The Morse index, topology and geometry of branched constant mean curvature surfaces.

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This project lies within the mathematical sciences section of EPSRC's remit. It involves a study of constant mean curvature (CMC) surfaces inside Riemannian three-manifolds. Whilst the subject is purely mathematical, CMC surfaces appear extensively in nature and the physical sciences, for example by describing the shape of soap films/bubbles, event horizons in general relativity and Fermi surfaces from condensed matter physics. Specifically the project will investigate the validity of a Morse index bound, from above, in terms of the genus and area of (branched) constant mean curvature (CMC) surfaces inside Riemannian three-manifolds. It will also provide bubble-compactness results for branched CMC hypersurfaces under suitable variational bounds. These results will provide insights in to the relationship between the Morse index, geometry and topology of CMC surfaces - a topic of significant interest due to the recent advances in existence theory of such objects. In particular this project will help to determine what types of CMC/minimal surfaces are admitted in a given space, and to classify those that do.