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

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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.