REVIEW 3 major objections 5 minor 10 references
The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double
T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read For surfaces with boundary, the Dirichlet-to-Neumann determinant factorizes into a boundary derivative term and the squared product of the genus-many period numbers.
desk verdict New period formula for the DN determinant, but the KV determinant step is not justified for non-sectorial H; needs major revision. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the boundary Hilbert transform H = ∂$γ^{{-1}}$Λ, a zero-order pseudodifferential operator whose essential spectrum is the pair {−i,+i} and whose discrete spectrum encodes the complex structure of M. The proof uses the Kontsevich–Vishik determinant det_Q for sectorial pseudodifferential operators and the path-independence and multiplicativity formula (10) for the two one-parameter families A(t)=∂γ and B(t)=θ(t)H+(1−θ(t))H₀, where H₀ is the Hilbert transform of the disk. A second variational argument for J(t)=−B(t)² shows that det_Q(H)/det_Q(H₀) equals DET(H), the product over the discrete spectrum; regularizer independence and the equality det_Q(Λ)=det_ζ(Λ) then deliver the main formula. The periods enter through Lemma 1, which relates the eigenvalues ±iμ_k to integrals of harmonic forms over l and τ∘l.
What would settle it
Compute both sides of formula (6) for a genus-one surface such as a flat torus with a disk removed: the ζ-regularized determinant of Λ on one side and the period number μ₁ on the other; if (μ₁)² does not equal det_ζ(Λ)/|Γ|, the factorization fails. Alternatively, exhibit an interpolating operator B(t) in the family whose spectrum is not contained in a closed sector, which would invalidate the Kontsevich–Vishik step.
Extended reading notes
Core claim
The paper's central claim is the identity det_ζ(Λ) = det_ζ(∂γ) · DET(H) for a surface (M,g) of genus g with connected boundary Γ. Here Λ is the DN map, ∂γ is differentiation along Γ, and H = ∂$γ^{{-1}}$Λ is the Hilbert transform of M. H has essential spectrum {−i,+i} and 2g discrete eigenvalues ±iμ_k with 0<μ_k<1; the paper defines DET(H) as the product of these discrete eigenvalues only, formally treating the essential eigenvalues as contributing 1. The numbers μ_k are periods of Abelian differentials on the Schottky double 2M, as made precise in Lemma 1. Combining the main identity with the known genus-zero value det_ζ(∂γ)=|Γ| and with the Ruelle zeta formula of [2] yields ((2πs)^{-g}R(s))|_{s=0} = (1−g)(μ₁⋯μ_g)². Thus a spectral invariant that previously required hyperbolic length spectra is determined by the period matrix of the double.
Load-bearing premise
The proof assumes that the family of operators interpolating between the disk Hilbert transform and the surface's Hilbert transform meets the analytic requirements of the determinant formalism it uses, namely sectoriality and trace-class derivatives, and that the formal removal of the essential eigenvalues in DET(H) is compatible with that formalism; only invertibility along the path is actually checked.
Editorial extensions
If this is right
- The conformal invariant det_ζ(Λ)/|Γ| is computable as (μ₁⋯μ_g)² from period data on the double, bypassing the Ruelle and Selberg zeta functions.
- Formula (7) gives a new relation between the length spectrum of the uniformized surface-with-boundary, through the Ruelle zeta value at zero, and the period matrix of the double.
- For genus g=1, the invariant reduces to μ₁², so the determinant is controlled by a single period ratio.
- The determinant of H is defined through its discrete spectrum only, making DET(H₀)=1 for the disk, consistent with det_ζ(Λ₀)=|Γ|.
- The equality det_Q(Λ)=det_ζ(Λ) shows that the Kontsevich–Vishik determinant with regularizer Λ reproduces the zeta-regularized DN determinant.
Reading between the lines
- If formula (6) holds, the same period product should control the quotient of zeta determinants under arbitrary conformal rescalings; a numerical test on a once-punctured torus could compare both sides using standard boundary integral solvers.
