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On non-homeomorphic surfaces with close DN maps

D. V. Korikov

TL;DR

The paper establishes that closeness of Dirichlet-to-Neumann maps between a fixed boundary surface and a topologically perturbed surface forces the Schottky double of the perturbed surface to have arbitrarily short closed geodesics, i.e., $\mathscr{L}(X'_s)\to0$. It combines inverse boundary data with the geometry of Riemann surfaces by leveraging the Hilbert transform associated to the DN map, Mumford’s compactness, and a careful quasiconformal deformation analysis to pass to a limit where an antiholomorphic involution arises on the common double. A Mandelstam-diagram framework is used to control the geometry of the doubles under perturbations, while a defect-operator/dimension argument yields a contradiction unless the geodesic length tends to zero. The results highlight inherent instability of topology-encoded data under DN-map perturbations and connect boundary measurements to global geometric and topological features of the underlying surfaces.

Abstract

Let $(M',g')$ be a genus $m'$ surface with boundary $Γ$ and the DN map $Λ'$. Introduce the Schottky double $X'$ of $(M',g')$ and denote by $\mathscr{L}(X')$ the length of the shortest closed geodesics in the hyperbolic metrics on $X'$. We prove that $\mathscr{L}(X')$ is small if $Λ'$ is close {\rm(}in $B(H^1(Γ;\mathbb{R});L_2(Γ;\mathbb{R}))${\rm)} to the DN map $Λ$ of some surface $(M,g)$ of lower genus $m<m'$ with the same boundary $Γ$: $$Λ'\toΛ\,\Longrightarrow \ \mathscr{L}(X')\to 0.$$

On non-homeomorphic surfaces with close DN maps

TL;DR

The paper establishes that closeness of Dirichlet-to-Neumann maps between a fixed boundary surface and a topologically perturbed surface forces the Schottky double of the perturbed surface to have arbitrarily short closed geodesics, i.e., . It combines inverse boundary data with the geometry of Riemann surfaces by leveraging the Hilbert transform associated to the DN map, Mumford’s compactness, and a careful quasiconformal deformation analysis to pass to a limit where an antiholomorphic involution arises on the common double. A Mandelstam-diagram framework is used to control the geometry of the doubles under perturbations, while a defect-operator/dimension argument yields a contradiction unless the geodesic length tends to zero. The results highlight inherent instability of topology-encoded data under DN-map perturbations and connect boundary measurements to global geometric and topological features of the underlying surfaces.

Abstract

Let be a genus surface with boundary and the DN map . Introduce the Schottky double of and denote by the length of the shortest closed geodesics in the hyperbolic metrics on . We prove that is small if is close {\rm(}in {\rm)} to the DN map of some surface of lower genus with the same boundary :
Paper Structure (10 sections, 3 theorems, 85 equations)

This paper contains 10 sections, 3 theorems, 85 equations.

Table of Contents

  1. DN map and Hilbert transform.
  2. Instability of surface topology under small perturbations of its DN map.
  3. Main result.
  4. 0. Basic facts and notions.
  5. 1. Applying Mumford's compactness theorem.
  6. 2. Local replacement of quasiconformal mapping.
  7. 3. Introducing Mandelstam variables.
  8. 4. Proof of the existence of an anti-holomorphic involution on $X$.
  9. 4. Proof that $X$ is a double of a surface with boundary.
  10. Completing the proof of the convergence $\mathscr{L}(X'_s)\to 0$ as $s\to\infty$.

Key Result

Proposition 1

Let $(M,g)$ be a surface of genus $m$ with boundary $(\Gamma,dl)$ (diffeomorphic to a circle) and DN map $\Lambda$. Let $\{(M_s,g_s)\}_s$ be a sequence of surfaces of genera ${\rm gen}(M_s)=m'>m$ with the same boundary $(\Gamma,dl)$ and DN maps $\Lambda_s$. Denote by $X$ and $X_s$ the doubles of $(M

Theorems & Definitions (4)

  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3