Constrained willmore surfaces

Acknowledgements. My deepest gratitude goes to Fran, Prof. Francis Burstall, my PhD supervisor, for his constant support and attention, for sharing with me some of his outstanding knowledge of mathematics and promising ideas, and for doing so with the enthusiasm, the articulation and the warmth that are characteristic of him. Fran is a great mathematician and an exceptional human being and it was a privilege and a pleasure to have worked with him for all these years.

Special thanks are due to Prof. John C. Wood and Dr. Udo Hertrich-Jeromin, my PhD examiners, for their thorough reading of my thesis and constructive comments to it and for the friendliness.

A special mention to all those whose path has crossed mine at some point of this long journey, who have greatly enriched this experience to me. And to the squirrels, ducks, spiders, the occasional mouse, et al., in Hyde Park, London, for the touch of innocence to my days.

A very special mention to my parents, for always being there, together and well, and to Tiago, my baby nephew, who, blissfully unaware of it, has changed the lives of so many people for so much better.

My PhD studies were financially supported by Fundação para a Ciência e a Tecnologia, Portugal, with the scholarship with reference SFRH/BD/6356/2001.

Preface. This work is based on the PhD Thesis with the same title submitted to the University of Bath, Department of Mathematical Sciences, on September 15, 2008 and defended viva voce on the (rather snowy) 2 nd of February 2009. The contents of the Thesis available from the University of Bath has been preserved, with the exception of Sections 8.2.3 and 8.2.4, where the original study of constrained Willmore spectral deformation and Bäcklund transformation of CMC surfaces has been extended to a general multiplier. This version has benefited from the rephrasing or reformulation of parts of some sections.