The shape of nonzero constant mean curvature surfaces embedded in Euclidean space.

This project consists of a thorough study of the geometry of nonzero constant mean curvature surfaces embedded in Euclidean space. One of the main goals of this project is obtaining some new and remarkable curvature estimates for simply-connected surfaces embedded in Euclidean space with nonzero constant mean curvature at points that are intrinsically sufficiently far away from the boundary. This can then be used to give a new characterisation of round spheres: round spheres are the only simply-connected surfaces embedded in Euclidean space with nonzero constant mean curvature. This characterisation is consonant to very classical and elegant results in the theory of constant mean curvature surfaces given by Hopf and Alexandrov, and more recent and groundbreaking results by Colding-Minicozzi and Meeks-Rosenberg. In addition, knowing the geometry of such surfaces in Euclidean space constitutes solid foundations from where to start an investigation of surfaces embedded in a 3-manifold with nonzero constant mean curvature. This is also part of the project.

It is hard to underestimate the impact of a thorough study of the geometry of constant mean curvature (CMC) surfaces. From a purely theoretical point of view the theory of CMC surfaces is classical and cuts across many branches of mathematics including the calculus of variations, partial differential equations, complex analysis and general relativity. In the past and in recent years knowledge of their geometry has been crucial in solving major mathematical conjectures in many other fields. From a more applied point of view the close connection between CMC surfaces and nature gives the subject a very concrete perspective and makes the theory very exciting. Soap films approximate with great accuracy minimal surfaces, CMC=0, while soap bubbles provide the analogous approximation for CMC surfaces. Thus questions about the possible shapes of CMC surfaces are also of great interest to engineers, physicists and material scientists. Thanks to the aforementioned diverse interactions enriching the theory of CMC surfaces, my project will be of great interest to mathematicians in many branches of mathematics other than geometric analysis and to a broad audience of scientists. Particularly in the U.K. there are many mathematicians who do research on geometric variational problems and who will benefit from these new results and tools. It is worth mentioning Haskins and Neves of Imperial College, Klingenberg of University of Durham, Micallef and Topping of University of Warwick, Wickramasekera of University of Cambridge et al.. The dissemination of the results in public lectures, seminars and conferences will be responsible for the main impact of the research. In order to spread the knowledge of CMC surfaces I have been and still am actively involved in the organisation of seminars related to this area of mathematics. During the year I spent as a Postdoctoral Fellow at Stanford University I was co-organiser of the geometry seminar. This was also the case while I was at University of Notre Dame as a Research Assistant Professor. Currently I am a Lecturer of Mathematics at King's College London and here I am once again organising the geometry seminar. In addition to being responsible for the geometry seminar at King's College London, my willingness to travel to meet my collaborators, to participate in conferences and workshops, to coordinate reading seminars and to mentor fellow students has also played and will play a significant role in making my research successful and more accessible to an audience of non-specialists. Besides publishing in leading mathematical journals in the past 3 years I have been invited more than 20 times to talk about my research at prestigious conferences some of them in Luminy, Banff and Oberwolfach and prestigious universities including University of Cambridge, Imperial College, M.I.T. and Princeton University. Finally, I was recently given the privilege to speak on behalf of the geometry group at the Postgraduate Open Day in Mathematics at King's College London. In front of a young audience of potential mathematicians I had the opportunity to show my enthusiasm for geometry and how it is an exciting field of research especially in recent years. Among the many mathematicians whom I would greatly benefit from discussing with, I intend to visit Bill Meeks of University of Massachusetts and Brian Smyth of University of Notre Dame; I am already involved in fruitful collaborations with both of them and I expect them to visit me several times during the next two years. I also plan to visit a few mathematicians with whom I have started collaborating such as Jacob Bernstein of Stanford University, Theodora Bourni of the Max Plank Institute and Christine Breiner of M.I.T.. Berstein, Bourni and Breiner have already visited me at King's College London in February and May 2010. Finally, it will be fruitful to visit my former Ph.D. Adviser, Bill Minicozzi of The Johns Hopkins University.