Some counterexamples for the special lagrangian curvature equation

Guanyu Tao; Guohuan Qiu

REVIEW 1 major objections 7 minor 1 cited by

Some counterexamples for the special lagrangian curvature equation

T0 review · 1 major / 7 minor · reviewed 2026-07-30 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Three counterexamples show that convexity and the critical phase are strictly necessary for interior regularity of the special Lagrangian curvature equation.

desk verdict Three explicit SLCE counterexamples that make Qiu–Zhou’s convexity and critical-phase assumptions necessary; 2D is clean and sharp, 3D is a controlled Mooney–Savin adaptation with one compressed collar identity. read the letter →

arxiv 2607.23592 v1 pith:52VUVEB3 submitted 2026-07-26 math.AP math.DG

Guohuan Qiu , Guanyu Tao This is my paper · ORCID

classification math.APmath.DG MSC 35J6053C4235B4549L2553A10

keywords specialLagrangiancurvatureequationviscositysolutionsaprioriestimatesfocalsetsLipschitzsingularitiesparallelsurfacesinteriorregularitycriticalphase

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 special Lagrangian curvature equation requires the principal curvatures of a hypersurface to have arctangents summing to a fixed phase. Recent interior theory gives curvature control for smooth graphical solutions when the graph is convex or the phase is critical, plus gradient estimates for every constant phase. This paper constructs three counterexamples proving those structural hypotheses cannot be removed. In two dimensions it builds an explicit Lipschitz viscosity solution with a genuine corner, and a family of smooth admissible solutions with uniformly bounded C¹ norm whose curvature blows up, with every uniform C^{1,β} bound failing for β > 1/3. In three dimensions and subcritical phase it builds a Lipschitz viscosity solution whose gradient jumps across an analytic surface. The examples isolate focal degeneration of parallel surfaces and dual-map fibre collapse as real geometric obstructions, not artefacts of proof technique.

What carries the argument

Parallel surfaces of constant positive Gauss curvature and their focal sets in two dimensions, and a dual Bellman–Legendre collapse of a rank-two phase core in three dimensions. The same Jacobi factor that makes the parallel map lose rank forces a principal curvature of the SLCE surface to blow up; after a coordinate change the dual gradient map collapses vertical fibres onto an analytic surface, which a local Legendre transform turns into a gradient jump.

What would settle it

Verify by direct computation that the reflected post-focal parallel-surface graph satisfies the viscosity inequalities for the constant-phase equation along the singular line, and that retaining the full P(x)^{1/2} congruence in the three-dimensional dual still yields a strict negative vertical derivative outside the core and a well-defined gradient jump after Legendre transform.

Watch

Extended reading notes

Core claim

The convexity assumption and the critical-phase restriction used for interior a priori estimates of the special Lagrangian curvature equation are strictly necessary. Without them there exist Lipschitz viscosity solutions that fail to be C¹, and sequences of smooth admissible solutions with uniform C¹ control whose second derivatives become unbounded, with sharp two-dimensional Hölder threshold exactly 1/3 in the focal direction.

Load-bearing premise

In the three-dimensional dual construction, the explicit spatial dependence of the curvature operator is only a small perturbation near the origin and does not destroy the determinant sign change, the inertia, or the vertical-fibre collapse needed for the gradient jump.

Editorial extensions

If this is right

Interior curvature estimates for the SLCE cannot follow from positive phase and uniform C¹ control alone on the nonconvex branch.
The exponent 1/3 is the exact borderline Hölder regularity for the two-dimensional higher-order focal degeneration.
Subcritical-phase Lipschitz viscosity solutions of the three-dimensional SLCE need not become C¹.
Adding flat variables extends the smooth two-dimensional curvature-blow-up examples to every dimension n ≥ 2 while keeping the numerical phase fixed.

Reading between the lines

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

The same focal Jacobi-factor mechanism is likely to obstruct curvature estimates for other fully nonlinear hypersurface equations whose linearization sees the normal map.
Because the constant-phase equation already produces singularities, the related optimal-transport problem with relativistic cost can lose regularity without any freedom in the densities.
A direct next test is whether critical-phase nonconvex solutions in dimension three still enjoy C¹ or curvature bounds, or whether a different obstruction appears.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The punchline is simple: Qiu–Zhou’s interior theory for the special Lagrangian curvature equation really does need convexity and the critical phase. The paper supplies three constructive counterexamples that close that question.

