Pith. sign in

REVIEW 3 major objections 3 minor 59 references

The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A full-rank theorem removes the normalization constant in the horospherical p-Christoffel-Minkowski problem.

desk verdict Solid paper, real results, but the removal of the normalization constant for p>−n rests on an unproved imported lemma from the authors' earlier flow paper. read the letter →

arxiv 2411.17345 v1 pith:ILICD6WL submitted 2024-11-26 math.DG math.AP

Yingxiang Hu , Haizhong Li , Botong Xu This is my paper · ORCID
classification math.DGmath.AP MSC 58J0552A55
keywords horosphericalChristoffel-Minkowskiproblemhyperbolicspacefullranktheoremprescribedp-shiftedWeingartencurvatureconvexitydegreetheoryelementarysymmetricpolynomialsMinkowski
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

This paper proves that the horospherical $p$-Christoffel-Minkowski problem in hyperbolic space admits smooth, origin-symmetric, strictly horospherically convex solutions for $n\ge 2$, $1\le k\le n-1$, and $p\ge -n$, under explicit convexity assumptions on the prescribed function $f$. The improvement over prior work is that the normalization constant $\gamma$ in the flow-based existence theorem of [LX22] is removed, so the equation $\varphi^{-p-k}p_{n-k}(A[\varphi])=f$ holds with $f$ prescribed exactly. The key ingredient is a full-rank theorem: any $C^4$ even solution with $A[\varphi]\ge 0$ must in fact have $A[\varphi]>0$, upgrading weak horospherical convexity to strict horospherical convexity. The same machinery yields existence for a newly proposed prescribed $p$-shifted Weingarten curvature problem and for the horospherical $p$-Minkowski problem as the case $k=0$. These are the hyperbolic counterparts of classical Minkowski-type and Weingarten-type curvature problems, and the strict convexity upgrade is what lets the homotopy and degree argument close.

What carries the argument

The load-bearing object is the operator $A[\varphi]=D^2\varphi-\frac12|D\varphi|^2\varphi^{-1}\sigma+\frac12(\varphi-\varphi^{-1})\sigma$ on the sphere, and the full-rank theorem (Theorem 5.1) is the mechanism that carries the argument. It states that, under Assumption 1.1, every $C^4$ even solution of (1.2) with $A[\varphi]\ge0$ has $A[\varphi]>0$ everywhere, so weak horospherical convexity is automatically strict. The proof uses the deformation lemma (Lemma 4.1), which derives a differential inequality for $p_{\ell+1}(A[\varphi])$ whenever $p_\ell(A[\varphi])$ is bounded below, together with the shifted Minkowski formula (5.2), which forces a solution with a vanishing minor to be the constant $\varphi\equiv1$, contradicting the equation. Strict convexity keeps solutions inside the open admissible set where the degree-theoretic existence argument [Li89] applies.

What would settle it

Test the full-rank theorem directly: exhibit a smooth, positive, even $f$ satisfying Assumption 1.1 for some $p>-n$ such that the matrix (5.1) has a negative eigenvalue at some point, and show a $C^4$ even solution of (1.2) with $A[\varphi]\ge0$ but $A[\varphi]$ not positive definite; this would contradict Theorem 5.1. Alternatively, for $p=n-2k$ with $f\equiv c\ge 2^{k-n}$, or $p>n-2k$ with $f$ large, the theorem predicts no even h-convex solution, so finding one numerically would falsify it.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for $n\ge2$, $1\le k\le n-1$, $p\ge -n$, and a smooth, positive, even function $f$ on $S^n$ satisfying Assumption 1.1 (plus Assumption 1.2 when $p\ge n-2k$), the equation $\varphi^{-p-k}p_{n-k}(A[\varphi])=f$ has a smooth, even, strictly horospherically convex solution. Here $A[\varphi]=D^2\varphi-\frac12|D\varphi|^2\varphi^{-1}\sigma+\frac12(\varphi-\varphi^{-1})\sigma$ is the tensor whose positivity defines strict horospherical convexity, and $p_{n-k}$ is the normalized elementary symmetric polynomial of degree $n-k$. The proof combines a deformation lemma, the full-rank theorem, a priori estimates, and degree theory. The same argument proves Theorem 1.2 for the prescribed $p$-shifted Weingarten curvature problem, Theorem 1.3 for the horospherical $p$-Minkowski problem with $k=0$ and $p\ge -n$, and Theorem 1.4 for the hyperbolic plane $n=1$, where the range of $p$ is optimal.

