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Determination of period matrix of double of surface with boundary via its DN map

Dmitrii Korikov

TL;DR

The paper tackles the problem of recovering the conformal class of a surface with boundary from its Dirichlet-to-Neumann map by passing to the Schottky double and extracting the $b$-period matrix $\mathbb{B}$, which uniquely encodes the double’s conformal structure via the Torelli theorem. It presents a concrete four-step algorithm that starts from boundary data of harmonic normal fields, enforces integer-period conditions for Abelian differentials, constructs a canonical dual basis in the normal space, and computes the period matrices of the base surface and its double; the entire process hinges on the Hilbert transform, Abelian differentials, and canonical homology bases. The authors further develop implementation details to handle noisy DN data, proving that the resulting $b$-period matrix $\mathbb{B}'$ satisfies $\|\mathbb{B}'-\mathbb{B}\|=O(\varepsilon)$ under a small noise bound $\varepsilon$, thereby establishing stability of the reconstruction. Overall, the work reduces the infinite-dimensional EIT problem to finite-dimensional coordinates in the Siegel space via $\mathbb{B}$, enabling robust conformal-invariant characterization and potential numerical visualization of the unknown surface from boundary measurements.

Abstract

As is well-known, a conformal class of a surface $M$ with boundary $Γ$ is determined by its DN map $Λ$. In the paper, the algorithm for determination of the $b$-period matrix $\mathbb{B}$ of the (Schottky) double of surface with boundary via $Λ$ is presented. Due to the Torelli theorem, $\mathbb{B}$ contains all information on the conformal class of $M$ except the proper way of attaching $Γ$ to it.

Determination of period matrix of double of surface with boundary via its DN map

TL;DR

The paper tackles the problem of recovering the conformal class of a surface with boundary from its Dirichlet-to-Neumann map by passing to the Schottky double and extracting the -period matrix , which uniquely encodes the double’s conformal structure via the Torelli theorem. It presents a concrete four-step algorithm that starts from boundary data of harmonic normal fields, enforces integer-period conditions for Abelian differentials, constructs a canonical dual basis in the normal space, and computes the period matrices of the base surface and its double; the entire process hinges on the Hilbert transform, Abelian differentials, and canonical homology bases. The authors further develop implementation details to handle noisy DN data, proving that the resulting -period matrix satisfies under a small noise bound , thereby establishing stability of the reconstruction. Overall, the work reduces the infinite-dimensional EIT problem to finite-dimensional coordinates in the Siegel space via , enabling robust conformal-invariant characterization and potential numerical visualization of the unknown surface from boundary measurements.

Abstract

As is well-known, a conformal class of a surface with boundary is determined by its DN map . In the paper, the algorithm for determination of the -period matrix of the (Schottky) double of surface with boundary via is presented. Due to the Torelli theorem, contains all information on the conformal class of except the proper way of attaching to it.
Paper Structure (23 sections, 9 theorems, 89 equations)

This paper contains 23 sections, 9 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. EIT problem.
  3. Main result.
  4. Comments.
  5. Preliminaries
  6. Complex structure.
  7. Harmonic fields.
  8. Hilbert transform.
  9. Double cover.
  10. Abelian differentials.
  11. Homology groups.
  12. Period matrices.
  13. Procedure for determination of period matrix of double cover of $M$ from its DN map
  14. Step 1. Determination of boundary data of harmonic normal vectors on $M$.
  15. Step 2. Determination of boundary data of harmonic normal vectors with integer periods on $M$.
  16. ...and 8 more sections

Key Result

Proposition 1

Let $\Lambda$ be a fixed DN map of some surface $(M,g)$ of genus $\mathfrak{g}$ with (known) boundary $\Gamma$. Then there are sufficiently small numbers $\varepsilon_0=\varepsilon_0(\Lambda)>0$ and $c_0=c_0(\Lambda)>0$ such that the implementation of the algorithm Steps 1-4 to any approximation $\L provides the matrix $\mathbb{B}'$ obeying where $\mathbb{B}$ is some $b$-period matrix of the doub

Theorems & Definitions (15)

  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • ...and 5 more