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REVIEW 4 major objections 4 minor 38 references

Cheeger type inequalities associated with isocapacitary constants on Riemannian manifolds with boundary

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Pith's one-line read This paper proves two-sided Cheeger-type bounds for Steklov eigenvalues: on a compact Riemannian manifold with boundary, the first nonzero Steklov eigenvalue σ1(M) is at least one quarter and at most twice the isocapacitary constant…

desk verdict A plausible and well-motivated transfer of Maz'ya's isocapacitary method to Steklov eigenvalues, but the lower-bound proof has a sign-level gap that likely forces a weaker constant; deserves serious refereeing with expected major revision. read the letter →

arxiv 2412.21008 v2 pith:JWOSZ45N submitted 2024-12-30 math.SP math.DG

Bobo Hua , Yang Shen This is my paper · ORCID
classification math.SPmath.DG MSC 58J5035P1553C21
keywords StekloveigenvalueDirichlet-to-NeumannoperatorisocapacitaryconstantCheegerinequalitycapacitycoareaformulahyperbolicsurfacebottomofspectrum
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

This paper proves a Cheeger-type sandwich for Steklov eigenvalues: on a compact Riemannian manifold with boundary, the first nonzero Steklov eigenvalue σ1(M) is bounded below by a quarter and above by twice the isocapacitary constant Γ∂(M). The same two-sided bound governs the bottom of the spectrum of the Dirichlet-to-Neumann operator when the manifold is non-compact with embedded boundary. The isocapacitary constant is a purely geometric quantity formed by minimizing the capacity of boundary sets per unit boundary measure. This gives a partial affirmative answer to the question of whether a Buser-type upper bound exists for Steklov spectra, using capacity rather than the classical Cheeger constant.

What carries the argument

The central object is the capacity Cap(A,B,M), the minimum Dirichlet energy of a function that is at least 1 on A and at most 0 on B. The isocapacitary constant Γ∂ is this capacity minimized over boundary sets and normalized by the smaller boundary measure. The proof machinery is a capacitary layer inequality: for a smooth function u, the integral over t of the capacity of the superlevel set {u≥t} relative to {u≤0}, weighted by t, is controlled by the Dirichlet energy of u on {u≥0}. This inequality comes from the coarea formula and Hardy's inequality applied to the distribution function ψ(t)=∫_0^t dτ / ∫_{M^τ_u}|∇u|ds. Applying this inequality to a Steklov eigenfunction converts level-set boundary masses into capacity estimates, giving the lower bound; the upper bound uses a direct test function built from a near-minimizer of the capacity ratio.

What would settle it

Compute the two sides of the printed Proposition 10 for u(x)=x on the interval [0,1]: Cap({x≥t},{x≤0}) = 1/t, so ∫$_0^{1}$ Cap dt diverges while the right side is 4, disproving that printed form. For the main theorem itself, compute σ1 and Γ∂ explicitly on a flat strip or thin cylinder; if the ratio σ1/Γ∂ ever falls outside [1/4,2], the two-sided bound fails.

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Extended reading notes

Core claim

The central claim is that the spectral quantity and the geometric quantity are comparable up to a universal factor. For a compact n-dimensional Riemannian manifold with smooth boundary, define Γ∂(M) = min Cap(A,B,M)/min{m(A),m(B)} over disjoint compact boundary sets A,B. The paper establishes (1/4)Γ∂(M) ≤ σ1(M) ≤ 2Γ∂(M). For a non-compact manifold with embedded smooth boundary, define Γ∂(M) = inf Cap(F,M)/m(F) over compact F⊂∂M; then (1/4)Γ∂(M) ≤ inf Spec(D_M) ≤ 2Γ∂(M). In both settings the bottom of the boundary spectral theory is thus controlled, up to the universal factor 8, by boundary capacities.

Load-bearing premise

The lower bound rests on the layer-capacity inequality, Proposition 10, which must compare an integral of t times capacity with the Dirichlet energy; as printed, the proposition lacks the factor t and is false for a linear function on an interval.

