pith. sign in

arxiv: 1906.10588 · v1 · pith:6JUB2WHXnew · submitted 2019-06-25 · 🧮 math.AP

A Liouville type theorem to 2-Hessian equations

Yan He , Haoyang Sheng , Ni Xiang This is my paper

Pith reviewed 2026-05-25 16:31 UTC · model grok-4.3

classification 🧮 math.AP
keywords Liouville theorem2-Hessian equations2-convex solutionsquadratic growthPogorelov estimategradient estimatefully nonlinear elliptic equations
0
0 comments X

The pith

Any 2-convex solution of σ₂(D²u)=1 with quadratic growth is a quadratic polynomial in dimensions n at least 3.

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

The paper proves a Liouville-type theorem for the 2-Hessian equation. It shows that every 2-convex entire solution with at most quadratic growth must in fact be a quadratic polynomial. The argument uses a Pogorelov-type estimate together with a global gradient bound to reach this conclusion. This settles an open question left in an earlier paper on the same topic.

Core claim

We prove that any 2-convex solution u of σ₂(D²u)=1 with a quadratic growth must be a quadratic polynomial in R^n for n≥3 by using a Pogorelov estimate and the global gradient estimate.

What carries the argument

The combination of the Pogorelov estimate and global gradient estimate applied to 2-convex solutions of the 2-Hessian equation under quadratic growth.

If this is right

  • The only 2-convex solutions with quadratic growth are quadratic polynomials.
  • This provides a positive answer to the unresolved issue raised in the cited reference.
  • Entire solutions cannot deviate from quadratic form while satisfying the equation and growth condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If similar estimates hold for higher k-Hessian equations, then analogous Liouville theorems would follow for those.
  • The result suggests that quadratic growth is a strong rigidity condition for convex solutions of fully nonlinear Hessian equations.
  • Testing the theorem in dimension 2 might reveal whether the dimension restriction is necessary.

Load-bearing premise

The Pogorelov estimate and global gradient estimate apply directly to 2-convex solutions of this equation under the quadratic growth assumption in dimension n≥3.

What would settle it

A counterexample would be any 2-convex function u on R^n (n≥3) satisfying σ₂(D²u)=1 everywhere, with |u(x)| bounded by C(1+|x|²), yet not equal to any quadratic polynomial.

read the original abstract

In this paper, we proved that any 2-convex solution $u$ of $\sigma_2(D^2u)=1$ with a quadratic growth must be a quadratic polynomial in $\mathbb{R}^n\ (n\geq 3 )$ by using a Pogorelov estimate and the global gradient estimate. And we give a positive answer to the unresolved issue in \cite{CX}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proves that any 2-convex solution u of σ₂(D²u)=1 with at most quadratic growth in R^n (n≥3) must be a quadratic polynomial. The argument proceeds by invoking a Pogorelov estimate to obtain an interior bound on second derivatives and a global gradient estimate to control the linear term, thereby reducing the solution to a quadratic polynomial and resolving an open question from [CX].

Significance. If the invoked estimates are valid for 2-convex solutions, the result supplies a positive answer to the unresolved issue in [CX] and extends classical Liouville theorems from the Monge-Ampère equation to the 2-Hessian equation under the natural 2-convexity condition.

major comments (2)
  1. [Proof of main theorem] Main proof (invocation of Pogorelov estimate): the manuscript applies the classical Pogorelov estimate directly to 2-convex solutions (λ(D²u) ∈ Γ₂) of σ₂=1, but provides no k=2-specific derivation or reference; the standard Pogorelov estimate was established for fully convex solutions of the Monge-Ampère equation, so the interior C² bound needed to close the quadratic-growth argument is not justified.
  2. [Proof of main theorem] Global gradient estimate step: the passage from quadratic growth to a uniform gradient bound is asserted via a named estimate whose applicability to merely 2-convex (rather than convex) solutions under the given equation is not verified in the text.
minor comments (1)
  1. [Introduction] The reference [CX] is cited but not expanded; a full bibliographic entry would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit justification of the estimates invoked in the proof. We address each major comment below and will revise the manuscript to incorporate the necessary clarifications and references.

read point-by-point responses

  1. Referee: [Proof of main theorem] Main proof (invocation of Pogorelov estimate): the manuscript applies the classical Pogorelov estimate directly to 2-convex solutions (λ(D²u) ∈ Γ₂) of σ₂=1, but provides no k=2-specific derivation or reference; the standard Pogorelov estimate was established for fully convex solutions of the Monge-Ampère equation, so the interior C² bound needed to close the quadratic-growth argument is not justified.

