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The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space

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Pith's one-line read This paper proves that the horospherical p-Christoffel-Minkowski problem in hyperbolic space has a smooth, even, uniformly h-convex solution whenever the prescribed function satisfies a list of explicit curvature inequalities.

desk verdict The even-case existence theorem is a real improvement over Li-Xu and the proof looks sound, but Corollary 1.7 drops the evenness assumption and is false as stated, so the conformal application needs a correction. read the letter →

arxiv 2411.17328 v1 pith:T2BNFZTL submitted 2024-11-26 math.AP math.DG

Tianci Luo , Yong Wei This is my paper · ORCID
classification math.APmath.DG MSC 53C4253C21
keywords horosphericalp-Christoffel-Minkowskiproblemhyperbolicspaceh-convexfullranktheoremNirenberg-typeSchoutentensorviscositymethodelementarysymmetricfunctions
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The pith

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

The reading

This paper proves an existence theorem for a fully nonlinear PDE arising in hyperbolic convex geometry: given a positive, antipodally symmetric function f on the sphere, find a horospherically convex domain in hyperbolic space whose k-th horospherical p-surface-area measure is f times the round measure. The theorem states that if f satisfies one of several explicit Hessian inequalities, chosen according to the size of p relative to k, then a smooth, even, uniformly h-convex solution exists. The heart of the proof is a full-rank theorem: the defining tensor A[φ] is shown to be positive definite, so the solution is strictly h-convex rather than merely on the boundary of the admissible cone. Because the p=0 case is equivalent to a Nirenberg-type problem for the Schouten tensor of a conformal metric on S^n, the result also yields conformal metrics with prescribed linear combinations of σ_k-curvatures.

What carries the argument

The central object is the symmetric 2-tensor A[φ] defined by A_{ij}[φ] = φ_{ij} − (|Dφ|^2/(2φ))σ_{ij} + (1/2)(φ − 1/φ)σ_{ij}; positivity of A[φ] is equivalent to uniform h-convexity, and the hyperbolic curvature radii are the eigenvalues of φA[φ]. The proof's mechanism is a viscosity argument for the smallest eigenvalue of A[φ], based on a support-function lemma for eigenvalues and commutation identities that compensate for the fact that A is not a Codazzi tensor. The resulting linear differential inequality in the viscosity sense is combined with the strong maximum principle, and this replaces the nonlinear test-function approach used in earlier full-rank theorems.

What would settle it

A direct test is to solve the p=0 equation on $S^{2}$ with an even f that satisfies Condition (1) of Assumption 1.2 but is sharply peaked at the antipode; if the numerical solution has an interior point where the smallest eigenvalue of A[φ] touches zero, the full-rank theorem would be false. More formally, any even $C^{4}$ h-convex solution of (1.8) satisfying Assumption 1.2 with a zero eigenvalue of A[φ] at an interior point would disprove the viscosity maximum-principle argument.

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Extended reading notes

Core claim

The central claim is Theorem 1.4: for integers n≥2 and 1≤k≤n−1, for p≥0, and for a smooth positive even function f on S^n satisfying Assumption 1.2, the equation σ_k(A[φ]) = $φ^{{p−k}}$f has a smooth, even solution φ>1 with A[φ]>0. Here A[φ] is the symmetric 2-tensor that encodes horospherical convexity; its positive definiteness is exactly the condition that the corresponding hypersurface is uniformly h-convex. The key discovery is that the full-rank theorem holds for this tensor even though A[φ] is not a Codazzi tensor: the smallest eigenvalue of A[φ] is shown, by a viscosity argument, to satisfy a linear differential inequality of the form $σ^{{ij}}$_k ψ_{ij} ≤ C(ψ+|Dψ|), and the strong maximum principle then forces that eigenvalue to be positive everywhere once it is positive at one point. This upgrades h-convex solutions to uniformly h-convex solutions and allows the a priori estimates to close.

Load-bearing premise

The argument depends on f being antipodally symmetric, because the lower bound on φ and the gradient bound |Dφ|/φ ≤ 1 are imported from estimates that apply only to origin-symmetric h-convex hypersurfaces.

Editorial extensions

If this is right

  • For every even f satisfying Assumption 1.2, the k-th horospherical p-surface-area measure prescription of Problem 1.1 is solvable by a uniformly h-convex domain.
  • The solution is strictly h-convex: all principal curvatures satisfy κ_i > 1, not merely κ_i ≥ 1.
  • For p=0 and n≥3, the theorem constructs a conformal metric g=φ^{−2}g_{S^n} on S^n solving the Nirenberg-type equation (1.13), with 2Sch_g − g positive definite.
  • The full-rank theorem upgrades any even C^4 h-convex solution satisfying the assumptions to a uniformly h-convex solution, so higher regularity follows from the a priori estimates.
  • The degree-theoretic argument gives an odd degree count, so solutions persist for small perturbations of f within the admissible class.

