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Constancy of the index for gradient mappings

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Pith's one-line read For a C^{1,1} function whose Hessian determinant is uniformly positive (or uniformly negative) almost everywhere, the Hessian's index — the number of negative eigenvalues — is constant almost everywhere on each connected domain.

desk verdict Solid proof of a long-open conjecture, with a real but easily patched gap in the negative-determinant clause. read the letter →

arxiv 2506.03906 v1 pith:G7BN2OQ3 submitted 2025-06-04 math.AP math.CVmath.DG

André Guerra , Riccardo Tione This is my paper
classification math.APmath.CVmath.DG MSC 58E0530C6535J96
keywords indexconstancyquasiregulargradientmapsfinitedistortioncriticalgroupsMorsetheoryMonge-AmpèreequationbranchsetHessiandeterminant
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 settles a 1992 conjecture of Šverák: if $u\in W^{2,\infty}_{\mathrm{loc}}(\Omega)$ on a connected open set $\Omega\subset\mathbb{R}^n$ has $\det D^2u(x)\ge\delta>0$ for almost every $x$ (or $\det D^2u\le-\delta<0$), then the index of $D^2u$, meaning the number of negative eigenvalues, is constant almost everywhere. This is trivial for smooth $u$ because $D^2u$ is continuous, but for merely $C^{1,1}$ functions the Hessian is only measurable, and an a.e. sign condition could in principle allow the index to jump. The proof goes through a more general theorem: if $f\in W^{1,n}_{\mathrm{loc}}$ has $Df\in\mathrm{Sym}(n)$ a.e., has finite distortion, and its distortion function $K_f=|Df|^n/\det Df$ lies in $L^p$ with $p>n-1$ (or $p\ge1$ when $n=2$), then $\mathrm{ind}(Df)$ is constant a.e. The new tool is a Morse-theoretic invariant, the critical groups, which remain well-behaved when the function is not twice differentiable and which recover the Hessian index at a.e. point.

What carries the argument

The machinery is the relative singular homology of sublevel sets. For a $C^1$ function $u$ with an isolated critical point at $x_0$, the critical groups $C_k(u,x_0)=H_k(\{u\le u(x_0)\}\cap U,\{u\le u(x_0)\}\cap U\setminus\{x_0\})$ measure how the topology of the sublevel sets changes across the critical level. A deformation lemma uses a pseudo-gradient flow to show that the sublevel set $\{u\le b\}$ can be deformed onto $\{u\le a\}$ without crossing critical values. At a point where $u$ is twice differentiable and $A=D^2u(x_0)$ is non-singular, $C_k(u,x_0)\cong\delta_{k,\mathrm{ind}(A)}\mathbb{Z}$, so the critical groups recover the Hessian index. On top of this, the proof relies on an external result: a non-constant finite-distortion map with $K_f\in L^p$ ($p>n-1$ for $n>2$, $p\ge1$ for $n=2$) is locally injective away from a null set, with the regular set open and connected.

What would settle it

Take a connected open set $\Omega$ and a function $u\in W^{2,\infty}_{\mathrm{loc}}(\Omega)$ with $\det D^2u\ge\delta>0$ a.e. (or the reverse inequality). If two Lebesgue points have different Hessian indices, the theorem is false. The known example $u(x_1,x_2)=x_1^3/|x_1|\,e^{x_2^2/2}$ on the unit ball shows that relaxing the uniform lower bound to merely positive determinant allows the index to jump across $\{x_1=0\}$, so the uniform bound is essential.

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Extended reading notes

Core claim

The central claim is that a uniform determinant bound on the Hessian forces a rigidity: the a.e. values of $D^2u$ cannot move between the connected components of the non-singular symmetric matrices. Theorem 1.1 states this for $u\in W^{2,\infty}_{\mathrm{loc}}$ with $\det D^2u\ge\delta>0$ or $\det D^2u\le-\delta<0$; Theorem 5.3 states the analogue for symmetric finite-distortion maps $f$ with $K_f\in L^p$ and $p$ as in (2.1). The proof shows that when $Du$ is locally injective the critical groups $C_k(u,x)$ are independent of $x$, that at any point of twice differentiability $C_k(u,x)\cong\delta_{k,\mathrm{ind}(D^2u(x))}\mathbb{Z}$, and that a known theorem on the branch set of finite-distortion maps removes local injectivity on a set of full measure. The result is nontrivial even for smooth gradients; in the smooth case it uses deep structure of quasiregular mappings.