    Authors: We acknowledge that the original Pogorelov estimate applies to the Monge-Ampère equation and that the manuscript does not contain a k=2-specific derivation. Generalizations of Pogorelov-type interior estimates to k-Hessian equations under the k-convexity condition exist in the literature. In the revised manuscript we will add an explicit reference to the appropriate result establishing the C² bound for 2-convex solutions of σ₂=1. revision: yes

  2. Referee: [Proof of main theorem] Global gradient estimate step: the passage from quadratic growth to a uniform gradient bound is asserted via a named estimate whose applicability to merely 2-convex (rather than convex) solutions under the given equation is not verified in the text.

    Authors: We agree that the applicability of the cited global gradient estimate to the 2-convex (rather than fully convex) setting requires verification in the text. In the revision we will either supply a reference confirming that the estimate extends to 2-convex solutions of σ₂=1 or include a short argument showing why the quadratic-growth assumption yields the uniform bound under the 2-convexity condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proof invokes external estimates

full rationale

The derivation proceeds by applying a Pogorelov estimate and global gradient estimate to 2-convex solutions of σ₂(D²u)=1 under quadratic growth, then concluding u is quadratic. These estimates are treated as established tools (with a citation to an external unresolved issue in CX), not derived or fitted within the paper itself. No self-definitional steps, no fitted inputs renamed as predictions, and no load-bearing self-citation chains appear in the abstract or described chain. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background from fully nonlinear PDE theory; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard definition of 2-convexity via eigenvalues of the Hessian and the elementary symmetric function σ₂
    Invoked to state the class of solutions considered.
  • domain assumption Existence and applicability of Pogorelov-type and global gradient estimates for the 2-Hessian equation under quadratic growth
    Cited as the tools that close the proof.

pith-pipeline@v0.9.0 · 5582 in / 1265 out tokens · 30053 ms · 2026-05-25T16:31:02.666171+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations

    math.AP 2019-07 unverdicted novelty 5.0

    Any k+1-convex-monotone solution to -u_t σ_k(D²u)=1 with quadratic growth on u(x,0) and 0<m1≤-u_t≤m2 is linear in t plus quadratic in x.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 1 Pith paper

  1. [1]

    Liouville property and regularity of a Hessian quotient equation

    Bao J.G., Chen J.Y., Guan B., Ji M.. Liouville property and regularity of a Hessian quotient equation. Amer. J. Math., 2003, 125, no. 2: 301-316

    work page 2003

  2. [2]

    Improper affine hyperspheres of convex type and a generalizati on of atheorem by K

    Calabi E.. Improper affine hyperspheres of convex type and a generalizati on of atheorem by K. J¨ orgens.Mich. Math. J., 1958, 5: 105-126

    work page 1958

  3. [3]

    The Dirichlet problem for nonlinear second- order elliptic equations

    Caffarelli L., Nirenberg, L., Spruck, J.. The Dirichlet problem for nonlinear second- order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math., 1985 155, no. 3-4: 261-301

    work page 1985

  4. [4]

    An extension to a theorem of J ¨ogens, Calabi, and Pogorelov

    Caffarelli L., and Li Y.Y.. An extension to a theorem of J ¨ogens, Calabi, and Pogorelov. Comm. Pure Appl. Math., 2003, 5, 56: 549-583 8 YAN HE, HAOYANG SHENG, NI XIANG

    work page 2003

  5. [5]

    and Yau S.T

    Cheng S.Y. and Yau S.T.. Complete affine hypersurfaces, part I. The completeness of affine metrics. Comm. Pure Appl. Math., 1986, 39: 839-866

    work page 1986

  6. [6]

    A Variational Theory of the Hessian Equation

    Chou K.S., Wang X.J.. A Variational Theory of the Hessian Equation. Comm. Pure Appl. Math., 2001, 9: 1029-1064

    work page 2001

  7. [7]