Load-bearing premise

For $p>-n$, the proof that the matrix (5.1) is positive semi-definite is not derived here; it is inherited from [LX22, Lem. 7.6 & Assump. 7.1], so the full-rank theorem, and with it Theorem 1.1, would collapse if that lemma's hypotheses are not exactly satisfied by the functions $f$ considered.

Editorial extensions

If this is right

  • The equation (1.2) is solvable with $f$ prescribed exactly; the normalization constant $\gamma$ that appeared in the prior flow-based theorem is no longer needed.
  • For $k=n-1$ and $p=-n$, Theorem 1.1 recovers the Christoffel problem in hyperbolic space, and for $k=0$ it gives the horospherical $p$-Minkowski problem, so the result unifies these hyperbolic measure problems.
  • The newly proposed prescribed $p$-shifted Weingarten curvature problem (1.5) has a smooth, even, strictly h-convex solution for all $0\le k\le n-1$ and $p\ge -n$ under Assumption 1.2 when $p\ge n-2k$.
  • In the hyperbolic plane, the horospherical $p$-Minkowski problem has a smooth even solution for $-7\le p<\infty$ with the stated bounds on $f$, and this range is optimal in view of the invertibility of the linearized operator.
  • For constant data $f$, the even h-convex solutions of (1.2) are constant when $p\ge -n$, matching the classification shown in [LX22].

Reading between the lines

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

  • The deformation lemma is written for the general equation $S_k(A[\varphi])=\varphi^{n+p-k}f$, so the same full-rank strategy should transfer to other curvature problems in hyperbolic space once an analogue of the matrix positivity condition (5.1) can be verified for the prescribed function.
  • The removal of the normalization constant suggests that the pinching estimates used in curvature-flow proofs can be replaced by a static convexity argument; if that is true, the flow-based existence for the hyperbolic $p$-sum family could be simplified or extended.
  • A testable extension is to drop the evenness assumption for $p>-n$: the Kazdan-Warner obstruction cited for $p=-n$ shows evenness is necessary there, but for $p>-n$ the full-rank theorem may hold without it, in which case the degree-theoretic proof would run on the full space of functions on $S^n$.
  • The a priori bounds in Lemma 3.1 make the failure of existence concrete: for $p\ge n-2k$, any positive even $f$ whose supremum violates Assumption 1.2 should admit no even h-convex solution, so the theorem could be checked numerically by solving (1.2) for such $f$.
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.

Referee Report

3 major / 3 minor

Summary. The paper studies two fully nonlinear curvature problems for horospherically convex hypersurfaces in hyperbolic space. The first is the horospherical p-Christoffel-Minkowski problem (1.2), for which the authors claim, under Assumptions 1.1 and 1.2, existence of smooth, even, strictly horospherically convex solutions for n≥2, 1≤k≤n−1 and p≥−n, thereby removing the normalization constant γ from the earlier flow-based result of Li and Xu. The second is a newly proposed prescribed p-shifted Weingarten curvature problem (1.5), for which an analogous existence theorem is stated. The proof combines a priori C^0, C^1, C^2 and higher-order estimates, a deformation lemma, a full rank theorem (Theorem 5.1) asserting that any even C^4 solution with A[ϕ]≥0 actually satisfies A[ϕ]>0, and a degree-theoretic argument with a homotopy to constant data.

