{"id":"0d8f0a5a-205e-4e6d-89e2-d74140a25db4","arxiv_id":"2411.17345","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Smooth even strictly horospherically convex solutions exist for the horospherical p-Christoffel-Minkowski problem and the new p-shifted Weingarten problem in hyperbolic space for p≥−n, under convexity bounds on f.","lead":"This paper proves existence of smooth symmetric solutions to two curvature equations for convex shapes in hyperbolic space, removing a normalization constant earlier work could not remove. It also introduces a new prescribed p-shifted Weingarten curvature problem and solves it under natural assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The full rank theorem inherits an unproved, flow-era PSD lemma from [LX22]; if [LX22, Lem. 7.6] does not apply to static even solutions of (1.2), the removal of the constant γ collapses.","rationale":"The reader identified the same load-bearing dependency: the proof of Lemma 5.1 for p > −n is not self-contained and rests entirely on [LX22, Lem. 7.6 & Assump. 7.1]. I read the surrounding argument in good faith and checked the main alternatives. The choice of ℓ in the proof of Theorem 5.1 can be justified by taking ℓ to be the minimal rank of A[φ] over S^n, since the equation gives S_{n−k}(A) > 0 and hence uniform ellipticity; this part is terse but not obviously broken. The sign conventions in Lemma 4.1 versus the matrix (5.1) also work out after substituting k = n−k. What remains genuinely load-bearing is the unverified import: the matrix (5.1) must be PSD for every even, weakly h-convex solution, not merely along the LX22 flow, and the paper gives no proof or precise restatement of the cited lemma. A further reason to flag this specifically is that for p ≥ −k the matrix (5.1) involves f^{−1/(n−k)} while Assumption 1.1(4)–(5) are stated for f^{−1/(n+p)}; the equivalence is nontrivial and is part of the imported content. Thus the central novelty, removing the constant γ, is conditional on a result the reader cannot verify from this manuscript. The appropriate verdict remains CONDITIONAL, so I recommend no change to the reader's judgment.","tokens_in":32459,"tokens_out":23568,"duration_ms":205864,"concrete_test":"Take p ≥ −k, set g = f^{−1/(n+p)} and h = f^{−1/(n−k)}, and attempt an independent proof that (5.1) ≥ 0 using only Assumption 1.1(4)–(5), evenness of φ, and |D log φ| < 1. If the proof goes through with no additional condition, the imported lemma is safe; if it requires a flow equation, a normalization of φ, or a further restriction on f, Lemma 5.1 is unsupported and Theorem 1.1 needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.1 is the pivotal step in Theorem 5.1: for p > −n it asserts that the matrix (5.1) is positive semidefinite for every even C^4 solution with A[φ] ≥ 0, and it does so by citing [LX22, Lem. 7.6 & Assump. 7.1] rather than giving a derivation. That lemma is not reproduced, and the paper does not verify that its hypotheses are satisfied in the static elliptic setting. The concern is concrete. First, [LX22, Eq. (7.39)] arose in a curvature-flow pinching estimate, and it is not shown that Lemma 7.6 is independent of parabolic evolution terms. Second, for p ≥ −k, (5.1) is written in terms of h = f^{−1/(n−k)}, whereas Assumption 1.1(4)–(5) are conditions on g = f^{−1/(n+p)}; the implication from inequalities on g to semidefiniteness of the h-dependent matrix is exactly the nontrivial content of the imported lemma and is not demonstrated here. Since Lemma 5.1 is the only bridge from weak h-convexity to strict h-convexity, any failure or missing hypothesis there invalidates the full rank theorem and hence Theorem 1.1's removal of the constant γ. This is a dependency gap, not a demonstrated error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":62,"tokens_out":23006,"duration_ms":371420,"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":[{"comment":"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).","section":"Section 5, proof of Lemma 5.1"},{"comment":"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}.","section":"Section 3, Lemma 3.1 (C^0-estimate)"},{"comment":"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.","section":"Section 6, proof of Theorem 1.1"}],"minor_comments":[{"comment":"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.","section":"Notation, Section 4"},{"comment":"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.","section":"Assumption 1.2"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the unproved import in Lemma 5.1: Theorem 1.1 for p>−n rests entirely on the applicability of [LX22, Lem. 7.6] to the static elliptic equation. If that lemma does not apply, the removal of γ for general p>−n collapses, although the p=−n case and the new Weingarten problem appear to be on firmer ground. The authors should be asked to fill this gap or to restrict the theorem to the cases proved in the paper. The self-citation pattern is heavy but not itself a reason for rejection; the technical dependency is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. The paper genuinely proves new existence results: it removes the normalization constant in the horospherical p-Christoffel-Minkowski problem for all 1≤k≤n−1, p≥−n, and it introduces the prescribed p-shifted Weingarten curvature problem with a first existence theorem. The second half, for the Weingarten problem, is more self-contained. But the first half has a load-bearing dependency gap. The full rank theorem (Theorem 5.1) rests on Lemma 5.1, and for p>−n the proof of Lemma 5.1 does not derive the required positive semidefiniteness; it cites [LX22, Lem. 7.6 & Assump. 