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Stationary points of conformally invariant polyconvex energies

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Pith's one-line read Stationary points of conformally invariant polyconvex energies in two dimensions are smooth outside a discrete set.

desk verdict This paper proves that stationary points of 2D polyconvex conformally invariant frame-indifferent energies are smooth off a discrete set and C1 when orientation-preserving, confirming the 2005 Astala-Iwaniec-Martin-Onninen conjecture for linear growth in distortion. read the letter →

arxiv 2606.10586 v1 pith:V65MT2J7 submitted 2026-06-09 math.AP math.CVmath.DG

André Guerra , Riccardo Tione This is my paper · ORCID
classification math.APmath.CVmath.DG
keywords stationarypointspolyconvexintegrandsconformalinvarianceframeindifferenceregularitytwodimensionsTeichmüllerproblemsdistortion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that for polyconvex integrands that are conformally invariant and frame indifferent, their stationary points in two dimensions are smooth outside a discrete set. This holds even for minimizers. Every orientation-preserving stationary point is C¹. This confirms a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005 for integrands with linear growth in the distortion.

What carries the argument

Polyconvex, conformally invariant and frame indifferent integrands whose stationary points satisfy the Euler-Lagrange equation.

What would settle it

An explicit construction of a stationary point in two dimensions that fails to be smooth outside a discrete set, or an orientation-preserving one that is not C¹, would disprove the claims.

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Extended reading notes

Core claim

In two dimensions, the corresponding stationary points are smooth outside a discrete set; every orientation-preserving stationary point is C¹. This confirms, in the case of integrands with linear growth in the distortion, a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005.

Load-bearing premise

The integrands under consideration are polyconvex, conformally invariant, and frame indifferent.

Editorial extensions

If this is right

  • The regularity result applies even to minimizers.
  • Orientation-preserving stationary points are C¹ everywhere.
  • The conclusions hold for integrands with linear growth in the distortion.
  • The stationary points arise in Teichmüller-type variational problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The methods may extend to other variational problems sharing conformal invariance.
  • The isolated singular points could admit further classification in concrete examples.
  • Analogous regularity statements might be pursued in higher dimensions under extra structural assumptions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 1 minor

Summary. The manuscript considers polyconvex integrands that are conformally invariant and frame indifferent. In two dimensions, it proves that the corresponding stationary points are smooth outside a discrete set; this result is new even for minimizers. It further shows that every orientation-preserving stationary point is C¹. The result confirms, for integrands with linear growth in the distortion, a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005.

Significance. If the result holds, it is significant because it establishes new regularity for stationary points (and even minimizers) of a structurally constrained class of 2D variational integrals, directly addressing a 2005 conjecture in the case of linear growth in distortion. The structural hypotheses (polyconvexity, conformal invariance, frame indifference) enable the conclusions and tie the work to Teichmüller-type problems. The absence of free parameters or ad-hoc reductions in the stated claim is a strength.

minor comments (1)
  1. The abstract states the main theorems clearly, but the introduction could benefit from a brief comparison table or paragraph contrasting the new result with prior partial results on the 2005 conjecture (e.g., which cases were already known for minimizers versus stationary points).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. The referee's summary correctly identifies the main results and their relation to the 2005 conjecture of Astala, Iwaniec, Martin, and Onninen.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper states a direct proof that stationary points of 2D polyconvex conformally invariant frame-indifferent integrands are smooth outside a discrete set and C¹ when orientation-preserving, confirming the 2005 Astala-Iwaniec-Martin-Onninen conjecture under linear growth in distortion. No load-bearing step reduces by definition, fitted input, or self-citation chain to the target result; the structural hypotheses are independent of the regularity conclusion, and the cited conjecture originates from unrelated authors. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all ledger entries are therefore empty.

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Cite this review

Pith. "Pith review of Stationary points of conformally invariant polyconvex energies." pith.science (2026). https://pith.science/paper/V65MT2J7

@misc{pith2026260610586,
  author       = {Pith},
  title        = {Pith review of: Stationary points of conformally invariant polyconvex energies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V65MT2J7}},
  note         = {Machine review of arXiv:2606.10586}
}
abstract

We consider polyconvex integrands that are conformally invariant and frame indifferent. In two dimensions, we prove that the corresponding stationary points are smooth outside a discrete set; this result is new even for minimizers. We further show that every orientation-preserving stationary point is $C^1$. Since such solutions are closely related to Teichm\"uller-type variational problems, our result also confirms, in the case of integrands with linear growth in the distortion, a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005.

Discussion (0). Continue with ORCID to comment.

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