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arxiv: 2604.13578 · v1 · submitted 2026-04-15 · 🧮 math.AP

The existence of (p, k)-convex hypersurfaces for a class of Hessian quotient type curvature equations

Jiabao Gong , Qiang Tu This is my paper

Pith reviewed 2026-05-10 13:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hessian quotient equationscurvature equations(p,k)-convex hypersurfacesstar-shaped hypersurfacesconstant rank theoremcontinuity methodinverse convexityfully nonlinear PDEs
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The pith

Closed star-shaped (p,k)-convex hypersurfaces exist and are unique for Hessian quotient curvature equations.

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

This paper aims to prove that there exist unique closed star-shaped hypersurfaces satisfying a class of curvature equations of Hessian quotient type, where the surfaces are at least (p,k)-convex. The approach combines a priori estimates on the solutions with the continuity method to connect to known cases. It further leverages the inverse convexity of the curvature operator to establish a constant rank result, ensuring the solutions can be made strictly convex. Readers interested in geometric analysis would care because these equations describe hypersurfaces with specific curvature conditions, and their solutions provide explicit constructions in differential geometry.

Core claim

By combining a priori estimates with the continuity method, the authors establish the existence and uniqueness of (p,k)-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered inverse convexity property of the operator sigma_k/sigma_l(Lambda), they prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations.

What carries the argument

The operator sigma_k/sigma_l(Lambda) and its inverse convexity property, which enables the constant rank theorem for strict convexity.

Load-bearing premise

The operator sigma_k/sigma_l(Lambda) must satisfy the inverse convexity property to allow the constant rank theorem to upgrade (p,k)-convexity to strict convexity.

What would settle it

Constructing or identifying a closed star-shaped solution to one of these equations that is (p,k)-convex but not strictly convex would disprove the constant rank theorem application.

read the original abstract

This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p}, k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered ``inverse convexity'' property of the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations.

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 manuscript claims to prove the existence and uniqueness of closed, star-shaped (p,k)-convex hypersurfaces for a class of Hessian quotient type curvature equations (generalizing the classical Hessian quotient operator) by combining a priori estimates with the continuity method, for both nonhomogeneous and homogeneous cases. It further exploits the recently discovered inverse convexity property of the operator σ_k/σ_l(Λ) to establish a constant rank theorem, thereby obtaining existence and uniqueness of strictly convex solutions.

Significance. If the a priori estimates hold and the inverse convexity property applies to the relevant ranges of p and k without further restrictions, the results would extend known existence theorems for Hessian quotient equations to this generalized setting, providing both (p,k)-convex and strictly convex hypersurfaces with uniqueness. The integration of standard continuity-method techniques with the cited inverse-convexity property represents a potentially useful contribution to fully nonlinear curvature problems, provided the technical details are verified.

major comments (2)
  1. Abstract and the section proving the constant rank theorem: the upgrade from (p,k)-convex to strictly convex solutions rests entirely on the inverse convexity property of σ_k/σ_l(Λ); the manuscript must explicitly verify or cite the precise parameter ranges of p and k for which this property holds in both the homogeneous and nonhomogeneous cases, as any gap would leave the strict-convexity claim unsupported while preserving the (p,k)-convex existence result.
  2. The a priori estimates section (central to the continuity-method argument): the estimates for (p,k)-convex hypersurfaces are load-bearing; the manuscript should confirm that all boundary cases for closed star-shaped hypersurfaces are covered, including any dependence on the specific form of the curvature equations.
minor comments (1)
  1. The abstract refers to the inverse convexity property as 'recently discovered'; the introduction or preliminaries should include a precise reference or self-contained statement of the property to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will make the suggested clarifications to strengthen the presentation.

read point-by-point responses

  1. Referee: Abstract and the section proving the constant rank theorem: the upgrade from (p,k)-convex to strictly convex solutions rests entirely on the inverse convexity property of σ_k/σ_l(Λ); the manuscript must explicitly verify or cite the precise parameter ranges of p and k for which this property holds in both the homogeneous and nonhomogeneous cases, as any gap would leave the strict-convexity claim unsupported while preserving the (p,k)-convex existence result.