7.1] and asserts the matrix (5.1) is exactly their Eq. (7.39). That lemma is not reproduced and the paper does not check that its flow-era hypotheses transfer to static even solutions of (1.2). If that lemma fails or its assumptions are not met, the strict convexity conclusion collapses and Theorem 1.1 loses its novelty. This is a genuine gap, not a manufactured one.\n\nThe first half also leans on uniqueness of constant solutions from [LX22] and [LW24]; that is acceptable as a benchmark but worth noting in the review. The citation pattern is self-referential, but not circular in the sense that the existence theorem is checked against the up-to-constant result. I would flag to the authors that the dependency needs to be made explicit.\n\nWhat the paper does well: the a priori estimates (C0, C1, C2), the deformation lemma, and the degree argument are coherent and carefully written. The full rank theorem follows the Guan-Ma/Chen architecture, and the authors correctly identify the new difficulty. I found a minor systematic typo in the formula for the minimum of ξq in Lemma 3.1; the q-form statement has an exponent error, but the p-form of Assumption 1.2 used in the theorems is correct, so it is cosmetic.\n\nWho should read it: people working on hyperbolic Brunn-Minkowski theory, curvature measures, or prescribed curvature equations. It deserves a serious referee. The right response is to send it to review, but the referee must demand that the authors either prove the imported lemma in the static setting or state Theorem 1.1 with the missing hypothesis made explicit. If the authors close that gap, this is a strong contribution.","headline":"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.","tokens_in":45,"tokens_out":2563,"would_cite":false,"duration_ms":86406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J05","52A55"],"pacs":[],"model":"deepseek-v4-flash","headline":"A full-rank theorem removes the normalization constant in the horospherical p-Christoffel-Minkowski problem.","keywords":["horospherical Christoffel-Minkowski problem","hyperbolic space","full rank theorem","prescribed p-shifted Weingarten curvature","horospherical convexity","degree theory","elementary symmetric polynomials","Minkowski problem"],"falsifier":"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.","tokens_in":32214,"feed_emoji":"📐","tokens_out":9231,"duration_ms":70089,"temperature":0.7,"pith_summary":"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.","feed_headline":"New full-rank theorem removes constant in hyperbolic curvature problem","feed_subtitle":"Smooth symmetric solutions exist for all p ≥ −n once weak convexity is upgraded to strict convexity.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $f$."],"forward_implications":["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]."],"supporting_citations":[{"why":"Supplies the prior flow-based existence theorem up to a constant and the lemma (7.6 with Assumption 7.1) that yields positive semi-definiteness of the matrix (5.1) for $p>-n$.","marker":"[LX22]"},{"why":"Proved the corresponding full-rank theorem for the special case $k=n-1$, $p=-n$, and supplies Lemma 4.3 used for the $p=-n$ case here.","marker":"[Che24]"},{"why":"Provides the deformation lemma estimates and the commutation formulas for elementary symmetric polynomials used in Lemma 4.1.","marker":"[GM03]"},{"why":"Supplies the degree theory for fully nonlinear elliptic operators on which the existence proofs of Theorems 1.1-1.4 rest.","marker":"[Li89]"},{"why":"Gives the shifted Minkowski formula used in the full-rank theorem to rule out a vanishing minor.","marker":"[HLW22]"},{"why":"Introduced the Christoffel problem in hyperbolic space and the Kazdan-Warner type obstruction showing the need for evenness when $p=-n$.","marker":"[EGM09]"}],"fun_headline_variants":["Hyperbolic curvature problems solved for all p ≥ −n","Smooth solutions in hyperbolic space for all p ≥ −n","Full-rank theorem yields new existence results in hyperbolic geometry","Horospherical p-curvature: existence for all p ≥ −n","Strict horospherical convexity achieved for all p ≥ −n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic curvature problems solved for all p ≥ −n","Smooth solutions in hyperbolic space for all p ≥ −n","Full-rank theorem yields new existence results in hyperbolic geometry","Horospherical p-curvature: existence for all p ≥ −n","Strict horospherical convexity achieved for all p ≥ −n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001056,"raw_usage":{"total_tokens":4421,"prompt_tokens":925,"completion_tokens":3496,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":3408}},"tokens_in":541,"tokens_out":3496,"duration_ms":21539,"temperature":1.0,"reasoning_tokens":3408,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:14:55.508021+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}