- The proof's restriction to the discrete spectrum suggests that any alternative regularized determinant for H must cancel the essential eigenvalues in exactly the same formal way, a consistency condition not addressed in the paper.
- The relation may extend to surfaces with several boundary components, where the double has a larger period lattice, but the paper does not treat that case.
- Because the μ_k depend only on the complex structure, formula (7) predicts the Ruelle zeta value at zero is a conformal invariant of the original surface, consistent with the known invariance of det_ζ(Λ)/|Γ|.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims a formula relating the zeta-regularized determinant of the Dirichlet-to-Neumann map of a surface with boundary to periods of holomorphic differentials on its double. The main identity is Eq. (6), detζ(Λ) = detζ(∂γ) · DET(H), where DET(H) is a finite product of the 2g discrete eigenvalues ±iμk of the Hilbert transform H of the surface. The authors derive Eq. (6) using Kontsevich-Vishik determinants of pseudodifferential operators and a variational formula for their logarithms. As a corollary, they obtain Eq. (7), relating the Ruelle zeta function of the uniformized surface at s=0 to (μ1⋯μg)^2. An appendix gives a proof of the period interpretation of the eigenvalues and an explicit equation (18) involving the period matrix of the double.
Significance. If the main identity is made rigorous, the paper gives an attractive and conceptually new expression for the conformal invariant detζ(Λ)/|Γ| purely in terms of periods of holomorphic differentials on the double. The corollary (7) is a concrete, falsifiable bridge between length-spectrum data and period data. The proof structure is elegant, and the appendix provides a self-contained derivation of the period lemma, including a nontrivial period-matrix condition. The manuscript has no free parameters. The main weakness is that the analytic foundations of the Kontsevich-Vishik determinant for the operators H and B(t) are not established; the central claim is therefore conditional on an unproved regularization hypothesis.
major comments (3)
- [Proof of formula (6), items i)-ii), Eqs. (10)-(11)] The multiplicativity formula (10) is applied to the families B(t)=θ(t)H+(1-θ(t))H0 and A(t)B(t)=θ(t)Λ+(1-θ(t))Λ0 without verifying the defining hypotheses of the Kontsevich-Vishik determinant. The paper checks only invertibility and trace-class properties of the derivatives; it does not prove that each B(t) is sectorial, that Q^{-s}log B(t) is trace class for Re s large, or that the relevant regularized traces admit analytic continuation to s=0. This is not a cosmetic gap: B(t) inherits the essential spectrum {−i,+i} from H0, and the introduction explicitly states that no zeta-regularized or Fredholm determinant of H can be defined. Since Eq. (11) is the step that produces a relation involving det_Q(H), the derivation of (6) is unsupported unless a separate regularization theorem for these non-sectorial operators is supplied.
- [Proof of formula (6), item iii), Eq. (12)] The identification det_Q(H)/det_Q(H0)=DET(H) rests on the ad hoc convention DET(H0)=1 and on the formal statement that the essential eigenvalues ±i contribute a unit determinant. This cancellation is not derived from the Kontsevich-Vishik determinant det_Q(H0); it is put in by hand in the definition (5). As written, Eq. (12) equates a finite product over discrete eigenvalues with a ratio of KV determinants whose existence and essential-spectrum contributions are not established. A rigorous treatment must either construct the KV determinants of H and H0 and compute their essential-spectrum part, or replace this step with a different regularization argument.
- [Proof of formula (6), item i), Eqs. (8)-(9)] The variational formula (9) requires that the boundary terms in the integration by parts vanish and that ∂_t can be interchanged with the analytic continuation and the Hadamard finite part. The text justifies this only by the conditional phrase 'if the right-hand is well-defined for ℜs>−ε'. For B(t), the right-hand side involves (λI-B(t))^{-1} in the presence of essential spectrum on the imaginary axis, and the required resolvent bounds are not proved. This is an independent aspect of the same analytic gap and affects the derivation of Eq. (12) even if sectoriality of B(t) is granted.
minor comments (5)
- [Abstract and Introduction] There are several typos: 'Rouelle' should be 'Ruelle', 'confromal' should be 'conformal', and 'pseudodifferenial' should be 'pseudodifferential'.
