Eigenvalue estimates and applications on weighted manifolds

A. C. Bezerra; F. Manfio; T. Castro Silva

REVIEW 4 major objections 5 minor 23 references

Eigenvalue estimates and applications on weighted manifolds

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that on compact weighted manifolds, the first Dirichlet and Neumann eigenvalues of the drifting Laplacian are strictly larger than $\mathrm{Ric}_h - c|\nabla h|^2$ under the condition $\mathrm{Ric}_h > c|\nabla h|^2$ and…

desk verdict The eigenvalue theorem is not proven as stated—only λ > inf S follows—and the stability theorem rests on it; the stability computation itself is real. read the letter →

arxiv 2412.09396 v4 pith:OT352DJC submitted 2024-12-12 math.DG

A. C. Bezerra , T. Castro Silva , F. Manfio This is my paper · ORCID

classification math.DG MSC 35P1553C2353C4258K25

keywords weightedmanifoldsBakry-ÉmeryRiccicurvaturedriftingLaplacianfirsteigenvalueDirichletandNeumannproblemsh-minimalhypersurfaceL_h-stabilityReillyformula

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 aims to establish strict lower bounds for the first nonzero eigenvalues of the Dirichlet and Neumann problems for the drifting Laplacian $\Delta_h$ on a compact weighted manifold $M_h = (M^n, g, e^{-h}dv)$, with the bounds expressed through the Bakry-Émery Ricci curvature $\mathrm{Ric}_h = \mathrm{Ric} + \nabla^2 h$. The central claim is that under the pointwise condition $\mathrm{Ric}_h > c|\nabla h|^2$ for some constant $c$, the first eigenvalue is strictly larger than $\mathrm{Ric}_h - c|\nabla h|^2$, provided the boundary has nonnegative weighted mean curvature (Dirichlet) or is convex (Neumann). This is meant to generalize earlier estimates of Ma and Du and, for constant weight, to give a strict Reilly-type Lichnerowicz–Obata bound $\lambda_1 > \mathrm{Ric}$. The same eigenvalue estimate is then applied to prove a sufficient curvature condition for a compact $h$-minimal hypersurface with boundary to be $L_h$-stable. A reader would care because the bounds connect the spectrum of natural diffusion operators on weighted spaces to the curvature quantity that governs optimal transport and gradient Ricci solitons.

What carries the argument

The engine of the paper is the weighted Reilly formula in Proposition 2.1, obtained by integrating the weighted Bochner formula, with the Hessian term controlled through $|\nabla^2 f|^2 \ge \frac{(\Delta_h f)^2}{m} - \frac{\langle\nabla f,\nabla h\rangle^2}{m-n}$ for $m>n$. Boundary control comes from the boundary integral $\frac12\int_{\partial M_h}\langle\nabla|\nabla f|^2,\eta\rangle\,da_h$, which is shown to be nonpositive for Dirichlet eigenfunctions under $H_h^{\partial M}\ge0$ and for Neumann eigenfunctions under convexity of the boundary. For Theorem 2, the key identities are the explicit formula for $L_h(fH)$ in Proposition 2.5, Young's inequality, and the variational characterization of $\lambda_{1,D}$, which together convert the eigenvalue estimate into the stability inequality.

What would settle it

On a Euclidean ball $B_R$ with density $e^{-\epsilon r^2}$, take $c$ small enough that $\mathrm{Ric}_h = 2\epsilon I > 4c\epsilon^2 r^2 I$ holds and the boundary has nonnegative weighted mean curvature; compute $\lambda_{1,D}$ numerically and compare it with the infimum and supremum of $2\epsilon - 4c\epsilon^2 r^2$ over $B_R$. An eigenvalue below the supremum refutes the pointwise claim, while an eigenvalue below the infimum refutes the weaker bound the proof actually supports.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: if an $n$-dimensional compact weighted manifold $M_h$ has nonempty smooth boundary, $\mathrm{Ric}_h > 0$, and $\mathrm{Ric}_h > c|\nabla h|^2$ for some constant $c$, then the first Dirichlet eigenvalue satisfies $\lambda_{1,D} > \mathrm{Ric}_h - c|\nabla h|^2$ when the weighted mean curvature of the boundary is nonnegative, and the first Neumann eigenvalue satisfies $\lambda_{1,N} > \mathrm{Ric}_h - c|\nabla h|^2$ when the boundary is convex. The proof derives a weighted Reilly-type inequality and then argues by contradiction, so the inequality is strict. Theorem 2 applies this bound, together with parallelism of $\nabla^2 h$ and the curvature lower bound (1.6), to conclude that a two-sided compact $h$-minimal hypersurface with $H\ne0$ and $H_h^{\partial M}\ge0$ is $L_h$-stable.