Editorial extensions

If this is right

  • On a compact manifold, the first nonzero Steklov eigenvalue is determined up to a universal factor of 8 by the isocapacitary constant Γ∂, so spectral gaps can be certified by boundary capacity computations.
  • The proof extends to Steklov–Dirichlet problems: for a submanifold N with mixed boundary data, the first eigenvalue ξ1(N,∂_I N) satisfies the analogous two-sided estimate (1/4)Γ_Y(N) ≤ ξ1 ≤ 2Γ_Y(N).
  • For compact hyperbolic surfaces, a boundary geodesic of length tending to zero forces the normalized first Steklov eigenvalue to satisfy limsup σ1(S_{g,1})·ℓ(γ) ≤ 4/3; if a surface with at least two boundaries has a boundary curve of length <1, then σ1(S_{g,n}) ≤ e + e^{-1} ≈ 3.086.
  • For the non-compact hyperbolic upper half-ball, the Dirichlet-to-Neumann operator has an explicit integral representation, and the bottom of its spectrum is exactly 2/π in dimension 2 and at least (∫_0^∞ cosh^{-(n-1)}s ds)^{-1} in dimension n≥3.

Reading between the lines

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

  • Beyond the paper: once Γ∂(M)>0 on a non-compact manifold, Theorem 2 gives a positive spectral gap above zero for the Dirichlet-to-Neumann operator, implying a form of boundary confinement that the authors do not single out.
  • Beyond the paper: the same capacity-versus-energy sandwich should transfer to weighted manifolds, where capacity is defined with a density, and the constants can be tested numerically on flat strips or warped cylinders.
  • Beyond the paper: the paper treats only the first eigenvalue, but the min-max principle suggests isocapacitary upper and lower bounds for higher Steklov eigenvalues σ_k with constants depending on k, which is a natural extension the authors do not state.
  • Beyond the paper: the hyperbolic-surface estimate σ1(S_{g,1})ℓ(γ) ≤ 4/3 + o(1) raises the question of whether 4/3 is asymptotically sharp for short geodesic boundaries, which the paper does not address.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper defines isocapacitary constants Γ∂(M) for a Riemannian manifold with boundary and claims Cheeger-type two-sided bounds: for compact M, 1/4 Γ∂(M) ≤ σ1(M) ≤ 2Γ∂(M), and for non-compact M, 1/4 Γ∂(M) ≤ inf Spec(DM) ≤ 2Γ∂(M). The proofs proceed through a capacity lemma (Proposition 10), a variational argument for the first Steklov eigenvalue, a limit argument for the non-compact case, and applications to compact hyperbolic surfaces with geodesic boundaries and to the hyperbolic half-ball. The manuscript is self-contained and the constants are computed, not fitted; the graph paper [20] is used only as an analogue, not as an input.

Significance. If the main theorems are correct, the paper gives a uniform control of the first Steklov eigenvalue (or the bottom of the Dirichlet-to-Neumann spectrum) by a purely geometric isocapacitary quantity, up to a universal factor of 8. This would partially answer the Buser-type question from Colbois–Girouard–Gordon–Sher and extend Maz'ya's isocapacitary framework to the Steklov setting. The applications to hyperbolic surfaces produce explicit upper bounds and eigenvalue decay statements that are concrete and checkable. The paper is readable and the overall strategy is credible, but the central lemma and one key application step need correction before the claims are established.

