{"id":"c4889172-7908-484f-9c43-51838d466dda","arxiv_id":"2607.17882","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For convex domains in the sphere and hyperbolic plane, the second Neumann eigenfunction has no interior critical points when μ2 D² ≤ j_{1,1}², and planar convex domains satisfy C(Ω) ≤ 2.4828.","lead":"This paper proves new conditions under which the second Neumann eigenfunction on convex domains in a sphere or hyperbolic plane has no interior critical points, with quantitative location bounds when such points exist. It also improves the known upper bound on the 'hot spots constant' for planar convex domains from about 3.16 to about 2.48.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The H² universal hot-spots bound (Theorem 5.10) rests on an unverified external area criterion, Hatcher's Corollary 1.2 (Area>33.35), which is neither proved nor stated in the introduction; if that criterion fails, Theorem 1.4(H) collapses.","rationale":"The reader's weakest assumption identifies the same external dependency, and my reading confirms it is the only substantive gap. The main critical-point theorems are internally coherent: Lemma 2.1's Sturm-Liouville argument for jμ'<0 under μ≤2 is valid; Proposition 2.4's nodal-domain argument is standard; Lemma 3.2 and Lemma 4.2 correctly convert monotonicity radii into Bessel-root bounds; and Theorem 5.2's Bessel-function optimization gives the Euclidean constant 2.4828 with no apparent algebraic error. I did not find a counterexample or circularity in Theorems 1.1-1.3 or Theorem 1.4(E). However, the paper's headline includes the first hot-spots constants in non-Euclidean space forms, and Theorem 1.4(H) depends on an uncited-proof area theorem from a separate preprint. Because that theorem is not reproduced and the introduction only mentions Hatcher's μ2≤1/4 condition, the universal constant for H² is not independently verifiable from this text. The paper should be accepted conditionally, with the Hatcher dependency resolved before the claim is treated as fully established.","tokens_in":21407,"tokens_out":35640,"duration_ms":322446,"concrete_test":"Retrieve arXiv:2605.21621 (Hatcher) and verify Corollary 1.2 verbatim: confirm it states that any bounded convex Ω⊂H² with Area(Ω)>33.35 has no interior critical points, and examine the proof to ensure it follows from an area-based estimate, such as a bound of the form Area>33.35 ⇒ μ2≤1/4, and does not depend on the present paper. As a numerical cross-check, compute the hyperbolic disk of area 33.35 and its second Neumann eigenvalue; if μ2>1/4 for that disk, then the claimed area criterion is inconsistent with the cited μ2≤1/4 theorem, and Theorem 5.10 needs an independent proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing step is in Theorem 5.10. After reducing to Area(Ω)<33.35, the proof splits D≤1 and D>1. The D>1 case is controlled by Lemma 5.9 and the area integral, but the whole argument starts by invoking 'Corollary 1.2 in [17]': every bounded convex Ω⊂H² with Area(Ω)>33.35 has second Neumann eigenfunction with no interior critical points, so C(Ω)=1. This is exactly the step that removes arbitrarily large domains, the case where r(θ) in Lemma 5.9 can be large and ∫e^{r/2} would otherwise grow exponentially. The manuscript neither reproduces the statement, the underlying area-to-eigenvalue bound, nor any estimate connecting Area>33.35 to μ2≤1/4; the introduction attributes only the μ2≤1/4 criterion to Hatcher. Theorem 1.4(H) therefore inherits an external, non-verified, and non-locally-checkable assumption. If Hatcher's Corollary 1.2 is misquoted, or if its constant 33.35 is incorrect, the claimed universal constant for convex hyperbolic domains is unsupported. This is a missing-support problem, not a disagreement with consensus; a direct check of the cited preprint settles it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies critical points of the second Neumann eigenfunction on bounded convex domains in the two-dimensional space forms S², H², and R². Theorem 1.1 proves that a convex domain Ω⊂S² contained in a hemisphere has no interior critical points when μ2(Ω)≤2. Theorem 1.2 establishes a unified diameter criterion μ2(Ω)D²≤j_{1,1}² for S² and H², and Theorem 1.3 gives quantitative lower bounds on the distance from any interior critical point to the boundary in terms of the diameter. Theorem 1.4 concerns the hot-spots constant C(Ω): an improved Euclidean upper bound of 2.4828, and universal, non-explicit bounds for convex domains in S² and H². The proofs combine radial comparison functions built from Bessel and Legendre functions, monotonicity