What is new is the geometry. In 2D they take a rotational surface of constant positive Gauss curvature, push it by the parallel distance −τ, and keep the post-focal branch. Reflection across the focal line gives an explicit Lipschitz viscosity solution that is not C¹ (Thm 1.1), with a clean expansion and a short test-function argument on the singular line. The same family, smoothed so the Jacobi factor vanishes quadratically, produces a sequence of smooth admissible graphs with uniform C¹ bound, curvature blowing up at one point, and the sharp threshold C^{1,1/3} (Thm 1.2). The limiting meridian profile is exactly |z|^{4/3}. Constants are tracked in the remarks; the branch condition is verified by the elementary Lemma 2.5. That part is elementary, self-contained, and hard to argue with.

The 3D subcritical example adapts Mooney–Savin’s Bellman–Legendre collapse. The dual operator carries explicit x-dependence through P(x)^{1/2}, but on the analytic core boundary the Hessians and the x-slot agree, so the dependence cancels exactly when comparing normal derivatives. The residual risk is one compressed second-normal-derivative identity for det when the kernel is tangent to ∂K; the cancellations are real (full Hessians and tangential third derivatives match, det Φ ≡ 0), but they are written in one line. If that sign flipped, the inertia and fibre collapse would fail. Nothing I checked contradicts the claim, and the subsequent Legendre jump and viscosity perturbation are standard.

Citations are appropriate: they cite their own estimates as the target to be shown sharp, and the classical parallel-surface / front literature and Mooney–Savin correctly. No circularity.

This is for people who work on fully nonlinear geometric PDE or calibrated geometry and care about the precise range of a priori estimates. It deserves a serious referee. I would accept it for peer review and would cite the 2D examples myself.

Referee Report

1 major / 7 minor

Summary. The paper constructs three explicit counterexamples for the special Lagrangian curvature equation (SLCE) ∑ arctan κ_i = Θ, aimed at showing that the hypotheses of the interior a priori estimates of Qiu–Zhou [20] — convexity and the critical phase Θ = (n−2)π/2 — are necessary. Theorem 1.1: in 2D, for every 0 < Θ < π/2 and every corner slope α > 0, a rotational constant-Gauss-curvature surface is offset to signed distance t = −τ (τ = cot Θ); the admissible post-focal branch is written as a one-sided graph and reflected across the focal line, yielding a Lipschitz viscosity solution with an exact corner, classical off the line, with the viscosity inequalities verified directly by test functions (subsolution vacuous, supersolution via the normal-curvature bound k_y ≤ −τ). Theorem 1.2: a family of smooth admissible 2D solutions with Jacobi factor J_{s,ε}(0) = −τε converging to a quadratic zero produces uniformly C¹-bounded graphs with |D²u_ε(0)| ≍ ε⁻¹, a uniform C^{1,1/3} bound, and blow-up of every C^{1,β} seminorm for β > 1/3; the limit has the exact focal profile (3c_Θ/4)|z|^{4/3}. Theorem 1.3: in 3D subcritical phase, a Mooney–Savin-type construction for the Legendre-dual operator F(x, D²w) = tr arctan(P(x)^{1/2}D²wP(x)^{1/2}) produces a Lipschitz viscosity solution whose gradient jumps across a compact analytic surface; the new difficulty relative to [15] is the explicit x-dependence via P(x)^{1/2}, handled through a strictly convex analytic core K = {ϑ_λ ≤ Θ*}, a Cauchy–Koval