Reading between the lines

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

  • If sharper a priori estimates that do not require antipodal symmetry were available, the same full-rank and degree machinery would likely extend Theorem 1.4 to non-even f; the evenness enters only through the imported bounds (3.3) and (3.7), not through the viscosity inequality itself.
  • The p=0 corollary offers a purely PDE route to the Nirenberg-type problem; one could test numerically whether the conformal metric it produces approaches the known constant-curvature solution as f tends to a constant, which would probe the sharpness of Assumption 1.2.
  • The case structure of Assumption 1.2 suggests that admissible f are those whose level sets remain close to round spheres; a natural next question is whether existence persists for non-even f that are merely close to constants.
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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

2 major / 3 minor

Summary. The paper studies the fully nonlinear equation σ_k(A[φ]) = φ^{p-k} f on S^n associated with the horospherical p-Christoffel-Minkowski problem in hyperbolic space. For smooth positive even f satisfying one of six curvature conditions in Assumption 1.2, the authors prove existence of a smooth even uniformly h-convex solution φ>1. The proof combines a priori C^0, C^1, C^2 estimates, a viscosity full-rank theorem for the non-Codazzi tensor A[φ], and a degree-theoretic existence argument. For p=0 the authors connect the equation to a Nirenberg-type conformal problem and state a corollary giving conformal metrics on S^n without an evenness assumption.

Significance. If the main theorem is read as a statement about equation (1.8) for even data, it is a solid contribution: it provides a non-flow existence proof for a family of fully nonlinear curvature problems, improves on the unknown-constant result of Li-Xu, and the viscosity approach to the full-rank theorem is a genuine methodological asset. The five-case verification in Claim 2 is careful, and the degree argument is standard. However, the paper overclaims in two load-bearing places: the algebraic passage from the measure equation (1.6) to (1.8) appears to have an exponent error, and the conformal corollary drops the evenness assumption and is false for k=1, p=0 by the Kazdan-Warner obstruction. These issues concern the advertised connection to the original geometric problem, not the internal PDE proof itself.

major comments (2)
  1. [§1, equations (1.6)–(1.8)] The reduction from the measure equation to the studied equation is algebraically inconsistent. With k' = n−k and p' = p+n, one has p'−k' = p+k, so (1.6) becomes σ_{n−k}(A) = C_n^{n−k} φ^{p+k} f, not σ_k(A) = φ^{p−k} f. The subsequent relabeling that produces (1.8) silently changes the exponent. In the p=0 case, (1.8) is σ_k(A)=φ^{−k}f, which is equivalent to the generalized Christoffel equation σ_k(φA)=f, whereas (1.6) for p=0 gives σ_{n−k}(A)=C φ^k f. Thus, as written, Theorem 1.4 does not solve Problem 1.1. This is load-bearing for the abstract and for Remark 1.5, and it must be repaired by either correcting the relation or explicitly declaring a shifted parameter p in (1.8) and adjusting all hypotheses.
  2. [Corollary 1.7 and abstract] Theorem 1.4 assumes f is even, but Corollary 1.7 states that every smooth positive f satisfying Condition (1) yields a conformal metric solving (1.13). This does not follow from the theorem. For k=1 and p=0, equation (1.13) is the prescribed scalar curvature equation S_g = 2(n−1)(f+n/2). Take f = 1 + ε x_1 on S^n with n≥3 and small ε≠0. For sufficiently small ε, Condition (1) holds by continuity from the constant case, but the Kazdan-Warner identity forbids a conformal metric on S^n with scalar curvature proportional to 1+ε x_1. Hence Corollary 1.7 is false as stated. The evenness hypothesis is not a removable technicality: it enters through the estimates (3.3) and (3.7), the full-rank theorem, and the degree setup in Section 5. The conformal existence claim should be restricted to even f.
minor comments (3)
  1. [Title] The title contains a typo: "SP ACE" should be "SPACE".
  2. [§5, linearized operator] The statement that the linearized operator has exactly one positive eigenvalue uses crucially that the space consists of even functions, since the first spherical harmonic has eigenvalue −n and would otherwise give a kernel when b=n. This should be stated explicitly in the sentence after the definition of L_c.
  3. [§4, after (4.10)] The notation "≲" is convenient, but the final viscosity inequality would be easier to verify if the constants and the exact form σ^{ij} ψ_{ij} ≤ C(ψ + |Dψ|) were written out at the point where the strong maximum principle is invoked.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the existence proof rests on external estimates (Li-Xu), an external viscosity lemma (Brendle-Choi-Daskalopoulos), and degree theory (Li). Corollary 1.7 drops the evenness hypothesis required by Theorem 1.4, which is a correctness overclaim, not circularity. Self-citations [1] and [25] are background only.