Load-bearing premise

The load-bearing premise is an external theorem asserting that a non-constant map of finite distortion with integrable distortion in the stated range is locally injective on an open, connected set of full measure; if the exception set where it fails to be locally injective were not measure zero, the argument could not transfer the constancy of critical groups to almost every point.

Editorial extensions

If this is right

  • Under $\det D^2u\ge\delta>0$ the index is constant a.e.; applying the theorem to $-u$ gives the case $\det D^2u\le-\delta<0$.
  • For any gradient map $f=Du$ with $Df\in\mathrm{Sym}(n)$ a.e., finite distortion, and $K_f\in L^p$ as in (2.1), the index of $Df$ is constant a.e., a statement that includes quasiregular gradient maps.
  • If $Du$ is locally injective, the critical groups $C_k(u,x)$ are independent of $x$; one consequence is a new proof that local injectivity of the gradient plus one locally supporting hyperplane forces strict convexity (Corollary 5.2).
  • Corollary 5.4 extends the index constancy to solutions $u\in W^{2,np}_{\mathrm{loc}}$ of (1.2); in dimension two the threshold cannot be lowered to $W^{2,q}$ with $q<2$.
  • The proof leaves open whether $u\in W^{2,n}_{\mathrm{loc}}$ solutions of (1.2) have open and discrete gradients; an affirmative answer would extend the result to that class.

Reading between the lines

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

  • The same mechanism likely forces the full signature of the Hessian to be constant, since the critical groups identify which component of non-singular symmetric matrices the Hessian occupies.
  • If the branch-set theorem can be relaxed to the endpoint $p=n-1$ or to $W^{2,n}$ potentials, index constancy would follow in those classes; Question 5.5 isolates exactly that missing input.
  • In higher dimensions, index rigidity appears as a branch-set phenomenon: any regularity class that guarantees small branch sets for symmetric finite-distortion maps will inherit index constancy.
  • A natural testable extension is to search for higher-dimensional $W^{2,q}$ maps with $q<n$ satisfying (1.2) whose index jumps, which would sharpen the optimal-regularity threshold.
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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

1 major / 4 minor

Summary. The paper proves a 1992 conjecture of Šverák: if u ∈ W^{2,∞}_{loc}(Ω) on a connected open set Ω ⊂ R^n satisfies det D²u ≥ δ > 0 a.e., then the index of the Hessian D²u is constant a.e. in Ω. The main engine is a more general theorem, Theorem 5.3, asserting that for a finite-distortion map f ∈ W^{1,n}_{loc} with Df symmetric a.e. and distortion exponent as in (2.1), the index of Df is constant a.e. The proof combines a branch-set theorem for finite-distortion maps, a new low-regularity Morse theory for critical groups, and a topological constancy result for critical groups when the gradient is locally injective. The paper also gives a new proof of Ball's theorem on strict convexity and discusses optimality and an open question about the W^{2,n} case.

Significance. If fully correct, this is a substantial contribution: it resolves in all dimensions a conjecture that had previously been known only for n ≤ 3, it supplies a new proof of Ball's theorem, and it exhibits critical groups as a robust topological invariant for low-regularity variational problems. The manuscript is largely self-contained in its topological and Morse-theoretic parts; the main external inputs are standard results from the theory of mappings of finite distortion and relative homology. The argument is honest about its limitations, including the optimality discussion and Question 5.5. The principal shortcoming is that one clause of the main theorem, the uniformly negative determinant case, is not covered by the proofs as written, although a short reduction appears to repair it.