Significance. If the constructions hold up — and after detailed checking I believe they do — this paper settles the sharpness of the Qiu–Zhou interior theory for the SLCE and identifies three genuinely distinct obstruction mechanisms (reflected first-order focal singularity, smooth higher-order focal concentration, and Bellman–Legendre collapse). Particular strengths worth naming: the examples are fully explicit and constructive, with all constants tracked in closed form (Remarks 3.1 and 4.1 give non-optimized but completely explicit admissible radii); Theorem 1.2 is not merely a failure of estimates but pins down the exact critical exponent 1/3 with an explicit limiting profile and constant c_Θ = (6/τK_0)^{1/3}, a falsifiable, parameter-free prediction; the viscosity conditions on the singular sets are verified by direct test-function arguments rather than appeal to stability alone (Theorem 1.1), and the 3D construction adapts Mooney–Savin to an operator with genuine x-dependence, with the potentially dangerous perturbation shown to cancel exactly on the core boundary. The paper also complements the Nadirashvili–Vlăduţ, Wang–Yuan, and Mooney–Savin singular-solution literature by carrying it ove

major comments (1)

[§5, Lemma 5.2 (proof, tangent-kernel case)] The sign det D²v < 0 on the exterior collar is the load-bearing input for the inertia (+,+,−), Lemma 2.4, and the fibre collapse in Lemma 5.3; a sign error here would invalidate Theorem 1.3. In the case where the kernel direction ξ of D²Φ_λ is tangent to ∂K, the identity ∂_{νν}(det D²v) = γκ_ξω < 0 is derived in a single sentence. I checked the identity and believe it is correct in substance — tangentially differentiated third derivatives of v and Φ_λ agree on ∂K because the full Hessians and the Cauchy data agree, the Φ-side of term II vanishes since det D²Φ_λ ≡ 0 identically, and the x-dependence of F cancels exactly on ∂K since the x-slot and Hessian slot coincide there. But the manuscript suppresses precisely these cancellation arguments (why term II vanishes, i.e. why G_{νν,kl} = 0 unless (k,l) = (ξ,ξ), and why differentiating v_{νν} − Φ^λ_{νν} = 0 twice along the boundary geodesic

minor comments (7)

[§2.3, Eq. (2.5)] The displayed formula for the normal curvature in the ∂_y direction is typographically garbled (missing fraction bar): it should read k_y = u_{yy}/(√(1+|Du|²)(1+u_y²)). Similar formatting artifacts occur elsewhere (e.g., 'notC 1' in the abstract, the double period at the end of the statement of Lemma 5.2).
[§5, Lemma 5.3 (proof)] The claim that H is a 'distance-expanding global diffeomorphism onto its image' is justified by det DH = cof(D²w)_{33} > 0 together with uniform closeness of DH to diag(2λ, 2λ, 1). Strictly, positive Jacobian plus closeness gives a local diffeomorphism; the global injectivity on U_0 uses the quantitative uniform closeness (e.g., a lower bound on the minimal singular value of DH). One sentence making this explicit would close a small gap.
[§5, proof of Theorem 1.3] In the subsolution approximation w_k = w − x_3²/k, the phrase 'by the strict ellipticity of F_{ij} evaluated on the rank-one segment' is opaque. Presumably one integrates d/dt F(x, D²w − t x_3²·(·)) along the segment joining D²w and D²w_k using F_{33} > 0; please spell this out.
[References] Reference [20] (Qiu–Zhou) is cited only as '2024' with no journal or arXiv identifier; since the paper's stated purpose is to demonstrate sharpness of that work's hypotheses, full bibliographic data should be supplied at revision.
[§5, Lemma 5.1, Eq. (5.2)] I verified the computation (5.2) independently via the spectral formula (including the divided-difference term, which contributes −16λ³/(1+4λ²) to the 11-entry); it is correct. Since this positivity is what makes the core K a compact strictly convex analytic body, a line indicating that the mixed/pair terms from ∂_kS(0) vanish or vanish in the (1,1)-entry would help the reader.
[§4, proof of Theorem 1.2] In the blow-up estimate for [Du_ε]_{C^{0,β}}, the displayed lower bound on |R'_ε(z♯_ε)| is typeset ambiguously ('≥ 1/2τε1/2'); it should read ≥ ε^{1/2}/(2τ). Also, it is worth stating explicitly that the quotient is evaluated between z = 0 and z♯_ε using (u_ε)_z(0,0) = 0.
[Figures 1–5] Figures 1–5 are genuinely helpful and the 'not to scale' disclaimers are appropriate. Consider adding one numerically computed meridian profile (the ODE r'' + K_0 r = 0 is explicit) to replace the 'local asymptotic model' curves in Figure 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: three constructive counterexamples stand alone; self-citation of Qiu–Zhou is only the positive theory being sharpened.