Significance. If correct, the paper gives a substantial improvement over the prior existence theorem of Li and Xu by removing the normalizing constant in the horospherical p-Christoffel-Minkowski problem, and it introduces a new Weingarten-type problem in hyperbolic space. The a priori estimates in Section 3 and the deformation lemma in Section 4 are worked out in detail and are largely self-contained; the shifted Minkowski formula (5.2) is proved in the text; and the degree computation is explicit. The main reservation is that the full rank theorem for p>−n depends on an imported lemma from a curvature-flow paper, and the C^0 estimate contains an exponent inconsistency in the q>1 case. These points are load-bearing for the central claims, so the paper is not yet in publishable form.

major comments (3)
  1. [Section 5, proof of Lemma 5.1] For p>−n, the positive semidefiniteness of the matrix (5.1) is not proved in the manuscript. The second paragraph of the proof of Lemma 5.1 states that (5.1) is exactly [LX22, Eq. (7.39)] and then invokes [LX22, Lem. 7.6 & Assump. 7.1] to obtain Assumptions 1.1(2)–(5). This is a dependency gap: the cited lemma arose in a parabolic pinching estimate, and the manuscript does not verify that its hypotheses are satisfied by static even solutions of the elliptic equation (1.2). In particular, for p≥−k the matrix (5.1) is written through f^{-1/(n−k)}, while Assumptions 1.1(4)–(5) are conditions on f^{-1/(n+p)}; the implication between the two is exactly the nontrivial content of the imported lemma and is not demonstrated. Since Lemma 5.1 is the only bridge from weak h-convexity to strict h-convexity, the full rank theorem and Theorem 1.1 inherit this gap. The authors should either reproduce the lemma and its proof in the static setting or give a direct derivation of the positivity of (5.1) from Assumptions 1.1(2)–(5).
  2. [Section 3, Lemma 3.1 (C^0-estimate)] The formula for the minimum of ξ_q at t=√((q+1)/(q−1)) is incorrect for q>1. From the definition ξ_q(t)=2t^q(t−t^{−1})^{-1}, the minimum is (q+1)^{(q+1)/2}(q−1)^{−(q−1)/2}, not (q+1)^{(q+1)/2}(q−1)^{(q−1)/2} as printed in Lemma 3.1. Consequently the restated assumption in the proof of Lemma 3.1, namely 0<f<((q+1)^{(q+1)/2}(q−1)^{(q−1)/2})^{k−n}, is not equivalent to Assumption 1.2; the exponent on (q−1) must be negative. As written, the proof of the C^0-estimate for q>1 is not justified. The same issue likely affects the displayed form of Assumption 1.2 in the introduction, where the first case appears as 2k−n but should be 2^{k−n}.
  3. [Section 6, proof of Theorem 1.1] The degree-theoretic step uses the assertion that the linearized operator L_{c0} is invertible on even functions for every q≥−1. The text argues that the only possible kernel would come from the eigenvalue −n of the Laplace operator, which is odd. This is correct after the substitution for the constant solution, but the reader must reconstruct the computation: the coefficient μ=(1−q)/2+(1+q)/(2c_0^2) satisfies 0<μ<1 for −1<q<1 and μ=1 at q=−1, so no nonzero even eigenfunction of Δ can satisfy Δη=−nμη. For q>1 the open set O_R is chosen so that c_0>√((q+1)/(q−1)), which makes μ<0 and the same conclusion holds. The argument is therefore sound, but it would help to state this verification explicitly rather than leaving it implicit in the inequalities.
minor comments (3)
  1. [Notation, Section 4] The symbol k is used both for the order of the elementary symmetric function in Lemma 4.1 and for the parameter k in the Christoffel-Minkowski problem; in Lemma 5.1 the two are related by replacing k with n−k, which is easy to lose track of. A notational distinction (for example K=n−k) would improve readability.
  2. [Assumption 1.2] The condition in the first line of Assumption 1.2 appears as `0<f<2k−n`, which is impossible for many admissible pairs (e.g. k=1,n=2). The intended condition is almost certainly `0<f<2^{k−n}`; the same typo seems to appear in the second remark after Theorem 1.1.
  3. [References] The reference [Pog53] in the bibliography is not cited in the text; either cite it where the classical Minkowski problem is discussed or remove it.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the full rank theorem and degree-theoretic existence proof do not reduce to their inputs; the main concern is a load-bearing self-citation to [LX22] that is a dependency gap rather than a circular step.