Load-bearing premise

The load-bearing premise is that negating the claimed pointwise inequality gives the reverse inequality at every point, so the proof really establishes only $\lambda_1 > \inf(\mathrm{Ric}_h - c|\nabla h|^2)$ rather than the pointwise comparison stated in the theorem.

Editorial extensions

If this is right

If the pointwise form of Theorem 1 is correct, then whenever $\mathrm{Ric}_h > c|\nabla h|^2$ and $H_h^{\partial M}\ge0$, the Dirichlet eigenvalue satisfies $\lambda_{1,D} > \mathrm{Ric}_h - c|\nabla h|^2$ at every point, and the analogous Neumann statement holds under convexity of the boundary.
Specializing to constant $h$ recovers strict Reilly-type bounds for the ordinary Laplacian, namely $\lambda_1 > \mathrm{Ric}$, for Dirichlet data with nonnegative mean curvature and for Neumann data on a convex boundary.
The authors note that for an appropriate choice of $c$ the estimate recovers the Ma–Du lower bound for drifting Laplacians, placing the new inequalities as strict refinements of previously known non-strict bounds.
Theorem 2 yields a sufficient condition for $L_h$-stability: a compact $h$-minimal hypersurface with nonnegative weighted boundary mean curvature, parallel Hessian of $h$, and $\mathrm{Ric}_h \ge 2[|A|^2 + c|\nabla h|^2 + (|\nabla^2 h|^2 + |\nabla H|^2)/H^2]$ is stable under compactly supported variations.

Reading between the lines

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

The contradiction argument in Theorem 1 actually appears to rule out only eigenvalues at or below $\inf(\mathrm{Ric}_h - c|\nabla h|^2)$; if that reading is right, the stated pointwise inequality is stronger than the proof establishes, and Theorem 2, which uses the pointwise form, inherits the gap.
The weighted Reilly inequality in Proposition 2.1 is not tied to the first eigenvalue; the same boundary-term analysis could be used to estimate higher eigenvalues or the fundamental tone of submanifolds with controlled boundary.
A concrete test of the stability criterion would be to compute the left-hand side of (1.6) for a family of model hypersurfaces (for instance, geodesic spheres in a warped product with a Gaussian weight) to see whether the threshold is sharp or only sufficient.
If the pointwise reading of Theorem 1 is abandoned, the stability conclusion might still survive if $\mathrm{Ric}_h - c|\nabla h|^2$ is replaced by its infimum in condition (1.6), suggesting the paper's framework is close to a correct statement.

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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

Dear colleague,

The paper has one new result that looks real—the L_h-stability criterion for h-minimal hypersurfaces under condition (1.6)—but the main eigenvalue theorem it rests on is not proven as stated. The proof only gets λ > inf(Ric_h − c|∇h|²), not the pointwise inequality λ > Ric_h − c|∇h|² that the statement and the stability application require.

The good part: Proposition 2.5, the computation of L_h(fH), is a serious piece of tensor analysis, and the stability condition (1.6) is new and geometrically meaningful. If the eigenvalue input were valid, Theorem 2 would be a nice application. The authors also correctly identify that their condition is different from Ma-Du's; they do not have the additive constant a, and with c = 1/(m−n) their bound is actually weaker, so the claim that Theorem 1 generalizes Ma-Du is not substantiated.

The soft spots are load-bearing. Theorem 1 compares a scalar λ to a tensor field; the only coherent reading is the pointwise quadratic-form inequality. The proof's contradiction starts by assuming the global statement λ ≤ S everywhere, which is not the negation of the pointwise claim. It rules out λ ≤ inf S, so at best it proves λ > inf S. Theorem 2's inequality (2.34) needs the integrand ½(Ric_h − c|∇h|²) − ... to be nonnegative pointwise, which only follows from the pointwise eigenvalue inequality. The gap is not a minor detail; the stability theorem is unsupported without it. Also, Theorem 1 omits c > 0, which the proof requires when choosing m ≥ n + 1/c (Theorem 2 states it correctly).

There are also presentation issues: typos like 'areises', 'preceeds', 'eingenvalue', inconsistent notation for Hessian inner products, and the abstract's 'We will present' reads like a talk announcement rather than a paper.

Who is this for? Geometers working on drifting Laplacians and f-minimal hypersurfaces. A careful reader will see the gap quickly, but the stability calculation itself might be useful for parts. I would not cite the results as stated. If the authors can salvage a correct weaker eigenvalue theorem (λ > inf S) and redo Theorem 2 with that or with a pointwise condition that is actually proved, the paper could become a modest contribution. In current form, it should not be accepted.

Recommendation: send to peer review only if you expect the referee to identify the gap and require major revision; the work is not ready for publication. If you want a cleaner desk decision, reject. My own verdict is reject, but I would let a referee confirm the gap because it is quantitative and fixable.