major comments (4)
  1. [§3, Proposition 10] Proposition 10 as printed is false. For M=[0,1] and u(x)=x, the set {u≥t} is [t,1] and {u≤0} is {0}; the capacity of these two sets is 1/t, so the left side ∫_0^1 Cap dt diverges while the right side is 4∫_0^1 |u'|^2 dx = 4. The proof in Eq. (14) integrates with respect to d(t(ψ)^2), which indicates that the intended statement is ∫_0^∞ t·Cap({u≥t},{u≤0},M) dt ≤ 2∫_{M_u≥0} |∇u|^2, equivalently ∫_0^∞ Cap d(t^2) ≤ 4∫|∇u|^2. This correction is local and repairable, but the lemma as stated is load-bearing for Eq. (19) and must be fixed.
  2. [§4, proof of Theorem 1, Eq. (19)–(20)] The lower-bound chain for a sign-changing Steklov eigenfunction is incomplete. With the corrected form of Proposition 10, the displayed argument yields σ1∫ f^2 ≥ 1/2 ∫_0^∞ t·Cap(...) dt ≥ 1/2 Γ∂(M) ∫_0^∞ t·m({f≥t}) dt = 1/2 Γ∂(M) · (1/2 ∫_{∂M} (f_+)^2 dσ), not 1/4 Γ∂(M)∫_{∂M} f^2 dσ. The identity used in the paper identifies ∫ t·m({f≥t}) dt with 1/2∫ f^2, but the correct identity is 1/2∫ (f_+)^2 because f changes sign. Assumption (18) controls the measures of the positive and negative sets but gives no control of ∫(f_+)^2 relative to ∫ f^2. Thus the claimed lower bound 1/4 Γ∂(M) does not follow without an additional argument handling f_+ and f_-, or a weaker constant.
  3. [§2.2 and §4, Theorem 2 and Lemma 12] The non-compact part silently assumes properties that are not stated. The construction of Hf via the limit over M∩B(p,r), the monotonicity of ξ1(Mr, ∂I Mr), and the equality Γ∂(M)=lim Γ∂(Mr,M) in Lemma 12 require that the geodesic balls B(p,r) have relatively compact intersection with M, which is automatic for complete Riemannian manifolds but not for arbitrary non-compact manifolds with embedded boundary. The completeness hypothesis should be added, or the statements should be restricted to a class of manifolds for which the exhaustion is valid.
  4. [§4, definition of Cap(F,M)] The definition of Cap(F,M) in the proof of Theorem 2 is misprinted: it reads 'f is taken over all smooth functions such that f ≡ 0 on F and f ∈ C_c^∞(M)', but the correct condition, as given in the introduction, is f ≡ 1 on F. With the printed condition, Cap(F,M)=0 for every F and Γ∂(M) becomes identically zero, which would make Theorem 2 vacuous. This is an obvious typo, but it affects the meaning of Lemma 12 and Theorem 2 and should be corrected.
minor comments (4)
  1. [Throughout] There are numerous typos, e.g., 'Huasdorff' for Hausdorff, 'Togther' for Together, 'exsit' for exist, and several instances of inconsistent spacing around equations. A careful proofreading pass is needed.
  2. [§2.2, Eq. after f_+ and f_-] The line defining f^- is misprinted as 'f − = 1/2 = (f − |f|)' and should read 'f_- = 1/2(f - |f|)'.
  3. [§5.2, Theorem 14 and surrounding text] The notation for the DtN operator is inconsistent: D^{n+1}_+ appears as D^{n+1}_+, DD^{n+1}_+, and DD^n_+ in different places. Please unify the notation.
  4. [§5.1, Case II] The statement 'σ1(Sg,n) ≤ e + e^{-1}' for l0 < 1 < ρ0 should specify how ρ0 is chosen in terms of l0; currently the inequality depends on the choice of ρ0 via the collar lemma, and the text should make the dependence explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the capacity constants are computed independently, and the target eigenvalue estimates are not assumed as inputs.

full rationale

The derivation chain is self-contained. The isocapacitary constants Γ∂(M) are defined directly from capacitary minimizers over boundary sets, and the proof of Theorem 1 does not assume σ1(M) in the definition of Γ∂(M). The lower bound uses a capacity-coarea comparison (Proposition 10) plus the variational characterization of σ1; the upper bound constructs explicit test functions from near-minimizing pairs (A,B), which is a standard argument rather than a circular one. The self-citation [20] is an analogue for graphs and is not used as a load-bearing premise in the proofs of Theorems 1 or 2. The non-compact case reduces to compact exhaustions via Lemma 5 and Lemma 12, with the limiting Γ∂ computed independently from the definition, and the DtN extension uses the standard Reed–Simon theorem. The specific example computations (hyperbolic surfaces and D^2_+) are independent explicit calculations. This review also notes that Proposition 10 as printed appears incorrect—for M=[0,1] and u(x)=x the left-hand side diverges while the right-hand side is finite—but this is a correctness defect in a supporting lemma, not a circularity: the flawed lemma does not make the theorem equivalent to its inputs. Therefore no circular step satisfying the required quote-and-reduction standard was found.