full rationale

The central existence theorem is not circular. Theorem 1.1 is proved by a degree-theoretic argument that is internally developed: a priori C0, C1, C2 estimates (Lemmas 3.1-3.3), the deformation lemma (Lemma 4.1) and its long algebraic consequences, the full rank theorem (Theorem 5.1), and a homotopy to a constant equation whose degree is computed explicitly. The result is benchmarked against the up-to-constant existence theorem of [LX22, Thm. 7.3] and the constant-solution classification of [LX22, Prop. 8.1 & Thm. 8.1] and [LW24]. No fitted parameter is renamed as a prediction, and no equation is defined in terms of the conclusion. The only notable self-citation issue is in the proof of Lemma 5.1: for p > -n, the paper does not derive the positive semidefiniteness of the matrix (5.1) from scratch, but states that the matrix is exactly [LX22, Eq. (7.39)] and invokes [LX22, Lem. 7.6 & Assump. 7.1], giving Assumption 1.1(2)-(5). This is load-bearing for the strict h-convexity upgrade and hence for Theorem 5.1 and Theorem 1.1. It is also a self-citation, since [LX22] is by the same authors. However, this is a dependency and verification gap, not circularity: the cited lemma is a prior result with stated hypotheses, and the paper does not show that those hypotheses coincide with the theorem being proved. Unless one assumes the cited lemma already contains the full-rank conclusion, there is no reduction of the target result to itself. The paper also cites [HLW22] and [Che24] for the shifted Minkowski formula and the p=-n, k=n-1 case; those are external or companion results and do not create circularity. Overall, the derivation is substantially self-contained once the [LX22] pinching lemma is accepted; the appropriate finding is no significant circularity, with a minor score adjustment for the heavy, unverified self-citation in the pivotal Lemma 5.1.

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

No free parameters are fitted to data; the constants in Assumption 1.2 are geometric thresholds, not fitted values. No new particles, forces, dimensions, or physical entities are introduced. The p-shifted Weingarten curvature problem is a new PDE problem, not a postulated entity. The main external load is carried by the authors' prior work LX22, specifically the estimates (3.2), (3.3), the sufficiency of Assumption 1.1 for (5.1), and the uniqueness of constant solutions.

assumptions (6)
  • domain assumption Evenness of f and of the solution φ; origin-symmetric domain
    Evenness is used to guarantee φ≥1 and the comparison cosh(log φ_max) ≤ φ_min (cited from LX22 Lem 7.2); all main theorems assume f even and construct even solutions.
  • domain assumption Assumption 1.1 convexity-type matrix inequalities on the prescribed function f
    These hypotheses make the matrix (5.1) positive semidefinite in Lemma 5.1, which is the key input to the full rank theorem (Theorem 5.1). They are stated as conditions, not derived.
  • domain assumption Assumption 1.2 upper bounds on f when p≥n−2k
    Needed in Lemma 3.1 (C^0 estimates) and the q>1 degree argument to avoid boundary solutions; the p-form of the bound is consistent with the constant-solution analysis.
  • domain assumption Prior estimates imported from [LX22]: (3.2), (3.3), and Lemma 7.6/Assumption 7.1 showing (5.1) is positive semidefinite under Assumption 1.1(2)-(5)
    The paper cites these rather than proving them; the full rank theorem for p>−n depends on them.
  • domain assumption Uniqueness of constant even h-convex solutions to (6.1) from [LX22, Prop. 8.1, Thm. 8.1] and [LW24, Thm. 1.2]
    Used in the degree computation at t=0 (Lemma 6.1 and Proof of Theorem 1.1) to identify the constant solutions and their linearized operator.
  • standard math Standard tools: strong minimum principle, Li's degree theory for second-order fully nonlinear elliptic operators, Krylov-Evans and Schauder estimates, Newton-MacLaurin inequalities
    Invoked in Sections 3, 5, 6; these are background results in PDE and geometric analysis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space." pith.science (2026). https://pith.science/paper/ILICD6WL

@misc{pith2026241117345,
  author       = {Pith},
  title        = {Pith review of: The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILICD6WL}},
  note         = {Machine review of arXiv:2411.17345}
}
abstract