    Rigidity theorems for the entire solutions of 2-Hessian equ ation

    Chen L., Xiang N.. Rigidity theorems for the entire solutions of 2-Hessian equ ation. To appear J Diff. Equa

    work page

  8. [8]

    A Liouville problem for the sigma-2 equation

    Chang, S.A., Yuan, Y.. A Liouville problem for the sigma-2 equation. Disc. Cont. Dyna. Syst., 2010, 28, no. 2 : 659-664

    work page 2010

  9. [9]

    Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds

    Guan B.. Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J., 2014, 163, no. 8: 1491-1524

    work page 2014

  10. [10]

    Second order parabolic differential equations

    Lieberman G.M.. Second order parabolic differential equations. World Scientific, 2005

    work page 2005

  11. [11]

    Global C 2 Estimates for Convex Solutions of Cur- vature Equations

    Guan P.F., Ren C.Y., Wang Z.Z.. Global C 2 Estimates for Convex Solutions of Cur- vature Equations. Comm. Pure Appl. Math. 2015, 68(8): 1287-1325

    work page 2015

  12. [12]

    Elliptic Partial Differential Equations of the Second Order

    Gilbarge D., Trudinger N.S.. Elliptic Partial Differential Equations of the Second Order . second edition, Springer, 1998

    work page 1998

  13. [13]

    Interior C 2 regularity of convex solutions to prescribing scalar curvature equations

    Guan P.F., Qiu G.H.. Interior C 2 regularity of convex solutions to prescribing scalar curvature equations. Duke Math. J., 2019, 168, no. 9: 1641-1663

    work page 2019

  14. [14]

    ¨Uber die L¨ osungen der Differential gleichung rt − s2 = 1

    Jorgens K.. ¨Uber die L¨ osungen der Differential gleichung rt − s2 = 1 . Math. Ann., 1854, 127: 130-134

    work page

  15. [15]

    An interior estimate for convex solutions and a rigidity theorem

    Li M., Ren C.Y., Wang Z.Z.. An interior estimate for convex solutions and a rigidity theorem. J. Func. Anal., 2016, 270: 2691-2714

    work page 2016

  16. [16]

    Hessian estimates for convex solutions to quadratic Hessian equation

    McGonagle M., Song C., Yuan Y.. Hessian estimates for convex solutions to quadratic Hessian equation. Ann. Inst. H. Poincar Anal. Non Linaire, 2019, 36, no. 2: 451- 454

    work page 2019

  17. [17]

    The Minkowski multidimensional problem

    Pogorelov A.V.. The Minkowski multidimensional problem. John Wiley, 1978

    work page 1978

  18. [18]

    Interior hessian estimates for sigma-2 equations in dimensi on three

    Qiu G.H.. Interior hessian estimates for sigma-2 equations in dimensi on three. preprint

    work page

  19. [19]

    Trudinger N. S.. On the Dirichlet problem for Hessian equations. Acta Math., 1995, 175, no. 2: 151-164

    work page 1995

  20. [20]

    Urbas J. I. E.. On the existence of nonclassical solutions for two classes o f fully non- linear elliptic equations. Indi. Univ. Math. J., 1990, 39, no. 2: 355-382

    work page 1990

  21. [21]

    and Yuan Y

    Warren M. and Yuan Y.. Hessian estimates for the σ2-equation in dimension 3. Comm. Pure Appl. Math., 2009, 62: 305-321

    work page 2009

  22. [22]

    On J ¨orgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Amp` ere equations

    Xiong J.G., Bao J.G.. On J ¨orgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Amp` ere equations. J. Diff. Equa., 2011, 250, no. 1: 367-385

    work page 2011

  23. [23]

    An extension of Jorgens-Calabi-Pogorelov theorem to parabolic Monge-Amp` ere equation

    Zhang W., Bao J.G., Wang B.. An extension of Jorgens-Calabi-Pogorelov theorem to parabolic Monge-Amp` ere equation. Calc. Var. Part. Diff. Equa., 2018, 57, no. 3: 57-90 F aculty of Mathematics and Statistics, Hubei Key Laborator y of Applied Mathe- matics, Hubei University, Wuhan 430062, P.R. China E-mail address : helenaig@hotmail.com; 907026694@qq.com; n...

    work page 2018