major comments (1)
  1. [Theorem 1.1 (second clause)] The sentence in Theorem 1.1 asserting the same conclusion when det D²u ≤ −δ < 0 is not proved by the arguments in §5. Theorem 5.3 and Corollary 5.4 are formulated only for maps of finite distortion with det Df > 0, and replacing u by −u changes the sign of det D²u only when n is odd; for even n, det(−D²u) = det D²u ≤ −δ, so the reduction does not work. The statement is therefore broader than the proved theorem. A short fix exists: apply Theorem 2.1 to g = S∘Du for a fixed reflection S; then det Dg = −det D²u > 0, B_g = B_{Du}, so Ω∖B_{Du} is a connected open set of full measure on which Du is locally injective, and Theorem 5.1 together with Proposition 4.11(iv) yields the desired constancy of the index. Please add this reduction or a separate proof for the negative-determinant clause.
minor comments (4)
  1. [Theorem 2.1, proof] The proof cites [30, I.4.11] for the fact that det Df(x₀) = 0 at every differentiability point x₀ of the branch set, but [30] is a monograph on quasiregular maps, whereas the theorem is applied to finite-distortion maps with only L^p-integrable distortion. Please provide a citation or a one-line justification showing this pointwise property holds in the needed generality, since it is the step that yields |B_f| = 0.
  2. [Corollary 4.10] The statement says γ depends only on u₁, but the proof also fixes r, the cutoff η, and the annulus where |Du₁| is bounded below. Please rephrase for clarity, e.g., γ depends on u₁ and these auxiliary choices.
  3. [Theorem 5.3, proof] When defining E, the sentence 'Again by Theorem 2.1 we see that E has full measure in Ω'' could be expanded: one uses det Df > 0 a.e. in Ω together with the a.e. existence of D²u for u ∈ W^{2,n}. This is implicit but worth spelling out.
  4. [Notation, §4] The paper alternates between ϕ_t(x) and ϕ(t,x) without comment. This is harmless, but a brief notational note would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the self-citations are survey-only, and the index-constancy theorem is a genuine critical-group computation plus standard external branch-set theory; the unproved negative-determinant clause of Theorem 1.1 is a completeness gap (correctness risk), not a circular step.

full rationale

Verification of the derivation chain. Theorem 1.3 is the quasiregular special case of Theorem 5.3 (the inclusion (1.4) forces det Df > 0 and K_f in L^infinity), and the positive half of Theorem 1.1 follows from Corollary 5.4 with p = infinity. Theorem 5.3's proof is a genuine reduction: it builds the potential u with Du = f, restricts to Omega' = Omega \ B_f (open, connected, full measure, with Du locally injective by Theorem 2.1), applies Theorem 5.1 to obtain constancy of the critical groups C_k(u_x, x), and converts critical groups into the index via Proposition 4.11(iv) on the full-measure set E of twice-differentiability points with det D^2u not equal to 0. No step is self-definitional: the index is the number of negative eigenvalues, C_k is a relative homology group, and the identity C_k = delta_{k,ind(A)}Z at a non-singular twice-differentiable point is computed (rescaling to the quadratic model q_A, C^1-stability via Corollary 4.10, deformation to a k-plane via Lemma 3.2), not assumed. The deformation machinery (Lemmas 4.2-4.8) and the excision/deformation-retract facts (Lemmas 3.1-3.2) are proved in the paper from the Eilenberg-Steenrod axioms. External inputs are Theorem 2.1 (branch-set size and det Df > 0 for finite-distortion maps, quoted from Hencl-Koskela [17] and Rickman [30]), Hatcher [16], and Ball's theorem [3], which the paper reproves rather than imports. None is authored by the present authors and none contains the target conclusion. Self-citations [8] (De Philippis-Guerra-Tione) and [14] (Guerra) occur only in the Section 1.1 survey and are never cited in Sections 2-5, so they are not load-bearing. There are no fitted parameters and no imported uniqueness claims. Completeness flag (correctness risk, not circularity): Theorem 1.1's second clause, 'The same conclusion holds if, instead of (1.2), we have det D^2u <= -delta < 0 a.e. in Omega', is not derived for even n, because the finite-distortion inequality |Df|^n <= K det Df forces det Df >= 0 a.e., replacing u by -u does not change the sign of det D^2u when n is even, and no orientation-reversing analogue of Theorem 5.3 is stated or proved. This is the opposite of circularity (the statement is broader than the proved theorem), so it does not raise the circularity score. Verdict: no significant circularity; score 1 reflects only the presence of two non-load-bearing self-citations.

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

The central claim rests on external results from algebraic topology (Eilenberg-Steenrod axioms) and geometric function theory (branch-set theorem for finite-distortion maps), plus standard facts about Sobolev potentials and Hessians. No free parameters or invented entities enter the proof; the argument is a parameter-free derivation from these inputs.