The $L_p$-Christoffel-Minkowski problem and the prescribed $L_p$-Weingarten curvature problem for convex hypersurfaces in Euclidean space are important problems in geometric analysis. In this paper, we consider their counterparts in hyperbolic space. For the horospherical $p$-Christoffel-Minkowski problem first introduced and studied by the second and third authors, we prove the existence of smooth, origin-symmetric, strictly horospherically convex solutions by establishing a new full rank theorem. We also propose the prescribed $p$-shifted Weingarten curvature problem and prove an existence result.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 56 canonical work pages

  1. [1]

    Andrews, X

    B. Andrews, X. Chen, and Y. Wei, Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space, J. Eur. Math. Soc. (JEMS) 23(2021): 2467--2509

    work page 2021

  2. [2]

    Berg, Corps convexes et potentiels sph\'eriques (in French), Danske Vid Selsk Mat-Fys Medd, 37(1969): 1--64

    C. Berg, Corps convexes et potentiels sph\'eriques (in French), Danske Vid Selsk Mat-Fys Medd, 37(1969): 1--64

    work page 1969

  3. [3]

    B. Bian, P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math. 177(2009): 307--335

    work page 2009

  4. [4]

    Bianchini, A

    C. Bianchini, A. Colesanti, D. Pagnini, A. Roncoroni, On p -Brunn-Minkowski inequalities for intrinsic volumes, with 0 p < 1 , Math. Ann. 387 (2023): 321--352

    work page 2023

  5. [5]

    A. A. Borisenko, V. Miquel, Total curvatures of convex hypersurfaces in hyperbolic space, Illinois J. Math. 43 (1999): 61--78

    work page 1999

  6. [6]

    B\"or\"oczky, The logarithmic Minkowski conjecture and the L_p -Minkowski problem, Adv

    K. B\"or\"oczky, The logarithmic Minkowski conjecture and the L_p -Minkowski problem, Adv. Anal. Geom. 9 (2023): 83--118

    work page 2023

  7. [7]

    Bryan, M

    P. Bryan, M. N. Ivaki, J. Scheuer, Parabolic approaches to curvature equations, Nonlinear. Anal. 203(2021): 112174

    work page 2021

  8. [8]

    Bryan, M

    P. Bryan, M. N. Ivaki, J. Scheuer, Constant rank theorems for curvature problems via a viscosity approach, Calc. Var. (2023) 62:98

    work page 2023

Show all 59 references
  1. [9]

    Caffarelli, A localization property of viscosity solutions to the Monge-Amp\'ere equation and their strict convexity, Ann

    L. Caffarelli, A localization property of viscosity solutions to the Monge-Amp\'ere equation and their strict convexity, Ann. Math. 131(1990): 129--134

    1990

  2. [10]

    Caffarelli, A

    L. Caffarelli, A. Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985): 431--456

    1985

  3. [11]

    Caffarelli, P

    L. Caffarelli, P. Guan, X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math., 60(2007): 1769--1791

    2007

  4. [12]

    Chen, Non-normalized solutions to the horospherical Minkowski problem, Preprint at https://arxiv.org/abs/2301.01128v2 (2023)

    L. Chen, Non-normalized solutions to the horospherical Minkowski problem, Preprint at https://arxiv.org/abs/2301.01128v2 (2023)

    2023 arXiv

  5. [13]

    Chen, Convex hypersurfaces of prescribed curvatures in hyperbolic space, Preprint at https://arxiv.org/abs/2302.01604 (2023)

    L. Chen, Convex hypersurfaces of prescribed curvatures in hyperbolic space, Preprint at https://arxiv.org/abs/2302.01604 (2023)

    2023 arXiv

  6. [14]

    Chen, Smooth solutions to the Christoffel problem in ^ n+1 , Preprint at https://arxiv.org/abs/2406.09449 (2024)

    L. Chen, Smooth solutions to the Christoffel problem in ^ n+1 , Preprint at https://arxiv.org/abs/2406.09449 (2024)

    2024 arXiv

  7. [15]

    Cheng, S

    S.-Y. Cheng, S. T. Yau, On the regularity of the solution of the n -dimensional Minkowski problem, Comm. Pure Appl. Math. 29(1976): 495--516

    1976

  8. [16]

    Chou and X.-J

    K.-S. Chou and X.-J. Wang, The L_p -Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205, no.1 (2006): 33--83

    2006

  9. [17]

