the Euclidean unit ball, and for conformal curves on closed targets it equates Brody, Kobayashi and Rω hyperbolicity.

What carries the argument

Energy-based Zalcman sequences: at a point of non-equicontinuity one constructs rescalings whose energies concentrate exactly at the critical threshold EN furnished by the small-mass isoperimetric inequalities of the target; the resulting equicontinuous sequence converges to a non-constant entire quasiregular curve.

Load-bearing premise

The quantitative small-mass isoperimetric inequalities for integral currents that hold once the target has a positive lower bound on injectivity radius and an upper bound on sectional curvature; these produce the critical energy threshold EN that makes the rescaling work.

What would settle it

Exhibit a closed calibrated manifold whose injectivity radius and sectional curvature satisfy the stated bounds, yet which admits a non-normal equibounded family of K-quasiregular curves from the unit ball while every entire K-quasiregular curve into it remains constant.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Solid GMT-based Miniowitz–Zalcman for quasiregular curves that cleanly settles the Broder–Iliashenko–Madnick hyperbolicity question and supplies usable examples.

read the letter

This paper gives a clean energy-based Miniowitz–Zalcman rescaling for K-quasiregular curves into n-calibrated manifolds (Theorem 1.1 / 6.2). From it they get the Brody–normality equivalence (1.2 / 6.3) and, for conformal curves on closed targets, the full equivalence of Brody, Kobayashi and Rω hyperbolicity (1.3), answering the question in BIM25. The intermediate Marty-type bound on derivatives (Prop. 7.2) is the right intermediate step. Section 8 then produces concrete entire curves through Special Lagrangians, associatives and Cayley submanifolds of closed special-holonomy manifolds, so the hyperbolicity notions are not vacuous.

What works well is the decision to rescale on energy concentration rather than Hölder seminorms. The quantitative geometry (small-mass isoperimetric inequalities under inj-rad and sec-curv bounds, Caccioppoli, reverse Hölder, energy gap) is classical comparison geometry plus the authors’ earlier push-forward machinery; once those are in place the Zalcman sequences (Prop. 6.6) and the equicontinuity criteria (Thm 5.1, Cor. 5.4) fall out cleanly. The proofs are self-contained and the citation pattern is honest: self-citations supply independent lemmas, not the target theorems.

The only soft spot is that the whole chain rests on the existence of a positive energy threshold EN coming from the isoperimetric profile. That is standard once δ is small enough, so it is not a hidden hypothesis that can fail inside the paper’s setting; it is just the usual price of working with currents in high codimension. For non-compact targets the statements correctly restrict to equibounded families. No circularity, no missing estimates, no data issues.

This is for people already working on quasiregular curves, calibrated geometry or hyperbolicity in special holonomy. It is not a broad-audience paper, but the tool is usable and the open question is settled. I would send it to a serious referee without hesitation; the math is solid enough that the referee’s job is mainly to check the GMT bookkeeping and the examples.

Referee Report

0 major / 5 minor

Summary. The paper establishes a Miniowitz–Zalcman rescaling principle (Theorem 1.1 / Theorem 6.2) for families of K-quasiregular curves into n-calibrated m-manifolds that satisfy an injectivity-radius lower bound and a sectional-curvature upper bound. The argument proceeds by constructing equicontinuous Zalcman sequences that concentrate a definite amount of energy (Proposition 6.6), using the small-mass isoperimetric profile of the target to produce a critical energy threshold EN and Hölder data β. From this the authors obtain a Brody-type characterization of normality (Theorem 1.2 / Theorem 6.3) and, for the subclass of conformal curves into closed calibrated manifolds, the equivalence of Brody, Kobayashi and Rω-hyperbolicity (Theorem 1.3), answering a question of Broder–Iliashenko–Madnick. An intermediate Marty-type theorem (Proposition 7.2) controls the derivatives of conformal curves. Section 8 supplies new examples of quasiregularly elliptic calibrated manifolds by realizing quasiregularly elliptic 3- and 4-manifolds as Special Lagrangian, associative, coassociative or Cayley submanifolds.