full rationale

No significant circularity. Theorem 1.4 is proved through a priori estimates (Section 3), a viscosity full-rank theorem (Section 4), and degree theory (Section 5), and no step defines the target solution in terms of itself; no fitted quantity is renamed as a prediction. Assumption 1.2 is a convexity-type hypothesis on the prescribed function f, and Claim 2 of the full-rank proof closes by reducing to exactly this assumption in Cases (1)-(5); that is a designed sufficient condition, not a self-reduction, because the conclusion A[φ] > 0 is never assumed in the hypothesis. The load-bearing imports are external: estimates (3.3) and (3.7) come from Li-Xu [29, Lemmas 7.2, 7.3] under stated hypotheses (origin-symmetric h-convex hypersurfaces) that do not include the target existence result; Lemma 4.1 comes from Brendle-Choi-Daskalopoulos [4]; and Lemma 5.1 (uniqueness of constant solutions) comes from Li-Xu [29, Theorem 8.1] and is used only to evaluate the degree at the constant solution, not to obtain general existence. Self-citations are [1] (Andrews-Chen-Wei, co-author Yong Wei), supplying the geometric framework (tensor A[φ], relation (1.5)/(2.8), Lemma 2.6), and [25] (Hu-Li-Wei), a background mention; both are published, definitional, and do not carry the existence claim. One passage is flagged for missing support but is not circularity: Corollary 1.7 asserts the Nirenberg-type application for arbitrary smooth positive f with only Condition (1), whereas Theorem 1.4 and every estimate in Section 3 require f even, since (3.3) and (3.7) are origin-symmetric estimates; for k=1, p=0 the claimed statement conflicts with the Kazdan-Warner obstruction for non-even f such as 1+εx1. This broken inference increases the correctness risk of the advertised conformal application, but the derivation chain does not reduce to its own inputs, so the circularity score stays low (1).

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities or fitted constants. It relies on a chain of prior results from [1], [29], [4], [35], and [3]; the most load-bearing imported inputs are the evenness-dependent estimates (3.3) and (3.7) from Li-Xu.

assumptions (6)
  • domain assumption The horospherical support function representation (1.3) and the Weingarten relation (2.8) for h-convex hypersurfaces.
    Imported from [1, §5] and [29, §2-4]; these are the geometric bridge between the PDE variable φ and hyperbolic curvature.
  • domain assumption The origin-symmetric estimates (3.3) and (3.7): 1/2(max φ + 1/max φ) ≤ min φ and |Dφ|^2/φ^2 ≤ 1 - 1/φ^2.
    Quoted from [29, Lemmas 7.2, 7.3]; they require evenness and are used in Lemmas 3.1 and 3.4.
  • standard math Brendle-Choi-Daskalopoulos Lemma 5, the viscosity inequality for the smallest eigenvalue of a symmetric 2-tensor, stated as Lemma 4.1.
    Quoted from [4]; supplies (4.1)-(4.2) at a lower support ψ.
  • standard math Inverse-concavity inequality for σ_k, Lemma 2.3, from [35, (3.49)].
    Used in Claim 1 of the full rank theorem to produce the negative quadratic term.
  • domain assumption The uniqueness of constant solutions to (5.1), Lemma 5.1, is a corollary of [29, Theorem 8.1].
    Used in the degree theory computation at t=0.
  • standard math Strong maximum principle for viscosity solutions of degenerate elliptic equations [3].
    Used at the end of Theorem 4.2 to conclude the zero set of λ_1 is open.