    Authors: We appreciate the referee highlighting the need for explicit parameter ranges. The inverse convexity property of σ_k/σ_l(Λ) is known to hold for p ≥ 1 and 1 ≤ l < k ≤ n (with minor adjustments for the homogeneous case where the right-hand side is a positive constant). These ranges are precisely those under which our (p,k)-convexity is defined and the constant rank theorem applies. We will add a dedicated remark immediately following the statement of the constant rank theorem, citing the source of the inverse convexity result and stating the applicable ranges separately for the homogeneous and nonhomogeneous equations. This revision will make the support for strict convexity fully transparent without altering the existing proofs. revision: yes

  2. Referee: The a priori estimates section (central to the continuity-method argument): the estimates for (p,k)-convex hypersurfaces are load-bearing; the manuscript should confirm that all boundary cases for closed star-shaped hypersurfaces are covered, including any dependence on the specific form of the curvature equations.

    Authors: The a priori estimates are derived via the maximum principle applied to the linearized operator on the closed star-shaped hypersurface, using the support function to control the geometry at all points. Because the hypersurfaces are compact and without boundary, there are no additional boundary cases to consider; the estimates hold uniformly under the (p,k)-convexity assumption and depend on the equation only through the ellipticity constants already controlled by the convexity parameters and the bounds on the right-hand side. To make this coverage explicit, we will insert a brief clarifying paragraph at the conclusion of the a priori estimates section summarizing that all closed star-shaped cases are included for the given class of equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; existence via continuity method independent of cited property

full rationale

The derivation combines standard a priori estimates with the continuity method to obtain existence and uniqueness of (p,k)-convex hypersurfaces for both equation types. The upgrade to strictly convex solutions invokes a constant rank theorem based on the inverse convexity property of σ_k/σ_l(Λ), which the abstract explicitly labels as recently discovered and therefore external. No equation or step in the abstract reduces the claimed existence result to a self-definition, a fitted input renamed as prediction, or a self-citation chain whose validity is presupposed by the present work. The central argument therefore retains independent content relative to external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in fully nonlinear elliptic PDE theory and geometric convexity. No new free parameters, invented entities, or ad-hoc axioms are introduced beyond the definition of the operator class and the (p,k)-convexity condition.

axioms (2)
  • domain assumption The operator σ_k/σ_l satisfies the inverse convexity property (cited as recently discovered).
    Invoked to obtain the constant rank theorem for strict convexity.
  • domain assumption Standard a priori estimates hold for the admissible class of (p,k)-convex hypersurfaces.
    Required for the continuity method to close.

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Works this paper leans on

47 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J

    B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608, 17-33 (2007)

    work page 2007

  2. [2]

    Berg, Corps convexes et potentials sph´ eriques, Det

    C. Berg, Corps convexes et potentials sph´ eriques, Det. Kgl. Danske Videnskab. Selskab. Math. fys. Medd, 37, 1-64, (1969)

    work page 1969

  3. [3]

    Caffarelli, L

    L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, I. Monge-Amp` ere equation, Comm. Pure Appl. Math., 37(3), 369-402, (1984)

    work page 1984

  4. [4]

    Caffarelli, L

    L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math., 155, 261-301, (1985)

    work page 1985

  5. [5]

    Caffarelli, L

    L. Caffarelli, L. Nirenberg, J. Spruck, Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equations, 1986, 1-26, (1986)

    work page 1986

  6. [6]

    X. Chen, Q. Tu, N. Xiang, A class of Hessian quotient equations in Euclidean space, J. Differ- ential Equations, 269(12), 11172-11194, (2020)

    work page 2020

  7. [7]

    X. Chen, Q. Tu, N. Xiang, The Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds, Int. Math. Res. Not., 2023(12), 10013-10036, (2023)

    work page 2023

  8. [8]

    Chu, A simple proof of curvature estimate for convex solution ofk-Hessian equation, Proc

    J. Chu, A simple proof of curvature estimate for convex solution ofk-Hessian equation, Proc. Amer. Math. Soc., 149(8), 3541-3552, (2021)

    work page 2021

  9. [9]

    J. Chu, H. Jiao, Curvature estimates for a class of Hessian type equations, Calc. Var. Partial Differ. Equ., 60(3), 90, (2021)

    work page 2021

  10. [10]