Significance. The rescaling principle unifies and extends classical Brody–Zalcman and Miniowitz lemmas to the calibrated setting of Harvey–Lawson, covering quasiregular mappings, pseudoholomorphic curves and Smith maps in a single framework. The equivalence of the three hyperbolicity notions for conformal curves settles an open question and supplies a clean analytic criterion (uniform derivative bounds) via the Marty analogue. The quantitative modulus-of-continuity estimates under curvature and injectivity-radius bounds, together with the energy-gap and reverse-Hölder inequalities, give concrete geometric control that is new even for quasiregular mappings. The examples in Section 8 systematically link the classification of quasiregularly elliptic manifolds to calibrated submanifolds of special-holonomy spaces, producing a large supply of non-hyperbolic targets. The proofs are self-contained once standard GMT comparison geometry is granted, and the energy-based construction of Zalcman sequences is a technical contribution of independent interest.

minor comments (5)

In the definition of the Finslerian pseudodistance d(N,ω) (page 4) the upper integral is used because measurability of t ↦ K(N,ω)(γ′(t)) is not obvious. A short remark that the upper integral coincides with the ordinary integral once the Marty bound of Proposition 7.2 is available would clarify the relation between the three hyperbolicity notions.
The constant EN defined in (3.1) is central to the Hölder estimates and the construction of Zalcman sequences. Adding a one-line reminder of its geometric meaning (half the n-dimensional isoperimetric mass bound, or the (n-1)-dimensional isoperimetric constant scaled by the comass lower bound) at its first appearance in Theorem 4.5 would help the reader.
Table 1 (Section 8) is a useful summary, but the caption could briefly indicate which ambient manifolds realize each check-mark (e.g., T6 for Special Lagrangian T3, Joyce’s G2 examples for associative S1×S2). This would make the table self-contained.
A few minor typographical inconsistencies appear: “thesecondauthorandPankkashowedin[PP25]” (page 4), missing spaces around some citations, and occasional switches between “ω-hyperbolic” and “Brody 1-hyperbolic”. These are easily cleaned in copy-editing.
In the proof of Proposition 7.2 the comparison between hyperbolic and Euclidean balls is standard but dense. A short reference to the precise statements in Vuorinen’s monograph (already cited) or a one-line display of the inclusion constants a and A would improve readability.

Circularity Check

1 steps flagged

No significant circularity: main rescaling and hyperbolicity equivalences are derived from geometric inputs and GMT tools; self-citations supply independent lemmas only.

specific steps

self citation load bearing [Corollary 4.3 and proof of Theorem 6.2 (via [Iko25])]
"As a consequence of the removability of singularities theorem [Iko25, Theorem 1.9], we obtain the following corollary for entire quasiregular curves. Corollary 4.3. ... Then E(F) ≥ M′ for every non-constant quasiregular curve F: Rn → (N, ω)."

The energy-gap statement for entire curves invokes the first author’s prior removability theorem. While the citation is load-bearing for this particular corollary, the main rescaling principle (Theorem 1.1 / Proposition 6.6) and the normality/Brody equivalences (Theorems 1.2–1.3) do not rely on it; they are proved directly from the isoperimetric profile and equicontinuity criteria developed in the present paper. The step is therefore only a minor, non-central self-citation.

full rationale

The derivation chain for Theorems 1.1–1.3 proceeds from standard comparison-geometry assumptions (injectivity-radius lower bound 2δ and sectional-curvature upper bound (π/(2δ))^{2}) that yield quantitative small-mass isoperimetric inequalities for integral currents (Section 3.1, constants EN, A, A′). These produce the energy gap (Corollaries 4.2–4.3), Hölder continuity (Theorem 4.5), weak reverse Hölder and Caccioppoli inequalities (Propositions 4.6–4.8), equicontinuity/normality criteria (Theorem 5.1, Corollary 5.4), and the energy-based construction of Zalcman sequences (Proposition 6.6). The rescaling limit is then extracted by Arzelà–Ascoli plus stability of quasiregularity (Corollary 5.3). All steps are self-contained once the classical isoperimetric profile is granted; no parameter is fitted to data, no quantity is defined in terms of the claimed conclusion, and no uniqueness theorem is imported solely from the authors to force the result. Self-citations ([Iko26b] for Sobolev push-forwards of currents, [Iko25] for removability, [PP25] for bubbling) appear only as supporting lemmas whose statements are independent of the Miniowitz–Zalcman principle and the Brody/Kobayashi equivalences. The examples in Section 8 likewise rely on external classifications (Jormakka, Heikkilä–Pankka et al.) and known calibrated embeddings (Bryant, Joyce). Consequently the central claims do not reduce by construction to their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

