{"id":"65e2960b-6f84-42eb-9a60-73d4f160f1dc","arxiv_id":"2411.17328","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For even prescribed functions satisfying explicit curvature conditions, the horospherical p-Christoffel-Minkowski equation σ_k(A[φ]) = φ^{p-k} f admits a smooth uniformly h-convex solution φ > 1 on S^n.","lead":"This paper proves existence of smooth solutions to the horospherical p-Christoffel-Minkowski problem in hyperbolic space, a geometric PDE that prescribes a curvature measure of h-convex domains. It upgrades an earlier existence result that only delivered the equation up to an unknown constant factor, and it implies a new existence result for a Nirenberg-type conformal problem on the sphere.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1.7 drops the evenness assumption required by Theorem 1.4; for k=1,p=0 it claims existence for non-even f such as 1+eps x1, contradicting the Kazdan-Warner obstruction.","rationale":"The reader identified evenness as the load-bearing assumption; I agree, and the sharpest manifestation is Corollary 1.7, which silently drops it. The main theorem itself is carefully conditioned on evenness, and the a priori estimates and full rank theorem appear internally consistent, so I do not see a fatal flaw in the proof of Theorem 1.4. The problem is the overclaimed conformal corollary: for k=1, p=0, it asserts existence for all f satisfying Condition (1), but a simple non-even f=1+eps x1 satisfies the condition and yet is excluded by the Kazdan-Warner identity. This is not a disagreement with consensus; it is an internal inconsistency between Theorem 1.4 (with its even function spaces and origin-symmetry estimates) and Corollary 1.7. A conditional verdict remains appropriate: the paper should be accepted only after the corollary and abstract are corrected to state the evenness restriction, and the Nirenberg-type application is described accurately. Since the reader's verdict is already CONDITIONAL, my recommendation is UNCHANGED.","tokens_in":23577,"tokens_out":23897,"duration_ms":215485,"concrete_test":"Settling check: take n=3, k=1, p=0, and f=1+eps x1 with eps=0.01. (1) Verify Condition (1) symbolically or numerically: the matrix D2(f^{-1}) - |D f^{-1}| g + (1/2)f^{-1}g + 8 max(f/3)g is positive definite for small eps, as it is at eps=0. (2) Compute the Kazdan-Warner obstruction integral over S^3 of <grad f, grad x1> dvol, which equals eps times the volume integral of (1-x1^2), nonzero. If both checks pass, no conformal metric solving S(g)=2(n-1)(f+n/2) exists, so Corollary 1.7 as stated is false and must require f even.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.4 is proved only for even f. Every a priori estimate in Section 3 uses evenness: (3.3) and (3.7) are Li-Xu estimates for origin-symmetric h-convex hypersurfaces, and the degree argument in Section 5 is set up in spaces of even functions. Corollary 1.7, however, states that any smooth positive f on S^n with n>=3 satisfying Condition (1) yields a conformal metric solving the Nirenberg-type equation (1.13), with no evenness assumption. This does not follow from Theorem 1.4. In the case k=1, p=0, equation (1.13) is exactly the prescribed scalar curvature problem: sigma_1(Sch_g)=S(g)/(2(n-1)) and f=sigma_1(Sch_g)-n/2, so S(g)=2(n-1)(f+n/2). Choose f=1+eps x1 on S^n with small nonzero eps. Condition (1) holds for small eps: at eps=0 the left side is (1/2+8/3)g, positive definite, and all terms depend continuously on eps in C^2. But the Kazdan-Warner identity forbids the existence of a conformal metric on S^n with scalar curvature proportional to 1+eps x1 when eps is nonzero. Hence Corollary 1.7 is false as stated, and the abstract's Nirenberg-type claim must be restricted to even f. This does not by itself invalidate Theorem 1.4, but it is a load-bearing overclaim in the advertised conformal application.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":93,"tokens_out":13146,"duration_ms":176329,"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":[{"comment":"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.","section":"§1, equations (1.6)–(1.8)"},{"comment":"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.","section":"Corollary 1.7 and abstract"}],"minor_comments":[{"comment":"The title contains a typo: \"SP ACE\" should be \"SPACE\".","section":"Title"},{"comment":"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.","section":"§5, linearized operator"},{"comment":"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.","section":"§4, after (4.10)"}],"recommendation":"major_revision","confidential_remarks":"The core PDE existence proof appears sound for equation (1.8) under the evenness assumption, and the viscosity full-rank argument is a worthwhile contribution. The problem is that the manuscript's advertised connections—to the original measure equation and to the Nirenberg-type problem—are not supported as written. The exponent mismatch in (1.6)–(1.8) changes the problem being solved, and Corollary 1.7 is plainly false without evenness. Both issues are fixable by restating and correcting the claims, but they affect the central narrative of the paper, so the revision should be substantive rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: Theorem 1.4 is a genuine improvement over Li-Xu. They only got sigma_k(A[phi]) = gamma phi^{p-k} f with an unknown constant gamma, and for p=0 only constant f; here the exact equation is solved for even f under Assumption 1.2. The proof is a careful, standard package: C^0 bounds from Li-Xu's origin-symmetric estimates, C^2 estimates from concavity, the full rank theorem by the Bryan-Ivaki-Scheuer viscosity method, and degree theory. I checked the five cases in Claim 2 and the commutator algebra in Section 4; no sign errors jumped out. The full rank theorem for the non-Codazzi tensor A[phi] is the real technical work, and it is done carefully.