Scattering boundary rigidity in the presence of a magnetic field

It has been shown in [10] 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 results in [10] to regions with the same scattering data, where only one is known a priori 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 cannot be modified in a small region while keeping all the curves of some fixed constant geodesic curvatures closed.

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Published 17 August 2012