Scattering Boundary Rigidity in the Presence of a Magnetic Field

TL;DR

This paper extends boundary rigidity results to regions with magnetic fields, showing simplicity can be deduced from boundary data, and applies this to a geometric rigidity problem involving constant curvature surfaces.

Contribution

It demonstrates that simplicity can be inferred from boundary and scattering data, even when only one region is initially known to be simple, and applies this to a geometric rigidity problem.

Findings

Simplicity can be read from boundary metric and scattering data.

Extended boundary rigidity results to regions with magnetic fields.

Proved that constant curvature surfaces cannot be locally deformed while preserving certain geodesic properties.

Abstract

It has been shown in \cite{DPSU} that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in \cite{DPSU} to regions with the same scattering data, where only one is known apriori to be simple. We will then use this results to resolve a local version of a question by Robert Bryant. That is, we show that a surface of constant curvature can not be modified in a small region while keeping all the curves of some fixed constant geodesic curvatures closed.