The low-lying spectrum of rank-varying sub-Riemannian manifolds
This thesis studies the relationship between the geometry of a space and the frequencies at which physical systems can vibrate. In particular, it focuses on the lowest frequencies associated with spaces whose geometry is more general than the familiar geometry of curved surfaces and higher-dimensional spaces. These spaces, known as sub-Riemannian manifolds, allow certain directions of movement to be restricted. This makes them useful for describing a wide range of mathematical and physical phenomena, but also means that many classical mathematical techniques no longer apply directly.
The thesis develops new methods for studying the lowest frequencies of such spaces and explores three main directions. First, it extends a classical result that relates the shape of a space to its lowest non-zero frequency. This provides new bounds in settings where the geometry can change from one point to another.
Second, these results are applied to the mathematical description of magnetic fields. A surprising connection allows magnetic fields to be studied using the same geometric framework. The new methods therefore provide information about the lowest energy levels of systems influenced by magnetic fields.
Finally, the thesis investigates how geometry can be simplified by changing the way distances are measured, focusing on two-dimensional spaces with singularities. A classical theorem describing the possible shapes of ordinary two-dimensional spaces is extended to this more general setting.
Together, these results show how ideas from geometry can be developed beyond the classical setting and applied to problems arising in mathematical physics. The thesis also provides an introduction to the mathematical background needed to understand these connections, bringing together ideas from geometry, spectral theory, and the study of magnetic fields.