radii of radial solutions, nodal-domain and variational arguments, and Green-formula identities with singular fundamental solutions of the Helmholtz equation.","tokens_in":21674,"tokens_out":18053,"duration_ms":154706,"significance":"If the proofs are completed, the paper makes substantial advances: it extends Miyamoto's Euclidean diameter criterion to the sphere and hyperbolic plane, provides the first hot-spots constants for convex domains in non-Euclidean space forms, and improves the Euclidean upper bound from 3.1642 to 2.4828 by a purely analytic method. The Euclidean bound is explicit, and the nodal and variational arguments in Sections 2-4 are detailed and mostly convincing. The main caveat is that Theorem 1.4(H) relies on an unstated external area criterion attributed to Hatcher [17]; until that dependence is made explicit or removed, the hyperbolic universal constant is conditional. The spherical and Euclidean components of the paper are, in my reading, sound and publishable after the identified revisions.","major_comments":[{"comment":"The proof of Theorem 5.10 begins by invoking \"Corollary 1.2 in [17]\" to the effect that every bounded convex Ω⊂H² with Area(Ω)>33.35 has a second Neumann eigenfunction with no interior critical points. This area criterion is never stated in the introduction or elsewhere in the manuscript, and the introduction attributes to Hatcher only the spectral criterion μ2(Ω)≤1/4. The area criterion is load-bearing because it is the only step that removes domains of arbitrarily large diameter, where the exponential factor e^{r/2} in Lemma 5.9 would otherwise make the integral in (5.20) uncontrollable. As written, Theorem 1.4(H) is conditional on an unverified external statement. Please state Hatcher's Corollary 1.2 explicitly and either prove it or give a fully specified self-contained reference; if the area constant or its statement is inaccurate, the claimed universal constant in Theorem 1.4(H) is unsupported.","section":"§5.3, Theorem 5.10"},{"comment":"The proofs of the universal upper bounds assert uniform boundedness of g(r,ν)=Q'_ν(cos r) sin² r (respectively Q'_ν(cosh r) sinh² r) over non-compact admissible parameter spaces. The only argument for the non-compact direction ν→∞ is an asymptotic formula for the Legendre function at large degree, stated without decay estimates in ν or a supporting reference (see the discussion after Eq. (5.16)). Since these theorems are the basis for Theorem 1.4(S) and part of Theorem 1.4(H), please supply precise uniform asymptotic bounds, or a compactness argument with explicit estimates, with references to standard sources such as DLMF.","section":"§5.2, Theorem 5.5 and §5.3, Proposition 5.8"}],"minor_comments":[{"comment":"In the proof of Theorem 5.5, \"Theorem 3.4\" should be \"Lemma 3.4\"; in the text following Proposition 5.3, \"Theorem 5.3\" should be \"Proposition 5.3\"; and in Corollary 5.7, \"Theorem 5.6\" should be \"Proposition 5.6\".","section":"§5.2-5.3 cross-references"},{"comment":"Hatcher's criterion is invoked with the strict inequality Area(Ω)>33.35, but the proof then splits into the cases Area(Ω)<A and D≤D* or D>D*; the equality case Area(Ω)=33.35 is not covered. Please clarify, for instance by using a non-strict version of the criterion or by perturbing A.","section":"§5.3, Theorem 5.10"},{"comment":"The limit at z→1+ is justified by appealing to \"the standard expansion (5.14)\", but Eq. (5.14) is the spherical expansion near z→1−. Please state the hyperbolic analogue near z→1+ explicitly.","section":"Lemma 5.9"},{"comment":"The displayed maxima are taken over the roots ξ of Qν, but the set of roots may be empty. Please state that in that case the maximum is understood as the boundary value (which is 1 in these corollaries), or otherwise adjust the statement.","section":"Corollaries 5.4 and 5.7"},{"comment":"The proof uses the bound μ2(Ω)D²≤4j_{0,1}² for convex planar domains as \"well known\" without a reference. Please add a citation for this eigenvalue bound.","section":"§5.1, Theorem 5.2"}],"recommendation":"major_revision","confidential_remarks":"The Euclidean and spherical parts of the paper are mathematically solid and make a clear contribution. The main risk to publication is the dependence of Theorem 1.4(H) on an unstated result from a recent arXiv preprint [17]; before acceptance, the editor should ensure that Hatcher's Corollary 1.2 is stated accurately and is available to readers, or that the authors provide a self-contained proof. The manuscript also has a large number of references to 2026 preprints, which is not itself a problem but may warrant a check of