    W. Dong, W. Wei, The Neumann problem for a type of fully nonlinear complex equations, J. Differential Equations, 306, 525-546, (2022) (p, k)-CONVEX HYPERSURF ACES FOR PRESCRIBED CUR V ATURE 31

    work page 2022

  11. [11]

    Dong, Curvature estimates for p-convex hypersurfaces of prescribed curvature, Rev

    W. Dong, Curvature estimates for p-convex hypersurfaces of prescribed curvature, Rev. Mat. Iberoam., 39(3), 1039-1058, (2023)

    work page 2023

  12. [12]

    Firey, The determination of convex bodies from their mean radius of curvature functions, Mathematika, 14(1), 1-13, (1967)

    W. Firey, The determination of convex bodies from their mean radius of curvature functions, Mathematika, 14(1), 1-13, (1967)

    work page 1967

  13. [13]

    Firey, Christoffel problems for general convex bodies, Mathematika, 15(1), 7-21, (1968)

    W. Firey, Christoffel problems for general convex bodies, Mathematika, 15(1), 7-21, (1968)

    work page 1968

  14. [14]

    Gauduchon, La 1-forme de torsion d´ une vari´ et´ e hermitienne compacte, Math

    P. Gauduchon, La 1-forme de torsion d´ une vari´ et´ e hermitienne compacte, Math. Ann., 267(4), 495-518, (1984)

    work page 1984

  15. [15]

    Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J

    C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom., 43, 612-641, (1996)

    work page 1996

  16. [16]

    Gerhardt, Curvature problems, Series in Geometry and Topology, International Press of Boston Inc., Somerville, 39, (2006)

    C. Gerhardt, Curvature problems, Series in Geometry and Topology, International Press of Boston Inc., Somerville, 39, (2006)

    work page 2006

  17. [17]

    J. Gong, Q. Tu, The Neumann problem for a class of degenerate Hessian quotient type equations, preprint, arXiv:2604.09044

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    B. Guan, P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156(2), 655-673, (2002)

    work page 2002

  19. [19]

    Guan, Topics in geometric fully nonlinear equations, Lecture Notes, (2002)

    P. Guan, Topics in geometric fully nonlinear equations, Lecture Notes, (2002)

    work page 2002

  20. [20]

    P. Guan, Y. Li,C 1,1 estimates for solutions of a problem of Alexandrov, Comm. Pure Appl. Math., 50(8), 789-811, (1997)

    work page 1997

  21. [21]

    P. Guan, C. Lin, X. Ma, The Christoffel-Minkowski problem II. Weingarten curvature equations, Chin. Ann. Math. Ser. B, 27(6), 595-614, (2006)

    work page 2006

  22. [22]

    P. Guan, J. Li, Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J., 161, 1927-1942, (2012)

    work page 1927

  23. [23]

    P. Guan, C. Ren, Z. Wang, GlobalC 2 estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68, 1287-1325, (2015)

    work page 2015

  24. [24]

    F. R. Harvey, H. B. Lawson Jr., Geometric plurisubharmonicity and convexity: an introduction, Adv. Math., 230(4-6), 2428-2456, (2012)

    work page 2012

  25. [25]

    F. R. Harvey, H. B. Lawson Jr., p-convexity, p-plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., 62(1), 149-169, (2013)

    work page 2013

  26. [26]

    Y. He, Q. Tu, N. Xiang, A class of prescribed Weingarten curvature equations for locally convex hypersurfaces with boundary inR n+1, J. Geom. Anal., 34(2), 48, (2024)

    work page 2024

  27. [27]

    C. Hu, X. Ma, C. Shen, On Christoffel-Minkowski problem of Firey’s p-sum, Calc. Var. Partial Differ. Equ., 21(2), 137-155, (2004)

    work page 2004

  28. [28]

    Huang, L

    Y. Huang, L. Xu, Two problems related to prescribed curvature measures, Discrete Contin. Dyn. Syst., 33(5), 1975-1986, (2013)

    work page 1975

  29. [29]

    Huisken, C

    G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183, 45-70, (1999)

    work page 1999

  30. [30]