full rationale

The paper’s load-bearing content is explicit geometric and analytic construction, not a derivation that recovers its inputs. Theorems 1.1–1.2 build admissible SLCE graphs from signed parallel surfaces of constant positive Gauss curvature (classical Bonnet/front geometry): the phase identity K+τH=1 is verified by direct substitution of the parallel-curvature formulas κ_i=λ_i/(1−tλ_i), and the Lipschitz corner / C^{1,1/3} focal scaling follow from the Jacobi factor J_s vanishing simply or quadratically. Theorem 1.3 adapts the external Mooney–Savin Bellman–Legendre collapse (cited [15]) to the dual operator F(x,D²w); Cauchy–Kovalevskaya, inertia (+,+,-), and vertical-fibre collapse are checked by direct computation (exact cancellation of x-dependence of P(x)^{1/2} on ∂K, not a fitted ansatz). Viscosity inequalities are verified by test-function arguments or stability of strict subsolutions. The only self-citation is Qiu–Zhou [20] as the a priori theory whose structural hypotheses (convexity, critical phase) the examples show are necessary—ordinary framing for a sharpness paper, not a premise inside the constructions. No fitted parameter is renamed a prediction, no uniqueness theorem is imported to forbid alternatives, and no quantity is defined in terms of the claimed output. Derivation chain is self-contained against external benchmarks (parallel-surface calculus, Legendre duality, CK theorem, Mooney–Savin).

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

Load-bearing content is standard geometric analysis plus the SLCE definition and the Qiu–Zhou positive theory being tested. No data-fitted constants. Construction parameters (Θ, α, ε, λ) are free choices inside open ranges, not fits. No new physical entities. The only domain-specific inputs are the graphical SLCE operator, the admissible phase branch, and classical tools (parallel surfaces, viscosity, Legendre transform, Cauchy–Kovalevskaya).

free parameters (3)

α > 0 (corner slope in Thm 1.1) = arbitrary positive real
Free geometric parameter selecting the one-sided slope of the reflected post-focal graph; any positive value works. Not fitted to data.
ε ∈ (0, 1/(2τ)] (smoothing scale in Thm 1.2) = sequence → 0
Family parameter driving Js,ε(0)=−τε → 0; used to produce curvature blow-up. Chosen in an open interval, not fitted.
λ < λ₀ = (1/2) tan(Θ*/2) (Bellman core scale in Thm 1.3) = sufficiently close to λ₀ from below
Amplitude of the rank-two phase core Φ_λ so that ϑ_λ(0)<Θ* and K={ϑ_λ≤Θ*} is a compact convex body. Chosen close to λ₀; not a data fit.

assumptions (7)

domain assumption Graphical SLCE is F_Θ(Du,D²u)=tr(arctan(P(Du)^{-1/2} D²u P(Du)^{-1/2}))−Θ=0, with F_Θ nondecreasing in the Hessian (Lemma 2.1), justifying the viscosity inequality directions in Def. 2.2.
Defines the equation under study; standard once the shape operator of a graph is written via P(p).
standard math For 0<Θ<π/2 and τ=cot Θ, under κ_i+τ>0 the SLCE is equivalent to K+τH=1 (Lemma 2.5).
Elementary tangent-addition identity used throughout the 2D constructions to verify the equation on parallel surfaces.
standard math Signed parallel surface of a constant-K₀ surface at distance t=−τ satisfies K+τH=1 wherever regular; singularities occur where the Jacobi factor 1+τλ_i=0.
Classical parallel-surface / Bonnet-type fact; load-bearing geometry for Thms 1.1–1.2.
standard math Legendre duality identity (Lemma 2.4): if D²w has inertia (+,+,-) then tr(arctan S[u])=π/2−tr(arctan S̃[w]) at corresponding points.
Converts dual Bellman solution into an SLCE solution of complementary phase; standard spectral arctan inversion.
standard math Cauchy–Kovalevskaya applies to F(x,D²v)=Θ* with non-characteristic boundary (F^{ij}ν_iν_j>0) to produce an analytic exterior collar solution matching Φ_λ Cauchy data (Lemma 5.2).
Used to build the dual potential outside the Bellman core in the 3D construction.
domain assumption Qiu–Zhou interior curvature estimates hold for smooth graphical SLCE solutions in the critical phase and in the convex case (cited positive theory).
The counterexamples are meaningful as sharpness statements only relative to this prior theorem; not used inside the constructions themselves.
standard math Viscosity stability under locally uniform limits: uk→u with F_Θ(Duk,D²uk)>0 implies u is a viscosity subsolution (used at end of Thm 1.3).
Standard viscosity-theory fact closing the 3D subsolution argument via the wk perturbation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Some counterexamples for the special lagrangian curvature equation." pith.science (2026). https://pith.science/paper/52VUVEB3

