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A singular profile for the relativistic heat cost and the special Lagrangian curvature equation

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Pith's one-line read A radially built solution of the relativistic transport equation is exactly C^{1,1/(2n−1)} and no smoother, and the same profile kills pure interior Hölder estimates for two-dimensional special Lagrangian curvature.

desk verdict Clean constructive counter-example: sharp C^{1,1/(2n-1)} radial profile for the relativistic cost, transferred to rule out pure interior C^{1,eta} estimates for 2D special Lagrangian curvature. read the letter →

arxiv 2607.26970 v1 pith:FT3MLVUZ submitted 2026-07-29 math.AP

Xiao-Tian Wu This is my paper · ORCID
classification math.AP MSC 35J9649Q2235B6553C4235J60
keywords optimaltransportationrelativisticcostMonge–Ampèretypeequationc-convexityinteriorregularitysingularsolutionspecialLagrangiancurvatureestimate
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 studies how regular the potentials of optimal transport can be when the cost is the relativistic heat cost, which forbids transport farther than a fixed distance a. Even with perfectly smooth constant data, the author builds an explicit radially symmetric generalized solution whose gradient is Hölder continuous of order exactly 1/(2n−1) and no better. The construction reduces the Monge–Ampère-type equation to a planar autonomous ODE whose phase variable vanishes to order 2n−1, producing the sharp modulus. In dimension two the same profile is realized as a uniform limit of smooth graphs solving the special Lagrangian curvature equation for every phase in (0, π/2). The limit shows that no pure interior C^{1,β} estimate can hold for any β larger than 1/3: uniform C^0 bounds alone do not control the Hölder regularity of the gradient. The result therefore exhibits the same critical exponent that appears for rough densities under the MTW condition, but here it arises from the geometry of the cost itself with smooth data.

What carries the argument

A rotationally symmetric ansatz u=√(r(t)²−|x'|²) reduces the equation to a second-order ODE for the meridian r; the ODE is rewritten as a smooth planar autonomous system in the phase variable s=ṙ. With initial data that force the radial factor A to vanish at the origin, s vanishes to order exactly 2n−1, giving the sharp Hölder exponent. A one-parameter perturbation of the initial radius removes the degeneracy and produces the approximating smooth solutions used for the curvature equation.

What would settle it

Directly recompute the Monge–Ampère measure of the constructed radial profile on a small ball and check whether |T_u(E)| equals λ|E|; independently, verify whether the graphs of the perturbed smooth solutions satisfy arctan κ1+arctan κ2=Θ with the stated parameters a=tan Θ and λ=1/cos²Θ.

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Extended reading notes

Core claim

For every dimension n≥2 there exists a generalized (Aleksandrov) solution of the relativistic Monge–Ampère equation on a ball that belongs to C^{1,1/(2n−1)} but fails to lie in C^{1,β} for every larger exponent. Transferring the same radial profile to dimension two yields a sequence of smooth graphical solutions of the special Lagrangian curvature equation that converge uniformly to a limit of class exactly C^{1,1/3}, so that equation admits no pure interior C^{1,β} estimate for any β>1/3.

Load-bearing premise

The passage from the relativistic equation to special Lagrangian curvature rests on an external dictionary that identifies smooth solutions of one equation with graphs solving the other; if that identification fails for these radial profiles, the curvature conclusion falls while the transport result stands.

Editorial extensions

If this is right

  • The critical Hölder exponent 1/(2n−1) is forced by the relativistic cost geometry even when the right-hand side is constant and smooth.
  • No pure interior C^{1,β} estimate (β>1/3) can hold for the two-dimensional special Lagrangian curvature equation with fixed phase.
  • Uniform C^0 control of graphical solutions does not prevent loss of gradient Hölder continuity at an interior point.
  • The same radial construction supplies an explicit family of singular profiles that can be used as test cases for any claimed interior estimate involving the relativistic cost.

Reading between the lines

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

  • Because the exponent tends to zero with dimension, the construction suggests that high-dimensional relativistic transport may lose all uniform gradient modulus of continuity under smooth data.
  • The parallel-surface collapse described in the geometric remark indicates that similar singularities may appear for other curvature equations linked to constant-Gauss-curvature offsets.
  • A natural next test is whether the same autonomous-system method produces singular profiles for other non-MTW costs that share a finite-speed cutoff.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper constructs an explicit family of radially symmetric generalized (Aleksandrov) solutions to the Monge–Ampère equation (1.3) associated with the relativistic cost c(x,y)=√(a²−|x−y|²). Among them is a solution that lies in C^{1,1/(2n−1)} on a ball but in no better Hölder class C^{1,β} for β>1/(2n−1) (Theorem 1.1). The construction reduces the PDE via the ansatz u=√(r(t)²−|x′|²) to a planar autonomous system (3.7) whose phase variable s=ṙ vanishes to order 2n−1, yielding the sharp gradient modulus. In dimension two the same profile is realized, after a standard initial-value perturbation, as a uniform limit of smooth graphical solutions of the special Lagrangian curvature equation arctan κ₁+arctan κ₂=Θ; consequently that equation admits no pure interior C^{1,β} estimate for any β>1/3 (Theorem 1.2).