- [Eq. (5)] The notation DET(H) is not a determinant in the standard operator sense; please state explicitly that it is a formal product over the discrete spectrum and clarify that DET(H0)=1 is a convention, not a derived value.
- [Proof of formula (6), item iii)] In the sentence containing 'detQ(I)=0', the intended statement is that log detQ(I)=0, equivalently detQ(I)=1; the current wording is misleading.
- [Appendix, proof of Lemma 1] The appendix begins with 'Suppose that Hf=-λf', so the symbol λ is the negative of the eigenvalue of H; this sign convention makes the comparison with Lemma 1 harder to follow and should be stated explicitly.
- [Introduction, Eq. (2)] The text says the product in the Ruelle zeta function is over primitive closed geodesics in (2M,h∞), but the surrounding sentence mentions both (M\Γ,h∞) and the double; please make clear which hyperbolic surface carries the geodesics used in Eq. (2).
Circularity Check
No significant circularity: Eq. (6) is proved from the Hilbert-transform decomposition and KV determinants rather than assumed, and Lemma 1 is reproved in the appendix; the noted sectoriality gaps are correctness risks, not circular reductions.
full rationale
The derivation chain does not assume its target. Eq. (6) is reached through the multiplicativity formula (10) applied to A(t)=∂γ and B(t)=H0+θ(t)H̃, followed by the computation detQ(H)/detQ(H0)=DET(H) in part iii). That computation uses only the smoothing nature of K=I+H^2=(H−H0)(H+H0) and the finite discrete spectrum of H; it does not invoke Eq. (6) or Eq. (7) as an input. Lemma 1 is attributed to [10] but is reproved in the appendix from the eigenvalue equation, and Eq. (18) is derived there, so the period interpretation of the μk is not assumed by definition. Corollary (7) is obtained by inserting the proved Eq. (6) into the external Guillarmou–Guillopé formula (2), not the reverse. The main caveat is analytic rather than circular: near Eq. (5) the paper states 'Due to the presence of the essential spectrum {−i,+i}, neither zeta-regularized nor Fredholm determinant of H can be defined,' yet formulas (11)–(12) use detQ(H). The required sectoriality and analytic-continuation hypotheses for the family B(t)=H0+θH̃ are checked only for invertibility, not for sectoriality; if those hypotheses fail, the identification in (12) would be an ad hoc definition rather than a Kontsevich–Vishik computation. This is an omitted verification that could threaten correctness, but it is not a reduction of the conclusion to the paper's inputs: detζ(Λ), detζ(∂γ), and DET(H) are independently defined quantities, and the paper does not fit one from the other. The self-citations [4] and [10] supply background spectral facts; [10] is reproven in the appendix, and neither of them assumes the target formula. Hence the circularity burden is low; score 2 reflects only the presence of non-load-bearing self-citation, not circular reasoning.
Assumptions & free parameters
assumptions (4)
- standard math Kontsevich-Vishik determinant and the multiplicativity formula (10) for sectorial pseudodifferential operators.
- domain assumption Spectral structure of the Hilbert transform H: essential spectrum {−i,+i} with eigenspaces of boundary traces of holomorphic and anti-holomorphic differentials, and discrete spectrum ±i μ_k with μ_k∈(0,1).
- domain assumption Guillarmou-Guillope formula (2) expressing the conformal determinant invariant via the Ruelle zeta function.