@misc{pith2026260723592,
  author       = {Pith},
  title        = {Pith review of: Some counterexamples for the special lagrangian curvature equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52VUVEB3}},
  note         = {Machine review of arXiv:2607.23592}
}

abstract

We construct three counterexamples for the special Lagrangian curvature equation (SLCE). First, in dimension two, we use a post-focal branch of a parallel surface with constant positive Gauss curvature to construct an explicit Lipschitz viscosity solution which is not $C^1$. Second, still in dimension two, we construct a sequence of smooth admissible solutions on a fixed rectangle with uniformly bounded $C^1$-norm but unbounded curvature at one point; furthermore, we show that any uniform $C^{1,\beta}$ estimate fails for $\beta > 1/3$. Third, in dimension three and in the subcritical phase, we construct a Mooney-Savin type Lipschitz viscosity solution whose gradient has a jump discontinuity across an analytic surface. These examples demonstrate the sharpness of the recent a priori estimates by Qiu and Zhou, revealing that their structural assumptions-convexity and the critical phase-are strictly necessary.

Figures

Figures reproduced from arXiv: 2607.23592 by the authors.

**Figure 1.** Geometry of the two-dimensional post-focal construction. The left panel is a schematic meridian-plane picture of the normal shift M = N − τν. The right panel displays the two local branches with their common tangent. The solid branch is the admissible post-focal branch retained in the proof; the curves are drawn from the local asymptotic model and are not to scale. Direct differentiation in the two principal directi… view at source ↗

**Figure 2.** Formation of the Lipschitz corner. The selected post-focal branch is first written as a one-sided graph and then reflected across z = 0. The common tangent becomes the two distinct one-sided slopes ±α. Center the coordinates at the focal point by x := R(s) cos θ − R0, y := R(s) sin θ, z := Z0 − Z(s). For −s0 < s < 0, (3.3), (3.1), and (3.2) give C8(−s) ≤ − d ds [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Leading-order local graph geometry. The left panel is the smooth one-sided post-focal graph. After reflection, the two smooth sheets meet continuously along the red curve z = 0, where the normal derivative has a jump. On Q := (−y0, y0) × (−z0, z0), define u(y, z) := u+(y, |z|). The formulas (u+)y = − y p R(z) 2 − y 2 , (u+)z = R′ (z)R(z) p R(z) 2 − y 2 show that u ∈ C 0,1 (Q). Moreover, (3.7) uz(y, 0 +) = αR0 q R2 0… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Schematic of the second-order focal degeneration. The limiting focal factor vanishes quadratically, which produces the cubic relation z ≍ −s 3 and hence the critical meridian profile x ≍ |z| 4/3 . The smooth profiles in the middle panel have the same transition scale a…

**Figure 5.** Figure 5: The principal sections of the limiting profile. Along y = 0, the derivative is exactly of order |z| 1/3 ; along z = 0, the profile remains an ordinary quadratic arc. Thus only the focal direction carries the critical loss of second-order regularity. Since |y|/Rε(z) ≤ 1…

**Figure 6.** Figure 6: The collapse underlying the three-dimensional example. The map H sends the Bellman core K to the convex body H(K). For each z ′ = (z1, z2) ∈ int E, the map T = ∇w ◦ H−1 collapses the vertical fibre of H(K) of length L(z ′ ) to one point of Γ. After Legendre transformat…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

A singular profile for the relativistic heat cost and the special Lagrangian curvature equation
math.AP 2026-07 accept novelty 7.0 of 10

An explicit radial generalized solution of the relativistic Monge–Ampère equation is exactly C^{1,1/(2n-1)}, and in 2D this yields smooth special-Lagrangian graphs converging to a C^{1,1/3} limit.

