New eigenvalue estimates involving Bessel functions

Fida EL Chami; Georges Habib; Nicolas Ginoux (IECL)

REVIEW 2 major objections 3 minor 43 references

New eigenvalue estimates involving Bessel functions

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Bessel functions give a sharp lower bound for boundary-to-bulk integrals on manifolds with nonnegative Ricci curvature and positive boundary mean curvature, with equality only on Euclidean geodesic balls; the estimate drives Dirichlet…

desk verdict The Bessel-function comparison in Theorem 3.1 is a genuine advance, but the even-dimensional proof has a concrete constant error that needs fixing before the paper can be accepted. read the letter →

arxiv 1908.02566 v2 pith:DVUPD2EH submitted 2019-08-07 math.DG math.APmath.SP

Fida El Chami , Nicolas Ginoux (IECL) , Georges Habib This is my paper · ORCID

classification math.DGmath.APmath.SP MSC 53C2753C2158J6035P1534B0933C10

keywords BesselfunctionseigenvalueestimatesDiracoperatorRobinLaplaciandifferentialformsboundarymeancurvatureYamabeisoperimetricinequalities

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

The paper establishes a sharp comparison for the ratio between the boundary integral and the bulk integral of a positive function on a compact manifold with boundary. Under the assumptions that the Ricci curvature is nonnegative and the boundary mean curvature is at least a positive constant $H_0$, any positive smooth $f$ with $\Delta f \le \lambda f$ and $\sqrt{\lambda}/H_0 < j_{n/2,1}$ satisfies $\int_{\partial M} f \ge \sqrt{\lambda}\, \frac{J_{n/2-1}(\sqrt{\lambda}/H_0)}{J_{n/2}(\sqrt{\lambda}/H_0)} \int_M f$. Equality forces the manifold to be a Euclidean geodesic ball of radius $1/H_0$. Because no boundary condition is imposed on $f$, this one inequality feeds directly into eigenvalue problems: it gives a direct proof of the isoperimetric eigenvalue inequalities for the Dirichlet and Robin Laplacians, new Dirac eigenvalue bounds that involve a Bessel zero alongside scalar curvature, and a Robin-type Laplacian on differential forms with a positive discrete spectrum bounded below by Bessel functions.

What carries the argument

The load-bearing mechanism is the comparison of a boundary-distance accumulated function $F(r)=\int_{\{\rho>r\}} f\,d\mu_g$ with the explicit solution of a Bessel equation. The mean value lemma expresses $F''$ through $\Delta f$ and the Laplacian of the distance function, and the volume comparison inequality bounds that Laplacian by $-\Theta'/\Theta\circ\rho$; under $K=0$ and $H\ge H_0$, $\Theta(r)=(1-rH_0)^{n-1}$. The substitution $s=1-rH_0$ turns the differential inequality $F''-\frac{\Theta'}{\Theta}F'+\lambda F\ge 0$ into a transformed Bessel equation, so the model solution is $(1-rH_0)^{n/2}$ times $J_{n/2}\left(\frac{\sqrt{\lambda}}{H_0}(1-rH_0)\right)$ (with $J_{-n/2}$ or $Y_{n/2}$ as the second independent solution). Comparing the first zero of that solution with the inner radius of $M$ yields the quotient bound in terms of $\frac{J_{n/2-1}}{J_{n/2}}$.

What would settle it

Compute the first Robin eigenvalue $\lambda_1(\tau,\Omega)$ numerically for a smooth, strictly convex Euclidean domain $\Omega$ that is not a disk, with $H_0$ the minimum of its boundary curvature, and compare it with the first Robin eigenvalue of the Euclidean disk of radius $1/H_0$. Corollary 3.8 asserts $\lambda_1(\tau,\Omega)\ge \lambda_1(\tau,B_{1/H_0})$ with equality only for the disk; a computed violation, for any $\tau>0$, would disprove the main theorem, while equality for a non-disk would disprove the rigidity statement.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: with $\mathrm{Ric}\ge 0$ and boundary mean curvature $H\ge H_0>0$, for every positive smooth $f$ satisfying $\Delta f\le \lambda f$, $\lambda>0$, and with $\sqrt{\lambda}/H_0$ below the first positive zero of $J_{n/2}$, the quotient $\int_{\partial M} f\,/\,\int_M f$ is bounded below by the Bessel quotient displayed above. Equality holds exactly when $M$ is isometric to the Euclidean ball of radius $1/H_0$, and for that ball with $\Delta f=\lambda f$ the inequality is an equality. The proof runs through the distributional second derivative of $F(r)=\int_{\{\rho>r\}} f$, a distance-function comparison that under the curvature hypotheses gives $\Theta(r)=(1-rH_0)^{n-1}$, and the change of variable $s=1-rH_0$, which turns the associated differential inequality into a Bessel-type equation whose first zero dominates the geometry. All subsequent Dirichlet, Robin, Dirac, Yamabe, and form-eigenvalue estimates are applications or direct corollaries of this quotient bound.