relevance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a real advance. It gives the first criteria for absence of interior critical points for second Neumann eigenfunctions on convex domains in the sphere and hyperbolic plane, proves quantitative location restrictions for such critical points, and improves the Euclidean hot-spots constant from 3.1642 to 2.4828 using a purely analytic method. The proofs of Theorems 1.1–1.3 and Theorem 5.2 are coherent and detailed; the monotonicity-radius estimates via Sturm–Liouville comparison and the nodal-set/variational argument are clean. The constants are explicit and no fitted parameter shows up. This genuinely extends the Miyamoto–Hatcher line to non-Euclidean settings.\n\nThe soft spot is exactly what the stress-test note identifies. Theorem 5.10, the universal bound for hyperbolic domains, rests on Hatcher's Corollary 1.2: a bounded convex domain in H^2 with area > 33.35 has second Neumann eigenfunction with no interior critical points. The paper cites this without stating it, and the introduction only mentions Hatcher's μ2 ≤ 1/4 condition, which is not the same. This is load-bearing: it is the step that removes arbitrarily large domains, and without it the exponential bound in Lemma 5.9 cannot be integrated over unbounded area. If Hatcher's result is correct, the proof works, but the reader cannot verify it from this text. This is a missing-support problem, not a subtle math error, and it should be fixable by stating the external theorem precisely and either proving it or pointing to a fully verified source.\n\nMinor caveats: the non-Euclidean universal constants in Theorems 5.5 and 5.10 are existence results, not explicit numbers—fine, but less satisfying than the Euclidean case. The references look solid and recent, with no obvious citation gaps.\n\nOverall this is a serious paper worth engaging. I would send it to peer review, asking the authors to clarify the Hatcher dependence and preferably make the non-Euclidean constants explicit or at least computable. The core Euclidean improvement and the spherical/hyperbolic critical-point criteria are valuable on their own.","headline":"Genuinely new critical-point criteria for second Neumann eigenfunctions in S^2 and H^2, plus an improved Euclidean hot-spots constant, but the H^2 universal bound leans on an unverified external area criterion.","tokens_in":22238,"tokens_out":2558,"would_cite":true,"duration_ms":23359,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B38","58J50","35J25","35J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The second Neumann eigenfunction has no interior critical points on convex domains in a hemisphere with $\\mu_2\\le 2$, and a unified diameter criterion covers the hyperbolic plane while the Euclidean hot-spots constant drops to 2.4828.","keywords":["hot spots conjecture","second Neumann eigenfunction","interior critical points","convex domains","space forms","Bessel functions","Legendre functions","hot spots constant"],"falsifier":"Compute the second Neumann eigenfunction on a convex geodesic polygon contained in a hemisphere with $\\mu_2(\\Omega)\\le 2$; a single such computation exhibiting an interior critical point would refute Theorem 1.1, and likewise a planar convex domain whose hot-spots constant exceeds $-1/J_0(j_{1,1})\\approx 2.4828$ would refute the Euclidean bound.","tokens_in":21203,"feed_emoji":"🔥","tokens_out":13000,"duration_ms":102064,"temperature":0.7,"pith_summary":"This paper studies the hot-spots conjecture for convex domains in the three two-dimensional constant-curvature spaces: the Euclidean plane, the sphere, and the hyperbolic plane. It tries to show that the second Neumann eigenfunction has no interior critical points—so its maxima and minima are on the boundary—under explicit conditions on the eigenvalue and diameter. The new sufficient conditions are $\\mu_2(\\Omega)\\le 2$ for convex domains contained in a hemisphere and the unified scale-invariant criterion $\\mu_2(\\Omega)D^2\\le j_{1,1}^2$ valid in both $\\mathbb{S}^2$ and $\\mathbb{H}^2$. When critical points do exist, the paper locates them: in a hemispherical domain each lies at least about $0.7967D$ from some boundary point. It also bounds the hot-spots constant, the largest possible ratio of interior to boundary supremum of a second eigenfunction, by $2.4828$ in the plane and by universal constants on the sphere and hyperbola.","feed_headline":"Hot spots forced to the boundary by one diameter inequality","feed_subtitle":"On sphere and hyperbola, convex domains with μ2D² ≤ j1,1² have no interior critical points; Euclidean constant falls to 