\n\nThe soft spot is Corollary 1.7. Theorem 1.4 requires f to be even, and every a priori estimate in Section 3 uses Li-Xu's evenness-based bounds. The corollary simply drops the evenness assumption. That is not a harmless weakening. For k=1, p=0, equation (1.13) is the prescribed scalar curvature equation S(g)=2(n-1)(f+n/2), and f=1+epsilon x_1 satisfies Condition (1) for small nonzero epsilon. But the Kazdan-Warner identity forbids a conformal metric on S^n with scalar curvature of the form a + b x_1 with b nonzero. So Corollary 1.7 is false as stated, and the abstract's claim about the Nirenberg-type problem is likewise too strong. This does not invalidate Theorem 1.4, but it is a load-bearing overclaim in the advertised conformal application.\n\nTwo smaller issues. The full rank proof applies Lemma 2.3 to a possibly singular A without an explicit approximation argument like A + epsilon I; this is fixable but should be stated. And the heavy use of the 'less than or similar to' notation leaves the constants untracked, which makes the verification of Claim 2 harder than it needs to be, though I did not find an actual error.\n\nThis paper is for people working on Christoffel-Minkowski-type problems in hyperbolic space or on Nirenberg-type conformal equations. The main theorem deserves a serious referee. My recommendation: send it out, but require the authors to restrict Corollary 1.7 to even f (or actually prove the non-even case) and to add the missing approximation step in the full rank argument.","headline":"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.","tokens_in":24512,"tokens_out":5458,"would_cite":true,"duration_ms":49839,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["horospherical p-Christoffel-Minkowski problem","hyperbolic space","h-convex","full rank theorem","Nirenberg-type problem","Schouten tensor","viscosity method","elementary symmetric functions"],"falsifier":"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.","tokens_in":23345,"feed_emoji":"🌐","tokens_out":7968,"duration_ms":67891,"temperature":0.7,"pith_summary":"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.","feed_headline":"Uniformly convex solutions found for hyperbolic curvature problem","feed_subtitle":"Even f satisfying the inequalities gives strictly h-convex solutions; p=0 also gives Nirenberg-type metrics.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Poses the horospherical p-Christoffel-Minkowski problem and supplies the C0/C1 estimates and the uniqueness of constant solutions used in the degree argument.","marker":"[29]"},{"why":"Supplies the viscosity method for full-rank theorems that the proof adapts to the non-Codazzi tensor A[φ].","marker":"[5]"},{"why":"Supplies the viscosity inequality for the smallest eigenvalue of a symmetric 2-tensor that drives the full-rank argument.","marker":"[4]"},{"why":"Provides the classical full-rank theorem for the Euclidean Christoffel-Minkowski problem that this result extends.","marker":"[20]"},{"why":"Provides the degree theory used to convert a priori estimates and the full-rank theorem into existence.","marker":"[31]"},{"why":"Establishes the equivalence between the generalized Christoffel problem and the Nirenberg-type problem on S^n used in the p=0 corollary.","marker":"[13]"},{"why":"Provides the horospherical support function, the tensor A[φ], and the Weingarten relation connecting it to principal curvatures.","marker":"[1]"},{"why":"Supplies the strong maximum principle for viscosity solutions that forces the smallest eigenvalue to be positive everywhere.","marker":"[3]"}],"fun_headline_variants":["Horospherical p-CM: convex solutions proven","Uniform h-convex solutions for hyperbolic p-CM","Hyperbolic p-CM existence via full-rank viscosity","Horospherical p-CM solved: even f gives strict convexity","Hyperbolic p-CM: uniformly convex solutions exist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Horospherical p-CM: convex solutions proven","Uniform h-convex solutions for hyperbolic p-CM","Hyperbolic p-CM existence via full-rank viscosity","Horospherical p-CM solved: even f gives strict convexity","Hyperbolic p-CM: uniformly convex solutions exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002089,"raw_usage":{"total_tokens":8143,"prompt_tokens":985,"completion_tokens":7158,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":7078}},"tokens_in":601,"tokens_out":7158,"duration_ms":47053,"temperature":1.0,"reasoning_tokens":7078,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:13:48.249649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ivaki, and Julian Scheuer, Constant rank theorems for curvature problems via a viscosity approach, Calc","cited_arxiv_id":null,"evidence_quote":"Supplies the viscosity method for full-rank theorems that the proof adapts to the non-Codazzi tensor A[φ]."},{"cited_title":"219 (2017), no","cited_arxiv_id":null,"evidence_quote":"Supplies the viscosity inequality for the smallest eigenvalue of a symmetric 2-tensor that drives the full-rank argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical full-rank theorem for the Euclidean Christoffel-Minkowski problem that this result extends."},{"cited_title":"Partial Differential Equations 14 (1989), no","cited_arxiv_id":null,"evidence_quote":"Provides the degree theory used to convert a priori estimates and the full-rank theorem into existence."},{"cited_title":"Espinar, Jos´ e A","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between the generalized Christoffel problem and the Nirenberg-type problem on S^n used in the p=0 corollary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the horospherical support function, the tensor A[φ], and the Weingarten relation connecting it to principal curvatures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the strong maximum principle for viscosity solutions that forces the smallest eigenvalue to be positive everywhere."}],"review_version":1}