    Ivochkina, Solution of the Dirichlet problem for curvature equations of orderm, Math

    N. Ivochkina, Solution of the Dirichlet problem for curvature equations of orderm, Math. USSR-Sb., 67(2), 317-339, (1990)

    work page 1990

  31. [31]

    Ivochkina, The Dirichlet problem for the equations of curvature of orderm, Leningrad Math

    N. Ivochkina, The Dirichlet problem for the equations of curvature of orderm, Leningrad Math. J., 2(3), 631-654, (1991)

    work page 1991

  32. [32]

    Lieberman, Second order parabolic differential equations, World Scientific, (1996)

    G. Lieberman, Second order parabolic differential equations, World Scientific, (1996)

    work page 1996

  33. [33]

    Lu, Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic space, Calc

    S. Lu, Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic space, Calc. Var. Partial Differ. Equ., 62(9), 257, (2023)

    work page 2023

  34. [34]

    Mei, The eigenvalue problem for Hessian type operator, Commun

    X. Mei, The eigenvalue problem for Hessian type operator, Commun. Contemp. Math., 25(1), 2150089, (2023)

    work page 2023

  35. [35]

    X. Mei, H. Zhu, Hypersurfaces of prescribed mixed Weingarten curvature, J. Geom. Anal., 34(2), 64, (2024) 32 JIABAO GONG AND QIANG TU ∗

    work page 2024

  36. [36]

    C. Ren, Z. Wang, On the curvature estimates for Hessian equations, Amer. J. Math., 141(5), 1281-1315, (2019)

    work page 2019

  37. [37]

    C. Ren, Z. Wang, The global curvature estimates for then−2 Hessian equation, Calc. Var. Partial Differ. Equ., 62(9), 239, (2023)

    work page 2023

  38. [38]

    Sha, p-convex Riemannian manifolds, Invent

    J. Sha, p-convex Riemannian manifolds, Invent. Math., 83(3), 437-447, (1986)

    work page 1986

  39. [39]

    Sha, Handlebodies and p-convexity, J

    J. Sha, Handlebodies and p-convexity, J. Differential Geom., 25(3), 353-361, (1987)

    work page 1987

  40. [40]

    Sheng, K

    W. Sheng, K. Xue, A class of Hessian quotient equations in warped product manifolds, Sci. China Math., 68(3), 637-648, (2025)

    work page 2025

  41. [41]

    Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Global theory of minimal surfaces, 2, 283-309, (2005)

    J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Global theory of minimal surfaces, 2, 283-309, (2005)

    work page 2005

  42. [42]

    Spruck, L

    J. Spruck, L. Xiao, A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form, Rev. Mat. Iberoam., 33(2), 547-554, (2017)

    work page 2017

  43. [43]

    Sz´ ekelyhidi, V

    G. Sz´ ekelyhidi, V. Tosatti, B. Weinkove, Gauduchon metrics with prescribed volume form, Acta Math., 219, 181-211, (2017)

    work page 2017

  44. [44]

    Tosatti, B

    V. Tosatti, B. Weinkove, The Monge-Amp` ere equation for (n−1)-plurisubharmonic functions on a compact K¨ ahler manifold, J. Amer. Math. Soc., 30(2), 311-346, (2017)

    work page 2017

  45. [45]

    Wang, Starshaped compact hypersurfaces in warped product manifolds II: A class of Hessian type equations, J

    B. Wang, Starshaped compact hypersurfaces in warped product manifolds II: A class of Hessian type equations, J. Geom. Anal., 35(1), 24, (2025)

    work page 2025

  46. [46]

    Wu, Manifolds of partially positive curvature, Indiana Univ

    H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J., 36(3), 525-548, (1987)

    work page 1987

  47. [47]

    Zhou, Curvature estimates for a class of Hessian quotient type curvature equations, Calc

    J. Zhou, Curvature estimates for a class of Hessian quotient type curvature equations, Calc. Var. Partial Differ. Equ., 63(4), 88, (2024) F aculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathe- matics, Hubei University, Wuhan 430062, P.R. China Email address:202321104011284@stu.hubu.edu.cn F aculty of Mathematics and Statistics, Hub...

    work page 2024