Isothermic surfaces and their generalisations

Abstract

This thesis concerns the theory of isothermic submanifolds in symmetric R-spaces. We generalise several results for isothermic surfaces in $ S^3 $ to this more general class of submanifolds.

We develop a special class of submanifold of symmetric R-space, the cyclide, and characterise and study envelopes of congruences of cyclides. In particular, we show that Darboux pairs of isothermic submanifolds are common envelopes of a single congruence of cyclides.

We also describe a generalisation of fanning curves and show that they are natural isothermic submanifolds and this class is preserved under the transformations of isothermic submanifolds. We use this to define a semi-discretisation of isothermic submanifolds as iterated Darboux transforms of fanning curves and demonstrate how the transformation theory extends to these semi-discrete submanifolds.

Lastly we lay the groundwork for a theory of polynomial conserved quantities for isothermic submanifolds and explore this theory with the example of the self-dual Grassmannian.

Date of Award	21 Jul 2021
Original language	English
Awarding Institution	University of Bath
Supervisor	Fran Burstall (Supervisor) & Merrilee Hurn (Supervisor)