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

TL;DR

This paper presents an algorithm to determine the period matrix of the double of a surface with boundary using its Dirichlet-to-Neumann map, providing a method to recover conformal class information.

Contribution

It introduces a novel algorithm linking the DN map to the period matrix of the surface's double, advancing inverse boundary value problem techniques.

Findings

Algorithm successfully computes the period matrix from DN map.

Period matrix encodes all conformal class information of the surface.

Method leverages Torelli theorem for conformal classification.

Abstract

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