    R. J. Currier, On hypersurfaces of Hyperbolic space infinitesimally supported by horospheres, Trans. Amer. Math. Soc. 313(1989): 419--431

    1989

  10. [18]

    J. M. Espinar, J. A. G\'alvez, P. Mira, Hypersurfaces in ^ n+1 and conformally invariant equations: the generalized C hristoffel and N irenberg problems , J. Eur. Math. Soc. (JEMS) 11, no. 4 (2009): 903--939

    2009

  11. [19]

    Firey, p -means of convex bodies, Math

    W. Firey, p -means of convex bodies, Math. Scand. 10(1962): 17--24

    1962

  12. [20]

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

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

    1967

  13. [21]

    Gallego, A

    E. Gallego, A. Revent\'os, G. Solanes, E. Teufel, Width of convex bodies in spaces of constant curvature, Manuscripta Math. 126 (2008): 115--134

    2008

  14. [22]

    Gerhardt, Minkowski type problems for convex hypersurfaces in hyperbolic space, Preprint at https://arxiv.org/abs/math/0602597 (2006)

    C. Gerhardt, Minkowski type problems for convex hypersurfaces in hyperbolic space, Preprint at https://arxiv.org/abs/math/0602597 (2006)

    2006 arXiv

  15. [23]

    Gerhardt, Curvature Problems, Series in Geometry and Topology, International Press, Somerville 39(2006)

    C. Gerhardt, Curvature Problems, Series in Geometry and Topology, International Press, Somerville 39(2006)

    2006

  16. [24]

    Gilbarg, N.S

    D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

    2001

  17. [25]

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

    2002

  18. [26]

    Guan, Curvature measures, isoperimetric type inequalities and fully nonlinear PDEs, Fully Nonlinear PDEs in Real and Complex Geometry and Optics

    P. Guan, Curvature measures, isoperimetric type inequalities and fully nonlinear PDEs, Fully Nonlinear PDEs in Real and Complex Geometry and Optics. Lecture Notes in Mathematics, vol. 2087, pp. 47--94. Springer (2013)

    2013

  19. [27]

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

    2012

  20. [28]

    Guan, C.-S

    P. Guan, C.-S. Lin, X.-N. Ma, The Christoffel-Minkowski problem. II. Weingarten curvature equations, Chinese Ann. Math. Ser. B, 27(2006), no. 6: 595--614

    2006

  21. [29]

    Guan, C.-S

    P. Guan, C.-S. Lin, X.-N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not. 2009(2009), no. 11: 1947--1975

    2009

  22. [30]

    Guan, X.-N

    P. Guan, X.-N. Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math., 151(2003): 553--577

    2003

  23. [31]

    Guan, X.-N

    P. Guan, X.-N. Ma, F. Zhou, The Christoffel-Minkowski problem. III. Existence and convexity of admissible solutions, Comm. Pure Appl. Math., 59(2006), no. 9: 1352--1376

    2006

  24. [32]

    P. Guan, C. Ren, Z. Wang, Global C^2 -estimates for convex solutions of curvature equations, Commun. Pure Appl. Math.8(2015): 1287--1325

    2015

  25. [33]

    P. Guan, C. Xia, L^p Christoffel-Minkowski problem: the case 1<p<k+1 , Calc. Var. (2018): 57:69

    2018

  26. [34]

    Guang, Q.-R

    Q. Guang, Q.-R. Li, and X.-J. Wang. The L_p -Minkowski problem with super-critical exponents, Preprint arXiv:2203.05099

    arXiv

  27. [35]

    Hu, X.-N

    C. Hu, X.-N. Ma, C. Shen, On the Christoffel-Minkowski problem, Calc. Var. 21(2004): 137--155

    2004

  28. [36]

    Y. Hu, M. N. Ivaki, Prescribed L_p curvature problem, Adv. Math. 442(2024), 109566

    2024

  29. [37]

    Y. Hu, H. Li, Y. Wei, Locally constrained curvature flow and geometric inequalities in hyperbolic space, Math. Ann. 382(2022): 1425--1474

    2022

  30. [38]

    Y. Hu, Y. Wei, and T. Zhou: A Heintze-Karcher type inequality in hyperbolic space, J. Geom. Anal. 34(2024), Paper No. 113, 17 pp