Reference graph

Works this paper leans on

28 extracted references · cited by 1 Pith paper

[20]

Guohuan Qiu and Xingchen Zhou,A priori interior estimates for special Lagrangian curvature equations, 2024

2024
[15]

15, 2929–2945

Connor Mooney and Ovidiu Savin,Non-c1 solutions to the special Lagrangian equation, Duke Mathe- matical Journal173(2024), no. 15, 2929–2945

2024
[1]

1–2, 353–374

Jérôme Bertrand and Marjolaine Puel,The optimal mass transport problem for relativistic costs, Calculus of Variations and Partial Differential Equations46(2013), no. 1–2, 353–374

2013
[2]

Caffarelli and Sandro Salsa, eds.), Lecture Notes in Mathematics, vol

Yann Brenier,Extended monge–kantorovich theory, Optimal Transportation and Applications (Luis A. Caffarelli and Sandro Salsa, eds.), Lecture Notes in Mathematics, vol. 1813, Springer, Berlin, 2003, pp. 91–121

2003
[3]

J. W. Bruce and T. C. Wilkinson,Folding maps and focal sets, Singularity Theory and its Applications (David Mond and James Montaldi, eds.), Lecture Notes in Mathematics, vol. 1462, Springer, Berlin, 1991, pp. 63–72

1991
[4]

4, 583–595

Jingyi Chen, Micah Warren, and Yu Yuan,A priori estimate for convex solutions to special Lagrangian equations and its application, Communications on Pure and Applied Mathematics62(2009), no. 4, 583–595

2009
[5]

Gaston Darboux,Sur la surface dont la courbure totale est constante, Annales scientifiques de l’École Normale Supérieure7(1890), 9–18
[6]

3, 387–408

Toshizumi Fukui and Masaru Hasegawa,Singularities of parallel surfaces, Tohoku Mathematical Journal 64(2012), no. 3, 387–408

2012

Show all 28 references

[7]

Gálvez, Antonio Martínez, and Francisco Milán,Linear Weingarten surfaces inR3, Monatshefte für Mathematik138(2003), no

José A. Gálvez, Antonio Martínez, and Francisco Milán,Linear Weingarten surfaces inR3, Monatshefte für Mathematik138(2003), no. 2, 133–144

2003
[8]

9, 1641–1663

Pengfei Guan and Guohuan Qiu,Interiorc2 regularity of convex solutions to prescribing scalar curvature equations, Duke Mathematical Journal168(2019), no. 9, 1641–1663

2019
[9]

8, 1287–1325

Pengfei Guan, Changyu Ren, and Zhizhang Wang,Globalc2-estimates for convex solutions of curvature equations, Communications on Pure and Applied Mathematics68(2015), no. 8, 1287–1325

2015
[10]

Lawson, H

Reese Harvey and Jr. Lawson, H. Blaine,Calibrated geometries, Acta Mathematica148(1982), 47–157

1982
[11]

Erhard Heinz,On elliptic Monge–Ampère equations and Weyl’s embedding problem, Journal d’Analyse Mathématique7(1959), 1–52

1959
[12]

2, 241–283

Grégoire Loeper,On the regularity of solutions of optimal transportation problems, Acta Mathematica 202(2009), no. 2, 241–283. COUNTEREXAMPLES FOR THE SLCE 23

2009
[13]

Trudinger, and Xu-Jia Wang,Regularity of potential functions of the optimal transportation problem, Archive for Rational Mechanics and Analysis177(2005), no

Xi-Nan Ma, Neil S. Trudinger, and Xu-Jia Wang,Regularity of potential functions of the optimal transportation problem, Archive for Rational Mechanics and Analysis177(2005), no. 2, 151–183

2005
[14]