Referee Report

4 major / 5 minor

Summary. The paper claims a lower bound for the first Dirichlet and Neumann eigenvalues of the drifting Laplacian on a compact weighted manifold with boundary in terms of the Bakry-Émery Ricci tensor: λ > Ric_h − c|∇h|² under the hypothesis Ric_h > c|∇h|² and suitable boundary conditions. It then uses this estimate to prove an Lh-stability criterion for compact h-minimal hypersurfaces with parallel Hessian of the weight. The proofs are based on a weighted Reilly-type formula (Proposition 2.1), a Bochner formula, and an eigenfunction argument; Theorem 2 combines the eigenvalue bound with an identity for L_h(fH) and Young's inequality.

Significance. If correct, the results would supply a new eigenvalue comparison with a curvature tensor and a stability criterion for weighted minimal hypersurfaces. The paper is self-contained and Proposition 2.1 is a standard weighted Reilly inequality that is plausibly correct; the proofs do not assume the desired conclusion, so circularity is not the issue. However, the main theorem is ill-posed as a comparison of a scalar eigenvalue with a tensor field, and the proof establishes at most λ > inf(Ric_h − c|∇h|²), not the pointwise inequality stated. Since Theorem 2 relies on the pointwise form inside an integral, the stability application is not established. The central claim as stated is therefore not credible.

major comments (4)

[§2.1, Theorem 1 and Eqs. (2.13)–(2.15)] Theorem 1 asserts λ_{1,D} > Ric_h − c|∇h|² with λ a real number and Ric_h a (0,2)-tensor; this inequality is not type-correct. If it is read as the pointwise quadratic-form inequality λ_{1,D} > Ric_h(v,v) − c|∇h|²(p) for every p and unit v, then the proof does not establish it. In the proof, the supposition 'λ_{1,D} ≤ Ric_h − c|∇h|²' is treated as a global pointwise assumption, and the contradiction obtained from (2.13)–(2.15) only rules out the possibility that the pointwise reverse inequality holds everywhere. The argument therefore implies at most λ_{1,D} > inf_{p,v}(Ric_h(v,v) − c|∇h|²(p)); the stated pointwise conclusion does not follow.
[§2.1, Eqs. (2.14)–(2.15)] The passage from (2.14) to (2.15) requires the coefficient of ∫|∇f|²|∇h|² to be nonnegative, i.e. c − 1/(m−n) ≥ 0. For c ≤ 0 this fails, and the condition m ≥ n + 1/c is vacuous when c is negative. Since Theorem 1 only says 'there exists a constant c' without any sign condition, the proof does not cover c ≤ 0; if the intended hypothesis is c > 0, it must be stated explicitly.
[§2.2, Eq. (2.34)] In deriving (2.34), the scalar λ_{1,D} is replaced inside the integrand by the function (Ric_h − c|∇h|²)/2. This replacement is legitimate only if the pointwise inequality λ_{1,D} > Ric_h − c|∇h|² holds on all of M. The proof of Theorem 1 supplies at most λ_{1,D} > inf(Ric_h − c|∇h|²), which does not control the sign of the integrand pointwise. Consequently the passage to (2.34) and the Lh-stability conclusion of Theorem 2 are unsupported.
[§2.2, Eqs. (1.6) and (2.34)] Both condition (1.6), 'Rich ≥ 2[|A|² + c|∇h|² + ...]', and the integrand in (2.34), '(Rich − c|∇h|²)/2', treat the Bakry–Émery tensor as a scalar function. If 'Rich ≥ ...' is meant as a quadratic-form inequality, the right-hand side must be tensorial, e.g. the scalar expression times the metric; if 'Rich' is meant as the infimum or the smallest eigenvalue of the tensor, that notion is not defined and is not the object used in Proposition 2.1. As written, the hypotheses and the application of Theorem 1 in Theorem 2 are mathematically ambiguous.

minor comments (5)

[Introduction and throughout] There are numerous typographical errors, including 'knwon', 'eingenvalue', 'areises', 'preceeds', and 'obtainded'; these should be corrected in any revision.
[§2.1, Eq. (2.11)] For f vanishing on ∂M, ∇f = f_η η, so the intermediate expression ∇f + f_η η equals 2f_η η and appears inconsistent with the following equality; the final boundary term is correct because of the factor 1/2 in (2.6), but the displayed chain of equalities should be rewritten.
[§2.1, Theorem 1 proof, item (2)] The notation ∇f = ∇̄f in the Neumann case is not defined; presumably the bar denotes the tangential gradient on the boundary, and this should be stated.
[Introduction, discussion after Theorem 1] The claim that Theorem 1 generalizes the Ma–Du estimate is not substantiated: with c = 1/(m−n), the conclusion would give λ > a, whereas Ma–Du gives λ ≥ ma/(m−n), which is stronger for a > 0.
[§1, Eq. (1.3)] The text states 0 < λ_1 ≤ λ_2 for the Neumann problem, but the Neumann spectrum contains the zero eigenvalue; presumably λ_{1,N} denotes the first nonzero Neumann eigenvalue, and this should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained, with the main defect a logical gap in Theorem 1 rather than circular reasoning.