Load-bearing premise

The whole chain rests on the combined assumption that the manifold's Ricci curvature is nonnegative and the boundary's inward mean curvature has a positive lower bound $H_0$; if either fails, the distance-function comparison that produces the Bessel equation no longer holds, and the bound collapses.

Editorial extensions

If this is right

The first Dirichlet eigenvalue on such a manifold satisfies $\lambda_1^D \ge H_0^2\, j_{n/2-1,1}^2$, with equality only for the Euclidean ball of radius $1/H_0$.
The first Robin eigenvalue satisfies $\lambda_1(\tau,M)\ge \lambda_1(\tau,B_{H_0})$, so the Euclidean ball minimizes the Robin spectrum among all manifolds with the same curvature and boundary-mean-curvature bounds.
Dirac eigenvalues under the CHI, gAPS, or mgAPS boundary conditions obey $\lambda^2 > \frac{n}{4(n-1)}\min_M S + \frac{n H_0^2}{2(n-1)}\tau_0^2$, where $\tau_0$ is a Bessel zero; the estimate remains nontrivial even when the scalar curvature vanishes at a point.
The Robin Laplacian on $p$-forms defined by $\iota^*(\nu\lrcorner d\omega)=\tau\,\iota^*\omega$ and $\iota^*(\nu\lrcorner\omega)=0$ is elliptic, self-adjoint, and has purely positive discrete spectrum, with first eigenvalue bounded below by a Bessel expression depending on the $p$-curvature of the boundary.
The same comparison yields gap and Gallot-Meyer-type estimates: $\lambda_{1,p}(\tau)-\lambda_{1,p-1}(\tau)\ge \frac{1}{p}\inf_M (W_M^{[p]}-T^{[p]})$, and under a positive curvature operator $\lambda_{1,p}(\tau)\ge p(n-p)\frac{c}{c-1}\gamma$ when $\tau\ge -\frac{c}{c-1}\sigma_p$.

Reading between the lines

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

The paper leaves implicit that, because the quotient bound needs no boundary condition on $f$, the same differential-inequality scheme should apply to eigenfunctions of any operator whose Bochner identity produces a pointwise inequality $\Delta |u|^2 \le \mu |u|^2$, not only to Laplacians and spinors.
The paper's restriction to $K=0$ is a convenience rather than a necessity: replacing $\Theta(r)=(1-rH_0)^{n-1}$ by the corresponding function for a nonzero Ricci lower bound should replace Bessel functions by the analogous special solutions, so the theorem should extend to manifolds with a lower Ricci bound $K\ne 0$.
The strictness of the Dirac and form estimates suggests the Bessel term acts as a genuine spectral gap; a numerical computation of the first Robin eigenvalue on a smooth non-spherical convex Euclidean domain should show a positive gap relative to the ball, and measuring how that gap scales with $\tau$ would test the sharpness of the constants $\tau_0$ and $\tau_1$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my take. The genuinely new result is Theorem 3.1, a sharp lower bound for the quotient of boundary to bulk integrals under Ric >= 0, H >= H0 > 0, and Delta f <= lambda f. This goes beyond Guerini-Savo and Raulot-Savo by solving the comparison ODE in Bessel functions rather than reducing to a constant-coefficient or subharmonic case. The applications to Dirichlet and Robin Laplacians, Dirac eigenvalues, and forms are credible. The bibliography looks appropriate; self-citations are for standard material. So the paper deserves serious referee time.

But there is a real problem in the proof of Theorem 3.1 in even dimensions. The stated constants A and B for the Y_{n/2} solution do not solve the linear system coming from y(0) = integral_M f and y'(0) = -integral_dM f. Correct Cramer's rule using Lommel's identity gives A = (pi/(2H0))(sqrt(lambda) Y_{n/2-1}(a) integral_M f - Y_{n/2}(a) integral_dM f) and B = (pi/(2H0))(J_{n/2}(a) integral_dM f - sqrt(lambda) J_{n/2-1}(a) integral_M f). The paper has A = H0/(2pi)(...) and B = H0/(2pi)(...), off by the positive factor H0^2/pi^2. The ratio -A/B is invariant under this rescaling, so the final inequality (7) may still be salvageable, but as written y does not satisfy the stated initial conditions and the comparison F >= y is not justified. A referee should ask for a corrected computation.