- standard math Tataru-Holmgren-Kovalevskaya uniqueness for elliptic Cauchy problems.
invented entities (1)
-
DET(H), the determinant of H defined by restricting to its discrete spectrum
Cite this review
Pith. "Pith review of The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double." pith.science (2026). https://pith.science/paper/V5D4REXS
@misc{pith2026260809737,
author = {Pith},
title = {Pith review of: The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double},
year = {2026},
howpublished = {\url{https://pith.science/paper/V5D4REXS}},
note = {Machine review of arXiv:2608.09737}
}
abstract
Let $(M,g)$ be a smooth orientable $2d$ Riemannian manifold of genus $\mathfrak{g}$ with Riemannian metric $g$ and connected boundary $\Gamma$. Let $\Lambda$ be the Dirichlet-to-Neumann map on $\Gamma$ and let ${\rm det}_\zeta(\Lambda)$ be its (modified, i. e. with zero mode excluded) $\zeta$-regularized determinant. It is well-known that the quantity ${\rm det}_\zeta(\Lambda)/|\Gamma|$ (where $|\Gamma|$ is the length of $\Gamma$) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ Guillarmou and Guillop\'e \cite{Guillarmou} found two explicit expressions for this invariant through the Rouelle and (respectively) the Selberg zeta-functions of the two surfaces of negative constant curvature from the conformal class of $(M,g)$: one is of infinite volume and complete whereas another has geodesic boundary. We present an elementary counterpart of the formulae of Guillarmou and Guillop\'e using the periods of holomorphic differentials on the double $2M$ of $M$ only. Our approach is based on the properties of the Hilbert transform of $M$ \cite{B,HilbKor} and the Kontsevich-Vishik-Friedlander-Guillemin regularization of the determinants of pseudodifferenial operators \cite{KV,Ww,F}. In particular, a connection between the length spectra of (uniformized) $M$, $2M$ and the periods of holomorphic differentials on $2M$ is established.
Reference graph
Works this paper leans on
- [1]
-
[2]
Colin Guillarmou, Laurent Guillop´ eThe determinant of the Dirichlet-to-Neumann map for surfaces with boundary.(2007),https://arxiv.org/abs/math/0701727
arXiv 2007
-
[3]
BelishevThe Calderon problem for two-dimensional manifolds by the BC-method
M.I. BelishevThe Calderon problem for two-dimensional manifolds by the BC-method. SIAM Journal of Mathematical Analysis 35(1), 172–182 (2003).https://doi.org/10.1137/ S0036141002413919
work page 2003
-
[4]
Dmitrii KorikovDetermination of period matrix of double of surface with boundary via its DN map.Canadian Journal of Mathematics (2025),https://doi.org/10.4153/ s0008414x25000264
work page 2025
-
[5]
Determinants of elliptic pseudo-differential operators
Maxim Kontsevich, Simeon VishikDeterminants of elliptic pseudo-differential operators. (1994),https://arxiv.org/pdf/hep-th/9404046
work page Pith review arXiv 1994
-
[6]
Richard A. WentworthGluing formulas for determinants of Dolbeault laplacians on Riemann surfaces.(2012),https://arxiv.org/abs/1008.2914 8 DMITRII KORIKOV AND ALEXEY KOKOTOV
arXiv 2012
-
[7]
Leonid FriedlanderOn Multiplicative Properties of Determinants.(2018),https://doi.org/ 10.48550/arXiv.1801.10606
work page Pith review arXiv doi:10.48550/arxiv.1801.10606 2018
-
[8]
G.GrubbFunctional calculus of pseudo-differential boundary value problems.Birkhaeuser, Boston, 1986
work page 1986
Show all 10 references
-
[9]
J.M. Lee, G. UhlmannDetermining anisotropic real-analytic conductivities by boundary mea- surements.Comm. Pure Appl. Math., 42, 1097–1112 (1989)
1989
-
[10]
Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, 27 Fontanka, St
Dmitrii KorikovOn non-homeomorphic surfaces with close DN maps.(2026),https:// arxiv.org/abs/2602.13236 St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, 27 Fontanka, St. Petersburg, Russia,https://orcid.org/0000-0002-3212-5874 Email a...
2026 arXiv
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.