assumptions (4)
  • standard math Eilenberg-Steenrod axioms for singular homology (homotopy invariance, excision, exactness, dimension).
    Section 3 lists properties (a)-(e) as the framework; used throughout Sections 4 and 5 for critical groups and deformation retracts.
  • domain assumption For a non-constant map f ∈ W^{1,n}_{loc} of finite distortion with K_f ∈ L^p, p > n−1 (p ≥ 1 if n=2), f is open, discrete, continuous, a.e. differentiable, det Df > 0 a.e., and the branch set B_f has topological dimension ≤ n−2 and measure zero; Ω \ B_f is open, connected, full measure.
    Theorem 2.1, quoted from Hencl-Koskela [17] and Rickman [30]. This is the bridge that lets the proof restrict to locally injective maps on a connected full-measure set.
  • standard math W^{2,p} functions have Hessians almost everywhere and are twice differentiable at almost every point.
    Used to define the full-measure set E in Theorem 5.3 where the Hessian exists and is non-singular.
  • standard math If f ∈ W^{1,n}(B^n, R^n) with Df symmetric a.e., then there exists a potential u ∈ W^{2,n}(B^n) with Du = f.
    Used in Theorem 5.3 to pass from the symmetric differential f to a scalar potential u; this is the Poincaré lemma for symmetric gradients.

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Pith. "Pith review of Constancy of the index for gradient mappings." pith.science (2026). https://pith.science/paper/G7BN2OQ3

@misc{pith2026250603906,
  author       = {Pith},
  title        = {Pith review of: Constancy of the index for gradient mappings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7BN2OQ3}},
  note         = {Machine review of arXiv:2506.03906}
}
abstract

We show that if the Hessian of a $C^{1,1}$ function has uniformly positive determinant almost everywhere then its index is locally constant, as conjectured by \v{S}ver\'ak in 1992. We deduce this result as a consequence of a more general theorem valid for quasiregular gradient mappings.

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Works this paper leans on

35 extracted references · 34 canonical work pages

  1. [1]

    Astala, T

    K. Astala, T. Iwaniec, and G. Martin.Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48). Princeton University Press, 2009. 4Up to the endpointp=n−1. 14

    work page 2009

  2. [2]

    Baernstein and L

    A. Baernstein and L. V. Kovalev. On H¨ older regularity for elliptic equations of non-divergence type in the plane.Ann. della Sc. Norm. - Cl. di Sci., 4(2):295–317, 2005

    work page 2005

  3. [3]

    J. M. Ball. Strict convexity, strong ellipticity, and regularity in the calculus of variations.Math. Proc. Cambridge Philos. Soc., 87(3):501–513, 1980

    work page 1980

  4. [4]

    R. Bott. Nondegenerate Critical Manifolds.Ann. Math., 60(2):248, 1954

    work page 1954

  5. [5]

    W. Cao, J. Hirsch, and D. Inauen.C 1, 1 3 − very weak solutions to the two dimensional Monge-Amp` ere equation.arXiv:2310.06693, pages 1–21, 2023

    work page Pith review arXiv 2023

  6. [6]

    Chang.Infinite Dimensional Morse Theory and Multiple Solution Problems

    K.-C. Chang.Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkh¨ auser, Boston, MA, 1993

    work page 1993

  7. [7]

    Conti, C

    S. Conti, C. De Lellis, and L. Sz´ ekelyhidi. h-Principle and Rigidity forC 1,α Isometric Embeddings. InNonlinear Partial Differ. Equations Abel Symp. 2010, pages 83–116. 2012

    work page 2010

  8. [8]

    De Philippis, A

    G. De Philippis, A. Guerra, and R. Tione. Unique continuation for differential inclusions.To Appear Ann. Inst. Henri Poincar´ e Anal. Non Lineaire., 2023

    work page 2023

Show all 35 references
  1. [9]

    D. Faraco. Milton’s conjecture on the regularity of solutions to isotropic equations.Ann. l’Institut Henri Poincare Anal. Non Lineaire, 20(5):889–909, 2003

    2003

  2. [10]

    Faraco and J

    D. Faraco and J. Kristensen. Compactness versus regularity in the calculus of variations.Discret. Contin. Dyn. Syst. - Ser. B, 17(2):473–485, 2012

    2012

  3. [11]

    Faraco, C

    D. Faraco, C. Mora-Corral, and M. Oliva. Sobolev homeomorphisms with gradients of low rank via laminates.Adv. Calc. Var., 11(2):111–138, 2018

    2018

  4. [12]

    Faraco and L

    D. Faraco and L. Sz´ ekelyhidi. Tartar’s conjecture and localization of the quasiconvex hull inR 2×2. Acta Math., 200(2):279–305, 2008

    2008

  5. [13]

    Fonseca and W

    I. Fonseca and W. Gangbo.Degree theory in analysis and applications. Oxford University Press, 1995

    1995

  6. [14]

    A. Guerra. Extremal rank-one convex integrands and a conjecture of ˇSver´ ak.Calc. Var. Partial Differ. Equ., 58(6):19 pp, 2019

    2019

  7. [15]