Other issues are minor. Theorem 3.1 assumes f > 0, but applications use nonnegative f = |psi|^2 or |omega|^2; this needs a short justification. In Section 5, the principal symbol for top-degree forms looks inconsistent with the definitions, and Proposition 5.3's regularity proof is sketched, with a density step that is not obvious. Inequality (26) relies on a private communication from Freitas; if it is essential for comparing the Dirac bounds, it should be proved or replaced by a checkable reference.

Bottom line: good paper, load-bearing but fixable error. Send it to peer review, keep it in the pipeline conditional on the even-dimensional constants being corrected.

Referee Report

2 major / 3 minor

Summary. The paper develops a comparison estimate for the quotient of the boundary integral to the bulk integral of a function f satisfying Δf ≤ λf on a compact Riemannian manifold with boundary, under Ricci curvature nonnegative and boundary mean curvature bounded below by a positive constant. The central result, Theorem 3.1, bounds this quotient from below by a quotient of Bessel functions and characterizes equality by Euclidean geodesic balls. The authors then apply this estimate to recover Faber-Krahn inequalities for the Dirichlet and Robin Laplacians, to obtain new lower bounds for Dirac eigenvalues under CHI, gAPS, and mgAPS boundary conditions, and to study a Robin-type Hodge Laplacian on differential p-forms, including ellipticity, self-adjointness, variational characterization, and eigenvalue estimates. The appendix collects the Bessel-function identities used in the proofs.

Significance. If the main comparison is valid, the paper offers an elegant and fairly unified way to derive several sharp spectral estimates from one Bessel-function comparison, including known Faber-Krahn results and new Dirac and form-eigenvalue bounds. The equality statements, the explicit nature of the bounds, and the careful treatment of the p-form Robin extension are genuine strengths. The proof is mostly standard comparison geometry, but the Bessel solution is a nontrivial addition. However, the proof of the main theorem contains a localized algebraic error in the even-dimensional case, and the theorem as stated does not formally cover the nonnegative functions used in several applications. These issues are fixable but currently block acceptance.

major comments (2)

[Section 3, proof of Theorem 3.1 (even-dimensional case)] The constants A and B displayed in the even-dimensional case do not solve the stated linear system. With ν=n/2, I_M=∫_M f dμ, I_∂=∫_∂M f dμ, and a=√λ/H0, Cramer's rule applied to J_ν(a)A+Y_ν(a)B=I_M and √λ J_{ν-1}(a)A+√λ Y_{ν-1}(a)B=I_∂ gives A_c=(π√λ/(2H0))Y_{ν-1}(a)I_M-(π/(2H0))Y_ν(a)I_∂ and B_c=(π/(2H0))J_ν(a)I_∂-(π√λ/(2H0))J_{ν-1}(a)I_M. The paper's constants are both equal to (H0²/π²) times these values, so the resulting function y(r) satisfies y(0)=(H0²/π²)I_M and y'(0)=-(H0²/π²)I_∂ rather than y(0)=I_M and y'(0)=-I_∂. For n=2, H0=1, λ=1, I_M=I_∂=1, this gives y(0)≈0.101 and y'(0)≈-0.101 instead of 1 and -1. Consequently the comparison F(r)≥y(r) is not established for the stated boundary data, and the derivation of (7) and its corollaries rests on an invalid step in even dimensions. Since A and B are multiplied by a common factor, the ratio -A/B and the first zero R0 are unchanged, so the final inequality may be salvageable; nevertheless the proof must be corrected and the equality case re-verified.
[Theorem 3.1 and applications in Sections 3-5] Theorem 3.1 is stated for a positive smooth function f, but it is applied to f=|ψ|², f=|ω|², and to Dirichlet eigenfunctions, which are only nonnegative and may vanish on substantial sets. Every step of the proof, in particular equations (2)-(8) and the equality-case argument using y(0)=∫_M f>0, only requires f≥0 and ∫_M f>0. The theorem statement should therefore be relaxed to 'nonnegative, not identically zero', or an approximation argument should be supplied. Without this change, Theorems 4.1, 4.3, and 5.5 are not formally consequences of Theorem 3.1 as written.

minor comments (3)

[Theorem 5.2, proof] The assertion that Courant's nodal domain theorem implies λ_{1,p}(τ) is simple and that every associated eigenfunction cannot change sign is not valid for p-forms when p≥1; the first eigenvalue of a Hodge Laplacian with Robin-type boundary conditions can be multiple. This simplicity claim is not used in the later eigenvalue estimates, so it should be removed or replaced by a correct statement such as a min-max description allowing repeated eigenvalues.
[Throughout] The symbol ν is used both for the inward unit normal field and for the order of Bessel functions. The context usually makes the meaning clear, but the dual use is a recurring source of possible confusion and should be disambiguated in the notation section.
[Section 2, equation (4)] The distributional inequality (4) is quoted from the comparison geometry literature without explicitly stating that it holds pointwise away from the cut locus and in the distributional sense across it. A one-sentence reminder of this regularity would help readers who are not specialists in the mean value lemma.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.1 is derived from Savo's mean value lemma and Heintze–Karcher comparison, with eigenvalue applications as genuine consequences.