2.4828.","key_machinery":"The carrying object is the radial solution of the Helmholtz equation centered at a point $p$: the function $j_\\mu(d(p,q))$, given by the Legendre function $P_\\nu(\\cos r)$ on $\\mathbb{S}^2$, and $k_\\mu(d(p,q))$, given by $P_\\nu(\\cosh r)$ on $\\mathbb{H}^2$, together with its monotonicity radius—the largest radius on which the radial function is strictly decreasing. Given a candidate interior critical point $p$, the paper forms $w(q)=u(p)\\Phi_\\mu(d(p,q))-u(q)$, which has $p$ as a nodal critical point; the nodal-set topology and the Neumann condition force two disjoint positive nodal domains whose Rayleigh quotients contradict the variational characterization of $\\mu_2$, unless $p$ lies where the radial function is decreasing. The quantitative input is a Sturm–Liouville lower bound $\\tau_\\mu\\ge j_{1,1}/\\sqrt{\\mu}$ for the monotonicity radius, paired with diameter-dependent upper bounds on $\\mu_2(\\Omega)$ from test functions on geodesic balls. For the hot-spots constant, the method is an integral representation of the interior maximum through the singular fundamental solution $(\\Delta+\\mu)\\Gamma=-\\delta_p$, followed by a sharp bound on $z|Y_1(z)+cJ_1(z)|$.","core_discovery":"The central claim is that the hot-spots conjecture holds for a broad, explicitly checkable class of convex domains in non-Euclidean two-dimensional space forms. If $\\Omega\\subset\\mathbb{S}^2$ is convex, lies in a hemisphere, and its second Neumann eigenvalue satisfies $\\mu_2(\\Omega)\\le 2$, then every second Neumann eigenfunction has no interior critical points; the same conclusion follows whenever $\\mu_2(\\Omega)D^2\\le j_{1,1}^2$, where $D$ is the geodesic diameter and $j_{1,1}$ the first positive zero of the Bessel function $J_1$, for both hemispherical and hyperbolic convex domains. When these conditions fail, the paper still restricts where critical points can lie. Using a purely analytic Green-function argument, it establishes that on bounded convex domains in $\\mathbb{R}^2$ the hot-spots constant $\\mathfrak{C}(\\Omega)$ is at most $-1/J_0(j_{1,1})\\approx 2.4828$, improving the previous Euclidean bound, and that universal, domain-independent hot-spots constants exist for convex domains in hemispherical $\\mathbb{S}^2$ and in $\\mathbb{H}^2$.","pith_inferences":["The same Green-function machinery should yield explicit, rather than merely existential, hot-spots constants on $\\mathbb{S}^2$ and $\\mathbb{H}^2$; the numerical search reported in the paper (value near $2.47$ for $\\nu\\le 10$) hints that the true universal constants may be close to the new Euclidean bound.","Because the spherical and hyperbolic estimates reduce to the Euclidean ones as the diameter tends to zero, the sharp hot-spots constant for all three space forms may be the same number $-1/J_0(j_{1,1})$; one could test this by computing $\\mathfrak{C}(\\Omega)$ for small geodesic caps or small hyperbolic polygons.","If Hatcher's area threshold $33.35$ can be sharpened, the hyperbolic universal constant in Theorem 5.10 improves accordingly; numerical experiments on convex hyperbolic polygons of increasing area could reveal the true threshold.","The monotonicity-radius construction suggests a route to interior-critical-point criteria for Robin boundary conditions or higher Neumann eigenvalues on convex domains in space forms, where the nodal-domain count changes but the comparison-function argument still applies."],"forward_implications":["Every convex domain contained in a hemisphere with $\\mu_2(\\Omega)\\le 2$ satisfies the hot-spots conjecture: generic Neumann heat flow has its extrema on the boundary.","The condition $\\mu_2(\\Omega)D^2\\le j_{1,1}^2$ provides a computable certificate valid in both $\\mathbb{S}^2$ and $\\mathbb{H}^2$: measure the second eigenvalue and the diameter, and if the product is small enough, interior critical points are impossible.","When the certificate fails, any interior critical point in a hemispherical convex domain must have distance at least $0.7967D$ from some boundary point, creating a quantitatively hot-spots-free layer near the boundary.","The Euclidean hot-spots constant on convex domains is at most $2.4828$, independent of shape, and universal constants exist for spherical and hyperbolic convex domains; this quantifies how far any counterexample to the conjecture can deviate from boundary extrema."],"supporting_citations":[{"why":"Supplies the Euclidean criterion $\\mu_2D^2\\le j_{1,1}^2$ and the comparison-function proof template that the paper generalizes to $\\mathbb{S}^2$ and $\\mathbb{H}^2$.","marker":"[32]"},{"why":"Supplies the