    C.-Y. Guo, S. Hencl, and V. Tengvall. Mappings of finite distortion: Size of the branch set.Adv. Calc. Var., 13(4):325–360, 2020

    2020

  8. [16]

    Hatcher.Algebraic Topology

    A. Hatcher.Algebraic Topology. Cambridge University Press, Cambridge, 2002

    2002

  9. [17]

    Hencl and P

    S. Hencl and P. Koskela.Lectures on Mappings of Finite Distortion, volume 2096 ofLecture Notes in Mathematics. Springer International Publishing, Cham, 2014

    2014

  10. [18]

    Inauen and M

    D. Inauen and M. Lewicka. The Monge-Amp` ere system in dimension two and codimension three. arXiv:2501.12474, pages 1–28, 2025

    2025 arXiv

  11. [19]

    Iwaniec, L

    T. Iwaniec, L. Kovalev, and J. Onninen. On injectivity of quasiregular mappings.Proc. Am. Math. Soc., 137(5):1783–1791, 2008

    2008

  12. [20]

    Kirchheim and L

    B. Kirchheim and L. Sz´ ekelyhidi. On the gradient set of Lipschitz maps.J. f¨ ur die reine und Angew. Math. (Crelles Journal), 2008(625):215–229, 2008

    2008

  13. [21]

    L. V. Kovalev and D. Maldonado. Mappings with convex potentials and the quasiconformal Jacobian problem.Illinois J. Math., 49(4):1039–1060, 2005

    2005

  14. [22]

    L. V. Kovalev and J. Onninen. On invertibility of sobolev mappings.J. fur die Reine und Angew. Math., 656(656):1–16, 2011

    2011

  15. [23]

    L. V. Kovalev, J. Onninen, and K. Rajala. Invertibility of Sobolev mappings under minimal hypothe- ses.Ann. l’Institut Henri Poincar´ e C, Anal. non lin´ eaire, 27(2):517–528, 2010

    2010

  16. [24]

    X. Lamy, A. Lorent, and G. Peng. Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture.Arch. Ration. Mech. Anal., 238(1):383–413, 2020

    2020

  17. [25]

    X. Lamy, A. Lorent, and G. Peng. On regularity and rigidity of 2×2 differential inclusions into non-elliptic curves.arXiv:2404.02121, 2024

    2024 arXiv

  18. [26]

    Lewicka, L

    M. Lewicka, L. Mahadevan, and M. R. Pakzad. The Monge–Amp` ere constraint: Matching of isome- tries, density and regularity, and elastic theories of shallow shells.Ann. l’Institut Henri Poincar´ e C, Anal. non lin´ eaire, 34(1):45–67, 2017

    2017

  19. [27]

    Mawhin and M

    J. Mawhin and M. Willem.Critical Point Theory and Hamiltonian Systems, volume 74 ofApplied Mathematical Sciences. Springer, New York, NY, 1989

    1989

  20. [28]

    Milnor.Morse Theory

    J. Milnor.Morse Theory. Princeton University Press, 1963. 15

    1963

  21. [29]

    M¨ uller and V.ˇSver´ ak

    S. M¨ uller and V.ˇSver´ ak. Convex integration for Lipschitz mappings and counterexamples to regular- ity.Ann. Math., 157(3):715–742, 2003

    2003

  22. [30]

    Rickman.Quasiregular Mappings

    S. Rickman.Quasiregular Mappings. Springer, Berlin, Heidelberg, 1993

    1993

  23. [31]

    ˇSver´ ak

    V. ˇSver´ ak. On regularity for the Monge-Amp` ere equation without convexity assumptions.Prepr. Heriot-Watt Univ., 1991

    1991

  24. [32]

    ˇSver´ ak

    V. ˇSver´ ak. New examples of quasiconvex functions.Arch. Ration. Mech. Anal., 119(4):293–300, 1992

    1992

  25. [33]

    ˇSver´ ak

    V. ˇSver´ ak. On Tartar’s conjecture.Ann. l’Institut Henri Poincar´ e Non Linear Anal., 10(4):405–412, 1993

    1993

  26. [34]

    Sz´ ekelyhidi

    L. Sz´ ekelyhidi. Rank-one convex hulls inR 2×2.Calc. Var. Partial Differ. Equ., 22(3):253–281, 2005

    2005

  27. [35]

    K. Zhang. On connected subsets ofM 2×2 without rank-one connections.Proc. R. Soc. Edinburgh Sect. A Math., 127(01):207–216, 1997. 16

    1997

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