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arxiv: 2603.15256 · v2 · pith:WTSGQAPEnew · submitted 2026-03-16 · 🧮 math.SP

Weyl asymptotics for singular metrics with a variable boundary degeneracy exponent

Yves Colin de Verdi\`ere (IF) , Charlotte Dietze (LJLL (UMR\_7598) , CNRS) , Emmanuel Tr\'elat (LJLL (UMR\_7598) , CaGE) This is my paper

Pith reviewed 2026-05-25 07:04 UTC · model grok-4.3

classification 🧮 math.SP
keywords Weyl asymptoticssingular metricsdegeneracy exponentFriedrichs LaplacianMorse-Bottspectral asymptoticsboundary degeneracy
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The pith

The leading term in Weyl asymptotics for the Friedrichs Laplacian on singular metrics with variable boundary degeneracy α is set by maxima of α when they exceed 2/(n+1), otherwise by truncated volume at distance λ^{-1/2}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for the Friedrichs Laplacian on a compact manifold with boundary carrying a singular metric of the form g = u^{-α} times a regular metric near the boundary, with α varying along the boundary, the leading asymptotic of the eigenvalue counting function depends on the global maximum of α relative to the critical value 2/(n+1). When this maximum exceeds the critical value, the leading term is controlled by the locations where α approaches its peak. When the maximum is at or below the critical value, the leading term instead follows the volume of the region lying at least distance λ^{-1/2} from the boundary. When the set where α attains its maximum is Morse-Bott, the paper supplies the explicit constant prefactor together with any logarithmic corrections.

Core claim

If the maximum α_max of α on M is strictly larger than the critical value α_c=2/(n+1), then the points where α is close to α_max govern the leading term in the Weyl asymptotics. If α_max≤α_c, then the leading term is governed by the truncated volume vol_g({dist(·,M)>λ^{-1/2}}). When the maximum set of α is Morse-Bott, we compute the associated constants and the logarithmic corrections.

What carries the argument

The variable degeneracy exponent α (with critical threshold α_c = 2/(n+1)) that switches the leading Weyl term between boundary-maxima dominance and truncated-volume dominance.

If this is right

  • When α_max > 2/(n+1), the leading coefficient in the Weyl law is determined by the local geometry near the maxima of α.
  • When α_max ≤ 2/(n+1), the leading coefficient equals the volume of the region lying at g-distance greater than λ^{-1/2} from the boundary.
  • When the set achieving α_max is Morse-Bott, the leading term includes an explicit constant factor and possible logarithmic corrections.
  • This supplies the first Weyl law for metrics whose degeneracy exponent varies along the boundary.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sharp transition at α_c = 2/(n+1) may reflect a general mechanism by which variable coefficients select between boundary and interior contributions in singular spectral problems.
  • The same regime switch could be tested in related settings such as magnetic Laplacians or Schrödinger operators with variable singular potentials.
  • Explicit model calculations on rotationally symmetric collars would allow direct verification of the computed constants and log terms.

Load-bearing premise

The metric must take the form g = u^{-α} times a regular metric in a collar neighborhood of the boundary, with α continuously differentiable and strictly less than 2.

What would settle it

Numerically compute the eigenvalue counting function for an explicit example (such as a model collar metric with α varying above and below 2/(n+1)) and compare the observed leading coefficient against the boundary-maxima prediction versus the truncated-volume prediction.

read the original abstract

We consider a compact smooth manifold $X$ of dimension $n+1$ with boundary $M=\partial X$. In a collar neighborhood of $M$, we assume that the metric has the form $g=u^{-\alpha}\bar g$, where $u$ is a boundary defining function, $\alpha\in C^1(M;[0,2))$ and $\bar g$ is a $C^1$ Riemannian metric up to $M$. Since $\alpha<2$, the boundary lies at finite $g$-distance and $(X,g)$ is a singular metric space. We study the Weyl asymptotics of the Friedrichs Laplacian $\triangle\_g$ when the degeneracy exponent $\alpha$ varies along $M$. If the maximum $\alpha\_{\mathrm{max}}$ of $\alpha$ on $M$ is strictly larger than the critical value $\alpha\_c=\frac{2}{n+1}$, then we prove that the points where $\alpha$ is close to $\alpha\_{\mathrm{max}}$ govern the leading term in the Weyl asymptotics. If $\alpha\_{\mathrm{max}}\leq\alpha\_c$, then the leading term is governed by the truncated volume $\vol\_g(\{\dist(\cdot,M)>\lambda^{-1/2}\})$. When the maximum set of $\alpha$ is Morse-Bott, we compute the associated constants and the logarithmic corrections. To the best of our knowledge, this is the first Weyl law in this setting with a boundary-dependent degeneracy exponent. The results highlight a sharp transition at $\alpha\_c$ between a boundary-dominated non-classical regime and a truncated-volume regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes Weyl asymptotics for the Friedrichs Laplacian on a compact (n+1)-dimensional manifold with boundary equipped with the singular metric g = u^{-α}¯g (α ∈ C¹(M;[0,2))) in a collar neighborhood. It proves a sharp transition at the critical value α_c = 2/(n+1): when α_max > α_c the leading term is governed by the maximum set of α, while when α_max ≤ α_c the leading term is given by the truncated volume vol_g({dist(·,M) > λ^{-1/2}}). Under a Morse-Bott assumption on the maximum set, explicit constants and logarithmic corrections are computed. This is presented as the first such result with variable boundary degeneracy exponent.