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

No fitted constants appear in the main theorems. The proof is self-contained apart from standard capacity, spectral theory, and the stated hyperbolic-geometry facts. The only non-standard input is the capacitary inequality in Proposition 10, whose printed form is ambiguous and needs correction.

assumptions (5)
  • standard math Coarea formula and layer-cake representation of Dirichlet energy for smooth functions on manifolds with boundary.
    Used in Section 3 (Lemma 8 and Proposition 10) to convert energy integrals into integrals over level sets of a function.
  • standard math Hardy inequality: ∫_0^a (t/ψ)^2 dψ ≤ 4∫_0^a (t')^2 dψ for absolutely continuous t with t(0)=0.
    Invoked in eq. (17) of Proposition 10 to obtain the factor 4 in the capacitary inequality.
  • standard math Reed-Simon Theorem X.23: a positive symmetric operator has a unique self-adjoint positive extension via its closable quadratic form.
    Used in Section 2.2 to define the DtN operator on non-compact manifolds and to get the variational formula (11).
  • domain assumption The exhaustion M_r = M∩B(p,r) consists of relatively compact domains with piecewise smooth boundary and exhausts M.
    Needed for Lemmas 4, 5, and 12 and Theorem 2; not stated explicitly, and it effectively requires completeness of M.
  • domain assumption Collar lemma for hyperbolic surfaces: a collar around a short boundary geodesic is a topological cylinder with metric ds²=dρ²+l0² cosh²ρ dt².
    Used in Section 5.1 to construct test functions Φ and Ψ; quoted from Buser, Theorem 4.1.1.

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Pith. "Pith review of Cheeger type inequalities associated with isocapacitary constants on Riemannian manifolds with boundary." pith.science (2026). https://pith.science/paper/JWOSZ45N

@misc{pith2026241221008,
  author       = {Pith},
  title        = {Pith review of: Cheeger type inequalities associated with isocapacitary constants on Riemannian manifolds with boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWOSZ45N}},
  note         = {Machine review of arXiv:2412.21008}
}
read the original abstract

In this paper, we study the Steklov eigenvalue of a Riemannian manifold (M, g) with smooth boundary. For compact M , we establish a Cheeger-type inequality for the first Steklov eigenvalue by the isocapacitary constant. For non-compact M , we estimate the bottom of the spectrum of the Dirichlet-to-Neumann operator by the isocapacitary constant.

Figures

Figures reproduced from arXiv: 2412.21008 by the authors.

Figure 1
Figure 1. In this article, we mainly consider the Steklov-Dirichlet problem (3) for the case that X = N and Y = ∂ IN. The equality (4) tells that ξ1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Assume ρ1 ≤ ρ0 is a fixed positive real number, consider functions f(u) = 1 cosh u , h(t) = sin 2πt [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Case I: n = 1. As showed in [5] (Example 4.5 in Page 39), the first Steklov eigenvalue σ1(Sg,1) has a universal lower bound as boundary length tends to infinity. Case II: n ≥ 2. Let γ1 and γ2 be two boundary curves corresponding to closed geodesics α1 and α2 respectively (see the second picture in [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Hence H n+1 R has another coordinate representation (u1, ..., un, r) such that for any point p = (x1, ..., xn, y) ∈ H n+1 R ,    ui(p) = xi for 1 ≤ i ≤ n − 1; un(p) = y; r(p) = dist(p, ∂H n+1 R ) = arccosh √ x2 n+y2 y . Direct calculation implies that under this…
Figure 5
Figure 5. Figure 5: On the other hand, we have P(p0, q) = P(τ (p0), τ (q))(31) = − 2 Vol(Sn) · 1 sinhn dist(p0, q) · ∂ dist(τ (p0), τ (r)) ∂ν |r=q = − 2 Vol(Sn) · 1 sinhn dist(p0, q) · ⟨∇dist(τ (p0), τ (q)), ν⟩ds2 = 2 Vol(Sn) · 1 sinhn dist(p0, q) · y(τ (p)) |τ (p)| . From (30) and (31), …

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