full rationale

The derivation chain is self-contained rather than circular. The main estimate (7) is obtained from the distributional differential inequality (6), which follows from Savo's mean value lemma [39, Thm. 2.5] and the Heintze–Karcher inequality (4). The solution y of the associated Bowman/Bessel equation is not defined in terms of the quotient being estimated; instead F(r) and y(r) solve the same initial-value problem with F(0)=∫M f and F'(0)=-∫∂M f, and the standard comparison principle yields F≥y. The desired inequality (7) is then extracted from the first zero of y and the monotonicity properties of Bessel-function ratios, not by fitting any parameter. The later Dirichlet, Robin, Yamabe, Dirac, and form-Laplacian bounds apply Theorem 3.1 to specific functions f (eigenfunctions, |ψ|^2, |ω|^2) and derive eigenvalue bounds by contradiction; these are genuine applications, not renamed inputs. External citations such as Savo's mean value lemma, Heintze–Karcher, Kasue's rigidity, and Raulot's Hijazi-type estimate are standard independent results; the self-citation to Ginoux's book [18] is used only for textbook spinor and boundary-condition facts and is not load-bearing in any circular way. The private-communication inequality (26) is an unverified external input but it is not used to derive the central quotient bound; it is a comparability claim between two derived estimates and does not make the derivation circular. A referee-flagged concern about the even-dimensional constants A and B in the proof of Theorem 3.1 is a correctness/calculation issue, not a circular reduction: no equation is defined in terms of its target, and no fitted parameter is renamed as a prediction. Overall, no circular step can be exhibited with a specific reduction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No arbitrary fitted constants or new physical entities appear. H0, tau0, tau1 are geometric or special-function constants, not tuned to data. The main dependencies on external results are standard comparison theorems and established spectral estimates.

assumptions (7)

standard math Mean value lemma of Savo (Eq. (2)): F''(r) = -∫_{rho>r} Delta f + rho_*(f Delta rho)(r) in the distributional sense.
Starting point for the differential inequality that drives the main estimate.
domain assumption Heintze-Karcher comparison (Eq. (4)): under Ric >= 0 and H >= H0, the distributional Laplacian of the distance function satisfies Delta rho >= -Theta'/Theta(rho) with Theta(r) = (1 - r H0)^(n-1).
Gives the explicit coefficient in the ODE and forces the K = 0 restriction; without H0 > 0 the Bessel change of variables s = 1 - r H0 is not valid.
standard math Sturm comparison via Wronskian: F >= y on [0, R0) when L[F] >= 0, L[y] = 0, F(0) = y(0), F'(0) = y'(0), and y > 0 on the interval.
The proof's comparison step is a Wronskian argument, not literally Gronwall; it is valid here.
standard math Bessel function identities, zeros, monotonicity properties (46)-(48), and interlacing of zeros from [1,43].
Used to solve the Bowman equation and to justify sign and monotonicity steps in Corollaries 3.4-3.8 and Theorems 4.1, 4.3, 5.5.
domain assumption Known boundary eigenvalue estimates for the Dirac operator from [22], including the inequality ∫_{partial M} <D^{partial M} psi, psi> <= 0 under CHI/gAPS/mgAPS boundary conditions.
Used in Theorem 4.1 to guarantee mu > 0 and to derive the boundary integral inequality (18).
standard math Raulot's Hijazi-type inequality (21) for CHI and MIT bag conditions.
Used in Corollary 4.4 to convert the Yamabe lower bound into a Dirac lower bound.
standard math Adequacy of Taylor's variational framework [42] for the Robin form-Laplacian (Proposition 5.3).
The paper defers several technical density and regularity details to Taylor's treatment.

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Pith. "Pith review of New eigenvalue estimates involving Bessel functions." pith.science (2026). https://pith.science/paper/DVUPD2EH

@misc{pith2026190802566,
  author       = {Pith},
  title        = {Pith review of: New eigenvalue estimates involving Bessel functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVUPD2EH}},
  note         = {Machine review of arXiv:1908.02566}
}

abstract

Given a compact Riemannian manifold (M n , g) with boundary $\partial$M , we give an estimate for the quotient $\partial$M f d$\mu$ g M f d$\mu$ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

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Reviewed August 14, 2026 · model on record in the stance chip above.

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