A Calder\'{o}n Problem for Beltrami Fields

TL;DR

This paper investigates how a boundary map for Beltrami fields on a 3D manifold encodes geometric information, enabling the reconstruction of the manifold from boundary data despite the absence of Green's functions.

Contribution

It introduces a boundary map for Beltrami fields, showing it determines the metric's Taylor series and allows manifold reconstruction, extending Calderón-type results to fluid mechanics.

Findings

The boundary map is a pseudodifferential operator of order zero.

The total symbol of this operator determines the boundary metric's Taylor series.

A real-analytic simply connected 3-manifold can be reconstructed from the boundary map.

Abstract

On a $3$-dimensional Riemannian manifold with boundary, we define an analogue of the Dirichlet-to-Neumann map for Beltrami fields, which are the eigenvectors of the curl operator and play a major role in fluid mechanics. This map sends the normal component of a Beltrami field to its tangential component on the boundary. In this paper we establish two results showing how this normal-to-tangential map encodes geometric information on the underlying manifold. First, we show that the normal-to-tangential map is a pseudodifferential operator of order zero on the boundary whose total symbol determines the Taylor series of the metric at the boundary. Second, we go on to show that a real-analytic simply connected $3$-manifold can be reconstructed from its normal-to-tangential map. Interestingly, since Green's functions do not exist for the Beltrami field equation, a key idea of the proof is to find an appropriate substitute, which turn out to have a natural physical interpretation as the magnetic fields generated by small current loops.