Significance. The work supplies a clean, fully explicit counter-example showing that the relativistic cost, which violates MTW, permits generalized solutions no smoother than Liu’s sharp C^{1,1/(2n−1)} threshold even when the right-hand side is constant. The same exponent therefore appears in two independent regimes: rough data with an MTW cost, and smooth data with a non-MTW cost. The two-dimensional application gives a concrete obstruction to pure interior curvature estimates for the special Lagrangian curvature equation, in contrast with the classical Monge–Ampère theory in the plane. The argument is constructive, parameter-free, and elementary once the autonomous system is set up; the only external input is the published Qiu–Zhou dictionary (Lemma 4.1). These features make the paper a solid contribution to the regularity theory of optimal transport and of special Lagrangian-type equations.

minor comments (5)
  1. [Title page] The running title on page 1 splits “CURVATURE” as “CUR V ATURE”; this should be corrected in production.
  2. [References] References [9] and [10] list the author only as “Liu” with no given name or initials; please supply the full bibliographic data for consistency with the rest of the bibliography.
  3. [§3.1, Remark 3.1] In Remark 3.1 the heuristic ṡ∼s^{−2(n−1)} is clear, but a one-line reminder that the same leading-order balance is later justified rigorously by Lemma 3.2 and (3.10) would help the reader who skips ahead.
  4. [Introduction / Note] The final Note on the independent work of Qiu–Tao is welcome; a single sentence in the introduction pointing to the different methods (all-dimensional relativistic profile versus parallel-surface construction) would make the relation to [16] more visible to the reader.
  5. [Lemma 3.6] Lemma 3.6(iii) uses η≤a/8 to obtain the crude bound |x′−A₀x′₀|<a/4; the argument is correct, but recording that any sufficiently small η works (the constant a/8 is only for convenience) would avoid the impression that the radius is rigidly constrained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: constructive ODE counter-example with independent asymptotics and external dictionary only

full rationale

The paper is a self-contained constructive counter-example. Theorem 1.1 reduces the relativistic Monge–Ampère equation under an openly stated radial ansatz to the autonomous system (3.7), whose right-hand side is smooth at the degeneracy A(0)=0; Picard–Lindelöf, the expansion A(s)=s²/2+o(s²), and inversion yield the sharp Hölder exponent 1/(2n−1) from the ODE, not by fitting or definitional renaming. c-convexity is proved from the reverse-triangle inequality (Lemma 2.6) plus a one-dimensional support inequality; the Monge–Ampère measure identity is an elementary slicing computation (3.15)–(3.16). Theorem 1.2 is a continuous-dependence perturbation of the same system plus the external published identification Qiu–Zhou [17, Lemma 6.1] (different authors), used only as a dictionary from smooth solutions of (1.3) to (1.4). No parameter is fitted to data and called a prediction; no load-bearing uniqueness or ansatz is imported from the author’s own prior work; the radial profile is an explicit construction in the Pogorelov spirit, not a circular self-definition. The derivation chain does not reduce any claimed output to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure-existence construction inside classical PDE/OT. It inherits the standard definitions of c-convexity and Aleksandrov solutions, ordinary Picard–Lindelöf theory, and one external dictionary lemma of Qiu–Zhou. No parameters are fitted to data; a and λ are equation coefficients chosen freely (and specialized to a=tan Θ, λ=1/cos²Θ only for the geometric application). No new physical or geometric entities are postulated.

assumptions (4)
  • standard math Picard–Lindelöf theorem: a locally Lipschitz autonomous ODE admits a unique C^∞ solution on a small interval
    Invoked to obtain the smooth solution of the phase-plane system (3.7) and its perturbations (4.1) near s=0.
  • domain assumption Definitions of c-convexity, c-normal map, and Aleksandrov/Monge–Ampère measure for a general cost (Villani, Figalli monographs)
    Section 2 adopts these as the meaning of ‘generalized solution’ of (1.3); the paper proves its profile satisfies them but does not re-derive the general theory.
  • domain assumption Qiu–Zhou Lemma 6.1: a smooth solution of the relativistic equation (1.3) with a=tan Θ and λ=1/cos²Θ has graph satisfying the special Lagrangian curvature equation (1.4)
    Quoted verbatim as Lemma 4.1; the entire transfer in Theorem 1.2 rests on this external identification.
  • standard math Continuous dependence of ODE solutions on initial data
    Used in §4 to pass to the limit ε→0 from the perturbed system (4.1) to the singular profile.

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Pith. "Pith review of A singular profile for the relativistic heat cost and the special Lagrangian curvature equation." pith.science (2026). https://pith.science/paper/FT3MLVUZ

@misc{pith2026260726970,
  author       = {Pith},
  title        = {Pith review of: A singular profile for the relativistic heat cost and the special Lagrangian curvature equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FT3MLVUZ}},
  note         = {Machine review of arXiv:2607.26970}
}
abstract

We study the interior regularity of generalized solutions to the Monge--Amp\`ere type equation governing optimal transportation for the relativistic cost $c(x,y)=\sqrt{a^2-|x-y|^2}$ on $\mathbb{R}^n$. We construct an explicit one-parameter family of radially structured generalized solutions on a ball and exhibit, among them, a solution that is of class $C^{1,\frac{1}{2n-1}}$ but of no better H\"older class: it fails to belong to $C^{1,\beta}$ for every $\beta>\frac{1}{2n-1}$. The construction reduces the equation to a planar autonomous system whose phase variable $s=\dot r$ vanishes to order $(2n-1)$ in the base variable. As an application, in dimension two we transfer the construction to the special Lagrangian curvature equation: for every phase $\Theta\in(0,\pi/2)$ we produce a sequence of smooth graphical solutions converging uniformly to a limit of class exactly $C^{1,1/3}$. Consequently the two-dimensional special Lagrangian curvature equation admits no pure interior $C^{1,\beta}$ estimate for any $\beta>\frac{1}{3}$.

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