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arxiv: 2602.13236 · v2 · submitted 2026-01-29 · 🧮 math.CV · math-ph· math.MP

On non-homeomorphic surfaces with close DN maps

D.V. Korikov This is my paper

Pith reviewed 2026-05-16 10:03 UTC · model grok-4.3

classification 🧮 math.CV math-phmath.MP
keywords Dirichlet-to-Neumann mapSchottky doublesystolegenussurface with boundaryhyperbolic metricoperator normRiemannian geometry
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The pith

If Dirichlet-to-Neumann maps of surfaces with the same boundary are close in operator norm, the systole of the Schottky double of the higher-genus surface must approach zero.

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

The paper proves that for a genus-m surface M with boundary Gamma whose DN map is close to the DN map of a lower-genus surface M* with the same boundary, the length of the shortest closed geodesic on the hyperbolic Schottky double 2M must go to zero. This implication holds regardless of the specific metrics chosen on the surfaces. A reader cares because the boundary operator then forces metric degeneration precisely when the topology differs, linking local boundary data to global interior geometry through the doubling construction.

Core claim

We prove that Sys(2M) is small if Lambda is close, in the operator norm, to the DN map Lambda* of some surface (M*,g*) of lower genus m*<m with the same boundary Gamma: ||Lambda - Lambda*||_{B(H^{1/2}(Gamma);H^{-1/2}(Gamma))} -> 0 implies Sys(2M) -> 0, where 2M is the Schottky double of (M,g) equipped with its hyperbolic metric.

What carries the argument

The Schottky double 2M of the bordered surface M, turned into a closed surface by gluing a copy across Gamma, whose shortest closed geodesic length Sys(2M) is measured in the hyperbolic metric on 2M; closeness of the DN maps on Gamma controls this length.

If this is right

  • DN maps of surfaces with different genera cannot remain at positive operator-norm distance unless the systole on the double stays bounded away from zero.
  • The boundary data encoded by the DN map detects a topological difference only when accompanied by metric collapse on the doubled surface.
  • The implication is quantitative in the operator norm between the trace spaces H^{1/2}(Gamma) and H^{-1/2}(Gamma).

Where Pith is reading between the lines

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

  • The result suggests that any inverse problem recovering genus from DN data must include a lower bound on systole to avoid degenerate cases.
  • Numerical checks could fix a positive lower bound on Sys(2M) and verify whether DN-map distances between different genera are then bounded away from zero.
  • The doubling technique may apply to other boundary operators whose norms control geodesic lengths on the closed double.

Load-bearing premise

The surfaces admit Riemannian metrics making the DN maps well-defined and allowing the systole to be measured on the hyperbolic structure of the Schottky double.

What would settle it

Construct two surfaces of different genera sharing the same boundary Gamma such that their DN maps satisfy ||Lambda - Lambda*|| < epsilon for arbitrarily small epsilon while Sys(2M) remains bounded below by a positive constant.

read the original abstract

Let $(M,g)$ be a genus $m$ surface with boundary $\Gamma$ and DN map $\Lambda$. Introduce the Schottky double $2M$ of $(M,g)$ and denote by $Sys(2M)$ the length of the shortest closed geodesics in the hyperbolic metrics on $2M$. We prove that $Sys(2M)$ is small if $\Lambda$ is close, in the operator norm, to the DN map $\Lambda_*$ of some surface $(M_*,g_*)$ of lower genus $m_*<m$ with the same boundary $\Gamma$: $$\|\Lambda-\Lambda_*\|_{B(H^{1/2}(\Gamma);H^{-1/2}(\Gamma))}\to 0\,\Longrightarrow \ Sys(2M)\to 0.$$

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 / 2 minor

Summary. The manuscript proves that if the Dirichlet-to-Neumann map Λ of a genus-m surface (M,g) with boundary Γ is close in the operator norm on B(H^{1/2}(Γ);H^{-1/2}(Γ)) to the DN map Λ* of a lower-genus surface (M*,g*) with the same boundary, then the systole Sys(2M) of the hyperbolic metric on the Schottky double 2M tends to zero.

Significance. If the central implication holds, the result links analytic closeness of boundary operators to geometric degeneration in the moduli space of the doubled surface. It strengthens rigidity statements for the DN map in two-dimensional inverse problems by showing that non-homeomorphic surfaces cannot have arbitrarily close DN maps without the conformal class degenerating. The approach via Schottky doubles and hyperbolic systoles is a natural and standard tool in the field.

minor comments (2)
  1. The definition of the operator norm and the precise Sobolev spaces H^{1/2}(Γ) and H^{-1/2}(Γ) should be recalled explicitly in the introduction or §2 for readers outside the immediate subfield.
  2. A short remark on the dependence of the DN map solely on the conformal class (rather than the specific metric) would clarify the setup before the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and the recommendation for minor revision. We address the referee summary below.

read point-by-point responses

  1. Referee: The manuscript proves that if the Dirichlet-to-Neumann map Λ of a genus-m surface (M,g) with boundary Γ is close in the operator norm on B(H^{1/2}(Γ);H^{-1/2}(Γ)) to the DN map Λ* of a lower-genus surface (M*,g*) with the same boundary, then the systole Sys(2M) of the hyperbolic metric on the Schottky double 2M tends to zero.

    Authors: This is an accurate summary of our central result. We appreciate the referee's recognition that the implication connects analytic closeness of boundary operators to geometric degeneration in the moduli space via Schottky doubles and hyperbolic systoles. No changes are required to this description. revision: no

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes an implication between operator-norm closeness of DN maps for surfaces of different genera and vanishing systole on the Schottky double. This is presented as a theorem derived from standard properties of the Dirichlet-to-Neumann operator, conformal classes, and hyperbolic metrics on doubles. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the stated result or abstract. The central claim retains independent content grounded in geometric analysis without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard constructions (Schottky double, DN map for the Laplace-Beltrami operator) whose details are not supplied; no free parameters or new entities are introduced in the statement.

axioms (2)
  • domain assumption The surfaces carry Riemannian metrics such that the DN map is the boundary operator for the Laplace-Beltrami equation.
    Implicit in the definition of Λ and Λ∗ and in the measurement of Sys(2M) on hyperbolic doubles.
  • standard math The Schottky double 2M admits a hyperbolic metric in which closed geodesics are well-defined.
    Required for the definition of Sys(2M).

pith-pipeline@v0.9.0 · 5428 in / 1316 out tokens · 25857 ms · 2026-05-16T10:03:50.634312+00:00 · methodology

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Reference graph

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14 extracted references · 14 canonical work pages

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