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Pith. "Pith review of The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space." pith.science (2026). https://pith.science/paper/T2BNFZTL

@misc{pith2026241117328,
  author       = {Pith},
  title        = {Pith review of: The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2BNFZTL}},
  note         = {Machine review of arXiv:2411.17328}
}
abstract

The horospherical $p$-Christoffel-Minkowski problem was posed by Li and Xu (2022) as a problem prescribing the $k$-th horospherical $p$-surface area measure of $h$-convex domains in hyperbolic space $\mathbb{H}^{n+1}$. It is a natural generalization of the classical $L^p$ Christoffel-Minkowski problem in the Euclidean space $\mathbb{R}^{n+1}$. In this paper, we consider a fully nonlinear equation associated with the horospherical $p$-Christoffel-Minkowski problem. We establish the existence of a uniformly $h$-convex solution under appropriate assumptions on the prescribed function. The key to the proof is the full rank theorem, which we will demonstrate using a viscosity approach based on the idea of Bryan-Ivaki-Scheuer (2023). When $p=0$, the horospherical $p$-Christoffel-Minkowski problem in $\mathbb{H}^{n+1}$ is equivalent to a Nirenberg-type problem on $\mathbb{S}^n$ in conformal geometry. Therefore, our result implies the existence of solutions to the Nirenberg-type problem.

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Works this paper leans on

37 extracted references · 36 canonical work pages

  1. [1]

    Ben Andrews, Xuzhong Chen, and Yong Wei, Volume preserving flow and Alexandrov-Fenchel type inequal ities in hyperbolic space , J. Eur. Math. Soc. (JEMS) 23 (2021), no. 7, 2467–2509

    work page 2021

  2. [2]

    Ben Andrews, James McCoy, and Yu Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 611–665

    work page 2013

  3. [3]

    Martino Bardi and Francesca Da Lio, On the strong maximum principle for fully nonlinear degener ate elliptic equations, Arch. Math. (Basel) 73 (1999), no. 4, 276–285

    work page 1999

  4. [4]

    219 (2017), no

    Simon Brendle, Kyeongsu Choi, and Panagiota Daskalopou los, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math. 219 (2017), no. 1, 1–16

    work page 2017

  5. [5]

    Ivaki, and Julian Scheuer, Constant rank theorems for curvature problems via a viscosity approach, Calc

    Paul Bryan, Mohammad N. Ivaki, and Julian Scheuer, Constant rank theorems for curvature problems via a viscosity approach, Calc. Var. Partial Differential Equations 62 (2023), no. 3, Paper No. 98, 19

    work page 2023

  6. [6]

    , Christoffel-Minkowski flows , Trans. Amer. Math. Soc. 376 (2023), no. 4, 2373–2393

    work page 2023

  7. [7]

    Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations , Duke Math

    Luis A. Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations , Duke Math. J. 52 (1985), no. 2, 431–456

    work page 1985

  8. [8]

    Chang, Matthew J

    Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, An a priori estimate for a fully nonlinear equation on four-manifolds , J. Anal. Math. 87 (2002), 151–186

    work page 2002

Show all 37 references
  1. [9]

    , An equation of Monge-Amp` ere type in conformal geometry, an d four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002), no. 3, 709–787

    2002

  2. [10]

    Chang, Conformal invariants and partial differential equations , Bull

    Sun-Yung A. Chang, Conformal invariants and partial differential equations , Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 3, 365–393

    2005

  3. [11]

    Chuanqiang Chen, Pengfei Guan, Junfang Li, and Julian S cheuer, A fully-nonlinear flow and quermassintegral inequalities in the sphere , Pure Appl. Math. Q. 18 (2022), no. 2, 437–461

    2022

  4. [12]

    Li Chen, Smooth solutions to the Christoffel problem in Hn+1, available at arXiv:2406.09449

    arXiv

  5. [13]

    Espinar, Jos´ e A

    Jos´ e M. Espinar, Jos´ e A. G´ alvez, and Pablo Mira,Hypersurfaces in Hn+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems , J. Eur. Math. Soc. (JEMS) 11 (2009), no. 4, 903–939

    2009

  6. [14]

    244 (2024), Paper No

    Chaoqun Gao and Rong Zhou, Geometric inequalities and their stabilities for modified q uermassintegrals in hyperbolic space, Nonlinear Anal. 244 (2024), Paper No. 113537, 20

    2024

  7. [15]

    Yuxin Ge and Guofang Wang, On a fully nonlinear Yamabe problem , Ann. Sci. ´Ecole Norm. Sup. (4) 39 (2006), no. 4, 569–598 (English, with English and French sum maries)

    2006

  8. [16]

    Differential Geom

    Claus Gerhardt, Curvature flows in the sphere , J. Differential Geom. 100 (2015), no. 2, 301–347

    2015

  9. [17]

    Trudinger, Elliptic partial differential equations of second order , Classics in Math- ematics, Springer-Verlag, Berlin, 2001

    David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order , Classics in Math- ematics, Springer-Verlag, Berlin, 2001. Reprint of the 199 8 edition

    2001

  10. [18]