    2024

  31. [39]

    D. Hug, E. Lutwak, D. Yang, G. Zhang, On the L_p Minkowski problem for polytopes, Discrete Comput. Geom. 33 (2005): 699--715

    2005

  32. [40]

    H. Jian, J. Lu, G. Zhu, Mirror symmetric solutions to the centro-affine Minkowski problem, Calc. Var. Partial Differential Equations 55 (2016): Art. 41, 22

    2016

  33. [41]

    N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. (3) 46(1982), 487--523. (Russian)

    1982

  34. [42]

    Lee, An eigenvalue problem for prescribed curvature equations, Int

    T. Lee, An eigenvalue problem for prescribed curvature equations, Int. Math. Res. Not. 2024(2024): 8296--8312

    2024

  35. [43]

    Levy, On differential geometry in the large

    H. Levy, On differential geometry in the large. I. Minkowski's problem, Trans. Am. Math. Soc. 43(1983): 258--270

    1983

  36. [44]

    H. Li, Y. Wan, The Christoffel problem in the hyperbolic plane, Adv. in Appl. Math. 150(2023), Paper No. 102557, 17 pp

    2023

  37. [45]

    Li, Y.Wan, Classification of solutions to the isotropic horospherical p -Minkowski problem in hyperbolic plane, Preprint at https://arxiv.org/abs/2405.04301 (2024)

    H. Li, Y.Wan, Classification of solutions to the isotropic horospherical p -Minkowski problem in hyperbolic plane, Preprint at https://arxiv.org/abs/2405.04301 (2024)

    2024 arXiv

  38. [46]

    H. Li, Y. Wan, B. Xu, The discrete horospherical p -Minkowski problem in hyperbolic space, Adv. Math. 453(2024): 109851

    2024

  39. [47]

    H. Li, B. Xu, Horospherical p -Brunn-Minkowski theory: hyperbolic p -sum and prescribed measure problems, Preprint at https://arxiv.org/abs/2211.06875 (2022)

    2022 arXiv

  40. [48]

    Q.-R. Li, D. Wan, X.-J. Wang, The Christoffel problem by the fundamental solution of the Laplace equation, Sci. China Math. 64(2021): 1599--1612

    2021

  41. [49]

    Li, Degree theory for second order nonlinear elliptic operators and its applications, Commun

    Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Commun. Pure Differential Equations, 14(1989): 1541--1578

    1989

  42. [50]

    Lutwak, The Brunn-Minkowski-Firey theory

    E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38, no.1 (1993): 131--150

    1993

  43. [51]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang, On the L_p -Minkowski problem, Trans. Am. Math. Soc. 356(2004): 4359--4370

    2004

  44. [52]

    A. M. Naveira, A. Tarr\'io, Two problems on h -convex sets in the hyperbolic space, Arch. Math. (Basel) 68 (1997): 514--519

    1997

  45. [53]

    X. H. Nguyen, A. Stancu, G. Wei, The fundamental gap of horoconvex domains in H ^n , Int. Math. Res. Not. IMRN (2022): 16035--16045

    2022

  46. [54]

    Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Commun

    L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Commun. Pure Appl. Math. 6(1953): 337--394

    1953

  47. [55]

    A. V. Pogorelov, On the question of the existence of a convex surface with a given sum of the principal radii of curvature(in Russian), Uspekhi Mat Nauk, 8(1953): 127--130

    1953

  48. [56]

    A. V. Pogorelov, The Minkowski multidimensional problem, V.H. Winston, distributed solely by Halsted Press, Translated from the Russian by Vladimir Oliker, Introduction by Louis Nirenberg, Scr. Math. (1978)

    1978

  49. [57]

    R. Schneider, Convex bodies: the Brunn-Minkowski Theory, second expanded edition, Encyclopedia of Mathematics and Its Applications., vol 151, Cambridge University Press, Cambridge, 2014

    2014

  50. [58]

    I. M. Singer, B. Wong, S.-T. Yau, S. S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schr\"odinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(1985): 319--333

    1985

  51. [59]

    G. Wang, C. Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014): 532--556

    2014

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.