Pure existence/uniqueness theorems in Riemannian geometry and geometric measure theory; no free parameters or ad-hoc entities. The only non-standard inputs are the quantitative isoperimetric constants that follow from the curvature/injectivity assumptions and the existence of calibrated submanifolds supplied by Bryant and Joyce.

axioms (4)

domain assumption Small-mass Euclidean isoperimetric inequalities for integral currents of dimension n-1 and n hold with constants depending only on injectivity-radius lower bound 2δ and sectional-curvature upper bound (π/(2δ))^{2} (Section 3.1).
Taken from Wenger and classical comparison geometry; used to define the critical energy EN that drives Hölder continuity and the energy-gap.
standard math Hardt–Lin regularity: weakly n-harmonic maps into Riemannian manifolds are C^{1,α} (used for conformal curves).
Invoked in Section 7 to obtain continuous differentials and the Marty bound.
domain assumption Bryant’s theorem: every closed real-analytic oriented 3-manifold embeds as a Special Lagrangian in some Calabi–Yau 6-fold.
Used in Section 8 to produce entire quasiregular curves factoring through Special Lagrangians.
domain assumption Existence of associative, coassociative and Cayley submanifolds of the topological types listed in Table 1 (Joyce, Bera, etc.).
Cited constructions supply the calibrated submanifolds through which the new entire curves factor.

pith-pipeline@v1.1.0-grok45 · 36957 in / 2690 out tokens · 32550 ms · 2026-07-13T03:38:52.284471+00:00 · methodology

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read the original abstract

We prove a Miniowitz--Zalcman rescaling principle for quasiregular curves into calibrated manifolds. We have two main applications. First, we introduce Brody hyperbolicity adapted to our setting and prove its equivalence to the normality of the family of quasiregular curves from the Euclidean unit ball into the target. When normality holds, we quantify the local modulus of continuity for quasiregular curves using an injectivity radius lower bound and a sectional curvature upper bound of the target. Second, in the special case of conformal curves into closed calibrated manifolds, we prove the equivalence of Kobayashi and Brody hyperbolicity. This answers a question posed by Broder--Iliashenko--Madnick. As an intermediate result, we prove an analogue of Marty's theorem from complex analysis in this setting. Additionally, we construct new examples of non-constant entire quasiregular curves factoring through a Special Lagrangian submanifold of a closed Calabi--Yau manifold, an associative submanifold of a closed $G_2$ manifold, and four-dimensional analogues thereof, providing obstructions to Brody hyperbolicity.

discussion (0)

Reference graph

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Nelson,A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list, J

[AGH+15] Ross Altman, James Gray, Yang-Hui He, Vishnu Jejjala, and Brent D. Nelson,A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list, J. High Energy Phys. (2015), no. 2, 158, front matter+48. MR 3321295 [AK00] Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1–80. MR 1794185 [Ale20] Jame...

2015

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Almgren, Optimal isoperimetric inequalities, Indiana Univ

[Alm86] F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), no. 3, 451–547. MR 855173 [BB11] Anders Björn and Jana Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich,

1986

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Methods Funct

MR 2867756 32 TONI IKONEN AND JONATHAN PIM [Ber06] Walter Bergweiler, Bloch’s principle, Comput. Methods Funct. Theory6 (2006), no. 1, 77–108. MR 2241035 [Ber26] Gorapada Bera, Associative submanifolds in twisted connected sumG2-manifolds, Int. Math. Res. Not. IMRN (2026), no. 8, Paper No. rnag070,

2006

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186 (2001), no

MR 5060833 [BH01] Mario Bonk and Juha Heinonen, Quasiregular mappings and cohomology, Acta Math. 186 (2001), no. 2, 219–238. MR 1846030 [BIM25] Kyle Broder, Anton Iliashenko, and Jesse Madnick, Hyperbolicity and Schwarz Lem- mas in Calibrated Geometry, arXiv e-prints (2025), arXiv:2507.16313. [Bro78] Robert Brody, Compact manifolds and hyperbolicity, Tran...

arXiv 2001

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Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent

MR 257325 [Gro85] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718 [Gro07] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, eng- lish ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007, Based on the 1981 French original, With appendices by M. K...

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Blaine Lawson, Jr., Calibrated geometries, Acta Math

MR 4757696 [HL82] Reese Harvey and H. Blaine Lawson, Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108 [HL87] Robert Hardt and Fang-Hua Lin, Mappings minimizing theLp norm of the gradient, Comm. Pure Appl. Math.40 (1987), no. 5, 555–588. MR 896767 [HP04] Ilkka Holopainen and Pekka Pankka, Mappings of finite distortion: global homeomor- ...