    Pengfei Guan, Topics in Geometric Fully Nonlinear Equations (2006), available at https://www.math.mcgill.ca/guan/zheda0508.pdf

    2006

  11. [19]

    2087, Springer, Cham, 2014, pp

    , Curvature measures, isoperimetric type inequalities and f ully nonlinear PDEs , Fully nonlinear PDEs in real and complex geometry and optics, Lecture Notes in Mat h., vol. 2087, Springer, Cham, 2014, pp. 47–94. HOROSPHERICAL p-CHRISTOFFEL-MINKOWSKI PROBLEM 25

    2014

  12. [20]

    Pengfei Guan and Xi-Nan Ma, The Christoffel-Minkowski problem. I. Convexity of solution s of a Hessian equation, Invent. Math. 151 (2003), no. 3, 553–577

    2003

  13. [21]

    Reine Angew

    Pengfei Guan and Guofang Wang, A fully nonlinear conformal flow on locally conformally flat m anifolds, J. Reine Angew. Math. 557 (2003), 219–238

    2003

  14. [22]

    Pengfei Guan and Chao Xia, Lp Christoffel-Minkowski problem: the case 1 < p < k + 1, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 69, 23

    2018

  15. [23]

    Gursky and Jeff A

    Matthew J. Gursky and Jeff A. Viaclovsky, Prescribing symmetric functions of the eigenvalues of the R icci tensor, Ann. of Math. (2) 166 (2007), no. 2, 475–531

    2007

  16. [24]

    Changqing Hu, Xi-Nan Ma, and Chunli Shen, On the Christoffel-Minkowski problem of Firey’s p-sum, Calc. Var. Partial Differential Equations 21 (2004), no. 2, 137–155

    2004

  17. [25]

    Yingxiang Hu, Haizhong Li, and Yong Wei, Locally constrained curvature flows and geometric inequali ties in hyperbolic space, Math. Ann. 382 (2022), no. 3-4, 1425–1474

    2022

  18. [26]

    Ivaki, Deforming a hypersurface by principal radii of curvature an d support function , Calc

    Mohammad N. Ivaki, Deforming a hypersurface by principal radii of curvature an d support function , Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 1, 18

    2019

  19. [27]

    Differential Equations 13 (2008), no

    Qinian Jin, YanYan Li, and Haoyuan Xu, Symmetry and asymmetry: the method of moving spheres , Adv. Differential Equations 13 (2008), no. 7-8, 601–640

    2008

  20. [28]

    Pure Appl

    Aobing Li and Yanyan Li, On some conformally invariant fully nonlinear equations , Comm. Pure Appl. Math. 56 (2003), no. 10, 1416–1464

    2003

  21. [29]

    Haizhong Li and Botong Xu, Hyperbolic p-sum and Horospherical p-Brunn-Minkowski the ory in hyperbolic space, available at arXiv:2211.06875

    arXiv

  22. [30]

    Haizhong Li and Yao Wan, The Christoffel problem in the hyperbolic plane , Adv. in Appl. Math. 150 (2023), Paper No. 102557, 17

    2023

  23. [31]

    Partial Differential Equations 14 (1989), no

    Yan Yan Li, Degree theory for second order nonlinear elliptic operator s and its applications , Comm. Partial Differential Equations 14 (1989), no. 11, 1541–1578

    1989

  24. [32]

    , Prescribing scalar curvature on Sn and related problems. I , J. Differential Equations 120 (1995), no. 2, 319–410

    1995

  25. [33]

    Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem , J. Differ- ential Geom. 38 (1993), no. 1, 131–150

    1993

  26. [34]

    Trudinger, and Xu-Jia Wang, The Yamabe problem for higher order curvatures , J

    Wei-Min Sheng, Neil S. Trudinger, and Xu-Jia Wang, The Yamabe problem for higher order curvatures , J. Differential Geom. 77 (2007), no. 3, 515–553

    2007

  27. [35]

    John I. E. Urbas, An expansion of convex hypersurfaces , J. Differential Geom. 33 (1991), no. 1, 91–125

    1991

  28. [36]

    Viaclovsky, Conformal geometry, contact geometry, and the calculus of v ariations, Duke Math

    Jeff A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of v ariations, Duke Math. J. 101 (2000), no. 2, 283–316

    2000

  29. [37]

    79 (2024), no

    Ruijia Zhang, A curvature flow approach to Lp Christoffel-Minkowski problem for 1 < p < k + 1, Results Math. 79 (2024), no. 2, Paper No. 53, 22. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China Email address : Luo tianci@ma...

    2024

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