full rationale

The paper's central estimate (Theorem 1) is derived from the weighted Bochner formula (2.1), the integration-by-parts identity (2.5), and the algebraic estimate (2.7); Proposition 2.1 restates this chain, and the proof of Theorem 1 applies it to a first eigenfunction, with boundary terms controlled by H_h ≥ 0 or convexity. No parameter is fitted from the eigenvalue data, and no conclusion is inserted as a hypothesis: the only external inputs are standard divergence and Bochner facts, plus cited background results used for comparison rather than as assumptions of the proof. The cited Lemma 2.3 from [3] is a standard hypersurface Laplacian decomposition; although one author of the present paper is an author of [3], the lemma is not special or contested, and its use does not make the derivation circular. Theorem 2 is a conditional application of Theorem 1 to the stability operator; whatever the merits of the eigenvalue bound, the stability conclusion is not obtained by assuming itself. The main defect identified by the reader, namely the scalar-versus-tensor comparison λ > Rich − c|∇h|² and the proof's use of the global pointwise negation, is a logical gap or type mismatch rather than a circular identification of a conclusion with an input. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof relies only on standard differential geometry tools. The constant c in the curvature hypothesis is an assumption, not fitted to data. No new objects are introduced.

assumptions (5)

standard math Weighted Bochner formula (2.1)
Used in Proposition 2.1; cited from Ma-Du [13].
standard math Divergence theorem for the measure e^{-h}dv
Used in (2.5) to integrate by parts on weighted manifolds.
standard math Variational characterization of the first Dirichlet eigenvalue
Used in (2.32) to compare the eigenvalue to weighted Dirichlet energy.
standard math Codazzi equation for hypersurfaces
Used in Proposition 2.5, equations (2.20) to (2.23).
standard math Young's inequality and Cauchy-Schwarz inequality
Used in Theorem 1 and Theorem 2 estimates to control cross terms.

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Pith. "Pith review of Eigenvalue estimates and applications on weighted manifolds." pith.science (2026). https://pith.science/paper/OT352DJC

@misc{pith2026241209396,
  author       = {Pith},
  title        = {Pith review of: Eigenvalue estimates and applications on weighted manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OT352DJC}},
  note         = {Machine review of arXiv:2412.09396}
}

read the original abstract

We will present an estimate for the first eigenvalue of the Dirichlet and Neumann problems in terms of the Bakry-\'Emery Ricci curvature for a compact weighted manifold. As an application we will establish a stability condition for a h-minimal hypersurface.

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Fonction de Green et valeurs propres du laplacien

Aubin, T. Fonction de Green et valeurs propres du laplacien. J. Math. Pures Appl. (9) 53 (1974), 347–371

work page 1974

[2] [2]

C., Manﬁo, F

Bezerra, A. C., Manﬁo, F. Rigidity and stability estimates for minimal submanifolds in the hyperbolic space. J. Math. Anal. Appl. 495 (2021), 124759

work page 2021

[3] [3]

P., Manﬁo, F

Cavalcante, M. P., Manﬁo, F. On the fundamental tone of immersions and submersions. Proc. Amer. Math. Soc. 146 (2018), no. 7, 2963–2971

work page 2018

[4] [4]

Diﬀusions hypercontractives

Bakry, D., ´Emery, M. Diﬀusions hypercontractives. Seminaire de proba- bilites, XIX, 1983/84, 177–206. Lecture Notes in Math., 1123, Sprin ger- Verlag, Berlin, 1985

work page 1983

[5] [5]

The relation between the Laplacian and the diameter for manifolds of non-negative curvature

Cheeger, J. The relation between the Laplacian and the diameter for manifolds of non-negative curvature. Arch. Math. 19 (1968), 558–560

work page 1968

[6] [6]

A lower bound for the smallest eigenvalue of the Laplacian

Cheeger, J. A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis (Sympos. in honor of Salomon Bochner, P rinceton Univ., Princeton, N.J., 1969), pp. 195–199

work page 1969

[7] [7]

Cheng, S. Y. Eigenvalue comparison theorems and its geometric appli- cations. Math. Z. 143 (1975), no. 3, 289–297. 15

work page 1975

[8] [8]

Stability and Compactness for complete f -minimal surfaces

Cheng, X., Mejia, T., Zhou, D. Stability and Compactness for complete f -minimal surfaces. Trans. Amer. Math. Soc. 367 (2015), no. 6, 4041– 4059

work page 2015