1982

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J.113 (2002), no

MR 4502762 [HR02] Juha Heinonen and Seppo Rickman, Geometric branched covers between generalized manifolds, Duke Math. J.113 (2002), no. 3, 465–529. MR 1909607 [IKM26] Anton Iliashenko, Spiro Karigiannis, and Jesse Madnick, Montel’s theorem and taut- ness in calibrated geometry, arXiv e-prints (2026), arXiv:2606.31393. [Iko25] Toni Ikonen, Quasiregular cu...

Pith/arXiv arXiv 2002

[8] [8]

MR 4947240 [Iko26a] , Entire conformal curves are affine or have super-Euclidean energy growth, Proc. Amer. Math. Soc.154 (2026), no. 9, 3837–3847. MR 5096373 [Iko26b] Toni Ikonen, Pushforward of currents under Sobolev maps, Journal of Functional Anal- ysis 290 (2026), no. 12, 111441. [IM01] Tadeusz Iwaniec and Gaven Martin, Geometric function theory and ...

2026

[9] [9]

MR 1859913 RESCALING PRINCIPLE AND HYPERBOLICITY 33 [IP24] Toni Ikonen and Pekka Pankka, Liouville’s theorem in calibrated geometries, arXiv e-prints (J. Anal. Math., to appear) (2024), arXiv:2410.02722. [IVV02] Tadeusz Iwaniec, Gregory C. Verchota, and Andrew L. Vogel,The failure of rank-one connections, Arch. Ration. Mech. Anal.163 (2002), no. 2, 125–16...

Pith/arXiv arXiv 2024

[10] [10]

Math., vol

MR 1787733 [Joy02] Dominic Joyce, On counting special Lagrangian homology 3-spheres, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 125–151. MR 1941627 [Joy07] Dominic D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, vol. 12, Oxford Unive...

2002

[11] [11]

MR 2292510 [Joy18] Dominic Joyce, Conjectures on counting associative 3-folds inG2-manifolds, Modern geometry: a celebration of the work of Simon Donaldson, Proc. Sympos. Pure Math., vol. 99, Amer. Math. Soc., Providence, RI, 2018, pp. 97–160. MR 3838881 [KO99] Boris S. Kruglikov and Marius Overholt, Pseudoholomorphic mappings and Kobayashi hyperbolicity,...

2018

[12] [12]

MR 277770 [Kra66] Vivian Yoh Kraines, Topology of quaternionic manifolds, Trans. Amer. Math. Soc. 122 (1966), 357–367. MR 192513 [LF76] Jacqueline Lelong-Ferrand, Geometrical interpretations of scalar curvature and reg- ularity of conformal homeomorphisms, Differential geometry and relativity, dath. Phys. Appl. Math., vol. Vol. 3, Reidel, Dordrecht-Boston...

Pith/arXiv arXiv 1966

[13] [13]

Synerg.7 (2021), no

MR 2954391 [OP21] Jani Onninen and Pekka Pankka, Quasiregular curves: Hölder continuity and higher integrability, Complex Anal. Synerg.7 (2021), no. 3, Paper No. 26,

2021

[14] [14]

MR 4289854 [Pan20] Pekka Pankka, Quasiregular curves, Ann. Acad. Sci. Fenn. Math.45 (2020), no. 2, 975–990. MR 4112271 [PP25] Pekka Pankka and Jonathan Pim, Nodal resolution of quasiregular curves via bubble trees, arXiv e-prints (2025), arXiv:2510.02927. [Pry19] Eden Prywes, A bound on the cohomology of quasiregularly elliptic manifolds, Ann. of Math. (2...

arXiv 2020

[15] [15]

MR3821514 [Ros10] Jean-Pierre Rosay, Uniqueness in rough almost complex structures, and differential inequalities, Ann. Inst. Fourier (Grenoble)60 (2010), no. 6, 2261–2273. MR 2791657 [Roy71] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), Lecture Notes in Math....

Pith/arXiv arXiv 2010

[16] [16]

MR 950174 [Wen07] Stefan Wenger, Flat convergence for integral currents in metric spaces, Calc. Var. Partial Differential Equations28 (2007), no. 2, 139–160. MR 2284563 [Zal75] Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 379852 Department of Mathematics, University of Fribourg, Chem...

2007