J. H. Michael and L. M. Simon,Sobolev and mean-value inequalities on generalized submanifolds ofRn, Communications on Pure and Applied Mathematics26(1973), 361–379

1973
[16]

5, 1179–1188

Nikolai Nadirashvili and Serge Vlăduţ,Singular solution to special Lagrangian equations, Annales de l’Institut Henri Poincaré C, Analyse non linéaire27(2010), no. 5, 1179–1188

2010
[17]

Aleksey Vasil’yevich Pogorelov,The minkowski multidimensional problem, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, DC, 1978, Translated from the Russian by Vladimir Oliker

1978
[18]

Porteous,The normal singularities of a submanifold, Journal of Differential Geometry5(1971), no

Ian R. Porteous,The normal singularities of a submanifold, Journal of Differential Geometry5(1971), no. 3–4, 543–564

1971
[19]

3, 579–605

Guohuan Qiu,Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three, American Journal of Mathematics146(2024), no. 3, 579–605

2024
[21]

2, 491–529

Kentaro Saji, Masaaki Umehara, and Kotaro Yamada,The geometry of fronts, Annals of Mathematics 169(2009), no. 2, 491–529

2009
[22]

2, 235–264

Weimin Sheng, John Urbas, and Xu-Jia Wang,Interior curvature bounds for a class of curvature equations, Duke Mathematical Journal123(2004), no. 2, 235–264

2004
[23]

1, 57–95

Graham Smith,Special Lagrangian curvature, Mathematische Annalen355(2013), no. 1, 57–95

2013
[24]

John I. E. Urbas,On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana University Mathematics Journal39(1990), no. 2, 355–382

1990
[25]

5, 1157–1177

Dake Wang and Yu Yuan,Singular solutions to special Lagrangian equations with subcritical phases and minimal surface systems, American Journal of Mathematics135(2013), no. 5, 1157–1177

2013
[26]

2, 481–499

,Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions, American Journal of Mathematics136(2014), no. 2, 481–499

2014
[27]

3, 305–321

Micah Warren and Yu Yuan,Hessian estimates for the sigma-2 equation in dimension 3, Communications on Pure and Applied Mathematics62(2009), no. 3, 305–321

2009
[28]

3, 751–770

,Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase, American Journal of Mathematics132(2010), no. 3, 751–770. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 Zhongguancun...

2010

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

[1] [20]

Guohuan Qiu and Xingchen Zhou,A priori interior estimates for special Lagrangian curvature equations, 2024

2024

[2] [15]

15, 2929–2945

Connor Mooney and Ovidiu Savin,Non-c1 solutions to the special Lagrangian equation, Duke Mathe- matical Journal173(2024), no. 15, 2929–2945

2024

[3] [1]

1–2, 353–374

Jérôme Bertrand and Marjolaine Puel,The optimal mass transport problem for relativistic costs, Calculus of Variations and Partial Differential Equations46(2013), no. 1–2, 353–374

2013

[4] [2]

Caffarelli and Sandro Salsa, eds.), Lecture Notes in Mathematics, vol

Yann Brenier,Extended monge–kantorovich theory, Optimal Transportation and Applications (Luis A. Caffarelli and Sandro Salsa, eds.), Lecture Notes in Mathematics, vol. 1813, Springer, Berlin, 2003, pp. 91–121

2003

[5] [3]

J. W. Bruce and T. C. Wilkinson,Folding maps and focal sets, Singularity Theory and its Applications (David Mond and James Montaldi, eds.), Lecture Notes in Mathematics, vol. 1462, Springer, Berlin, 1991, pp. 63–72

1991

[6] [4]

4, 583–595

Jingyi Chen, Micah Warren, and Yu Yuan,A priori estimate for convex solutions to special Lagrangian equations and its application, Communications on Pure and Applied Mathematics62(2009), no. 4, 583–595

2009

[7] [5]

Gaston Darboux,Sur la surface dont la courbure totale est constante, Annales scientifiques de l’École Normale Supérieure7(1890), 9–18

[8] [6]

3, 387–408

Toshizumi Fukui and Masaru Hasegawa,Singularities of parallel surfaces, Tohoku Mathematical Journal 64(2012), no. 3, 387–408

2012