Significance. If the proofs are correct, the work supplies the first Weyl law in this singular-metric setting with spatially varying degeneracy, together with a precise regime separation based on the integrability threshold of the volume form u^{-α(n+1)/2} du. The explicit constants under the Morse-Bott hypothesis and the reduction to the constant-α case constitute concrete, falsifiable predictions that advance the spectral theory of singular metrics.

minor comments (3)
  1. [§2] §2, definition of the truncated volume: the precise cutoff function or the exact meaning of dist(·,M) > λ^{-1/2} should be stated with an explicit reference to the geodesic distance induced by g.
  2. [Introduction] The statement of the Morse-Bott condition on the maximum set of α (used for the constants and log corrections) appears only in the abstract and introduction; a self-contained definition in the main text would improve readability.
  3. [§1] Notation: the symbol vol_g is used both for the full volume and the truncated volume; a distinct notation for the truncated quantity would avoid ambiguity in the statements of Theorems 1.1 and 1.2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance as the first Weyl law with spatially varying boundary degeneracy, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage. We remain available to address any additional queries or minor suggestions the referee may wish to provide.

Circularity Check

0 steps flagged

No circularity: asymptotics derived from metric form and volume integrability

full rationale

The derivation separates regimes at the explicit integrability threshold α_c = 2/(n+1) for the volume element u^{-α(n+1)/2} du. When α_max > α_c the local contribution near the maximum set dominates by direct comparison of the divergent integral; when α_max ≤ α_c the truncated-volume cutoff at distance λ^{-1/2} yields the leading term by the same volume computation. The Morse-Bott hypothesis is used only after the leading-term statement, solely to extract constants and log corrections. No fitted parameters are renamed as predictions, no self-citation chain is invoked to justify the regime split or the collar form, and the result is not equivalent to its inputs by definition. The paper is therefore self-contained against the stated geometric assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard assumptions of Riemannian geometry and elliptic operator theory; the abstract introduces no new free parameters, invented entities, or ad-hoc axioms beyond the stated metric form and manifold hypotheses.

axioms (2)
  • domain assumption X is a compact smooth manifold of dimension n+1 with boundary M
    Opening sentence of the abstract.
  • domain assumption Metric g = u^{-α} ¯g near M with α ∈ C^1(M; [0,2)) and ¯g C^1 Riemannian up to M
    Core modeling assumption that defines the singular metric and enables the Friedrichs Laplacian.

pith-pipeline@v0.9.0 · 5857 in / 1508 out tokens · 65131 ms · 2026-05-25T07:04:09.484654+00:00 · methodology

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Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Bott, Nondegenerate critical manifolds , Ann

    R. Bott, Nondegenerate critical manifolds , Ann. Math. 60 (1954), 248–261

    work page 1954

  2. [2]

    Cairns, On the triangulation of regular loci , Ann

    S. Cairns, On the triangulation of regular loci , Ann. Math. 35 (1934), 579–587

    work page 1934

  3. [3]

    Colin de Verdi` ere, C

    Y. Colin de Verdi` ere, C. Dietze, M. V. de Hoop, E. Tr´ elat, Weyl formulae for some singular metrics with application to acoustic modes in gas giants , arXiv:2406.19734 (2024), to appear in Ann. Inst. H. Poincar´ e

    work page arXiv 2024

  4. [4]

    Chitour, D

    Y. Chitour, D. Prandi, L. Rizzi, Weyl’s law for singular Riemannian manifolds , J. Math. Pures Appl. 181 (2024), 113–151

    work page 2024

  5. [5]

    Dietze, Spectral estimates for singular systems , PhD thesis, https://edoc.ub.uni- muenchen.de/35192/ (2025)

    C. Dietze, Spectral estimates for singular systems , PhD thesis, https://edoc.ub.uni- muenchen.de/35192/ (2025)

    work page 2025

  6. [6]

    Edelsbrunner, D

    H. Edelsbrunner, D. R. Grayson, Edgewise subdivision of a simplex , Discrete & Computational Geometry 24 (2000), no. 4, 707–719

    work page 2000

  7. [7]

    Freudenthal, Simplizialzerlegungen von beschr¨ ankter Flachheit, Ann

    H. Freudenthal, Simplizialzerlegungen von beschr¨ ankter Flachheit, Ann. Math. 43 (1942), no. 2, 580–582

    work page 1942

  8. [8]

    Menikoff, J

    A. Menikoff, J. Sj¨ ostrand, On the eigenvalues of a class of hypoelliptic operators , Math. Ann. 235 (1978), 55–85

    work page 1978

  9. [9]

    Munkres, Elementary Differential Topology, MIT lectures (1961)

    J. Munkres, Elementary Differential Topology, MIT lectures (1961)

    work page 1961

  10. [10]

    M. Reed, B. Simon. Methods of Modern Mathematical Physics IV: Analysis of Oper ators, Academic Press, 1978

    work page 1978

  11. [11]

    H. C. Whitehead, On C1-Complexes , Ann. Math. 41 (1940), no. 4, 809–824

    work page 1940

  12. [12]

    Wong, Asymptotic Approximations of Integrals , SIAM, 2001

    R. Wong, Asymptotic Approximations of Integrals , SIAM, 2001. 17

    work page 2001