hyperbolic no-critical-point result for $\\mu_2\\le 1/4$ and the area>$33.35$ criterion cited in Theorem 5.10.","marker":"[17]"},{"why":"Supplies the inequality $\\mu_2\\le \\lambda_1$ for convex domains in $\\mathbb{S}^2$, used to rule out nodal loops.","marker":"[1]"},{"why":"Supplies the same Neumann-below-Dirichlet inequality in $\\mathbb{H}^2$, needed in the hyperbolic critical-point lemmas.","marker":"[31]"},{"why":"Supplies the spherical-cap first Dirichlet eigenvalue bound used for the diameter-dependent upper bound on $\\mu_2(\\Omega)$.","marker":"[3]"},{"why":"Supplies the geodesic-ball eigenvalue comparison used in the hyperbolic diameter bound of Lemma 4.5.","marker":"[7]"},{"why":"Provides the previous general hot-spots-constant bound in dimension two that the analytic method improves.","marker":"[30]"},{"why":"Provides the previous sharp Euclidean bound $3.1642$ that is improved to $2.4828$.","marker":"[13]"},{"why":"Introduces the hot-spots constant and its first universal bound; the present proof is an analytic alternative to its probabilistic approach.","marker":"[40]"}],"fun_headline_variants":["Diameter inequality pins hot spots to boundary","μ2D² bound: no interior hot spots","Boundary hot spots by one checkable inequality","Hot spots conjecture extended to sphere and hyperbola","Euclidean hot spots constant sharpened to 2.4828"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a borrowed fact not proved here: every bounded convex domain in the hyperbolic plane with area larger than $33.35$ already has its second eigenfunction's extrema on the boundary, and if that fact fails the paper's universality argument for $\\mathbb{H}^2$ collapses.","fun_headline_variants_meta":{"raw":{"variants":["Diameter inequality pins hot spots to boundary","μ2D² bound: no interior hot spots","Boundary hot spots by one checkable inequality","Hot spots conjecture extended to sphere and hyperbola","Euclidean hot spots constant sharpened to 2.4828"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00033,"raw_usage":{"total_tokens":1896,"prompt_tokens":1056,"completion_tokens":840,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":765}},"tokens_in":672,"tokens_out":840,"duration_ms":7576,"temperature":1.0,"reasoning_tokens":765,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:36:57.559208+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the second Neumann eigenfunction on a convex geodesic polygon contained in a hemisphere with $\\mu_2(\\Omega)\\le 2$; a single such computation exhibiting an interior critical point would refute Theorem 1.1, and likewise a planar convex domain whose hot-spots constant exceeds $-1/J_0(j_{1,1})\\approx 2.4828$ would refute the Euclidean bound.","supporting_citations":[{"cited_title":"hot spots","cited_arxiv_id":null,"evidence_quote":"Supplies the Euclidean criterion $\\mu_2D^2\\le j_{1,1}^2$ and the comparison-function proof template that the paper generalizes to $\\mathbb{S}^2$ and $\\mathbb{H}^2$."},{"cited_title":"Hot spots in convex hyperbolic planar domains with small eigenvalues","cited_arxiv_id":"2605.21621","evidence_quote":"Supplies the hyperbolic no-critical-point result for $\\mu_2\\le 1/4$ and the area>$33.35$ criterion cited in Theorem 5.10."},{"cited_title":"Ashbaugh, H.A","cited_arxiv_id":null,"evidence_quote":"Supplies the inequality $\\mu_2\\le \\lambda_1$ for convex domains in $\\mathbb{S}^2$, used to rule out nodal loops."},{"cited_title":"Mazzeo, Remarks on a paper of Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues, Internat","cited_arxiv_id":null,"evidence_quote":"Supplies the same Neumann-below-Dirichlet inequality in $\\mathbb{H}^2$, needed in the hyperbolic critical-point lemmas."},{"cited_title":"Baginski, Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap, Quart","cited_arxiv_id":null,"evidence_quote":"Supplies the spherical-cap first Dirichlet eigenvalue bound used for the diameter-dependent upper bound on $\\mu_2(\\Omega)$."},{"cited_title":"Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115","cited_arxiv_id":null,"evidence_quote":"Supplies the geodesic-ball eigenvalue comparison used in the hyperbolic diameter bound of Lemma 4.5."},{"cited_title":"Mariano,H","cited_arxiv_id":null,"evidence_quote":"Provides the previous general hot-spots-constant bound in dimension two that the analytic method improves."},{"cited_title":"Steinerberger, An upper bound on the hot spots constant, Rev","cited_arxiv_id":null,"evidence_quote":"Introduces the hot-spots constant and its first universal bound; the present proof is an analytic alternative to its probabilistic approach."}],"review_version":2}