# Constant mean curvature surfaces of revolution versus Willmore surfaces of revolution: A comparative study with physical applications

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This thesis studies some special types of surfaces of revolution and their real world applications. The main two cases hereby considered are the constant mean curvature surfaces of revolution (also called Delaunay surfaces) and Willmore's surfaces of revolution, respectively. We first present some geometric results on Delaunay CMC surfaces which correspond to certain classes of ordinary differential equations. We present the original construction of Delaunay surfaces, based on roulettes of conics, after which we characterize these geometric objects as solutions to specific ODEs. We present a few physical models of Delaunay surfaces arising as liquid bridges between two vertical walls - which are proved to be unduloidal surfaces, by using Calculus of Variations. We numerically computed the profile curves of these surfaces and provided some numerical models for them. By contrast, we studied Willmore surfaces as minimizers of the Willmore energy (or bending energy). In particular, we have studied some Willmore surfaces of revolution which come in as solutions to BVP problems consisting of the Willmore equation, together with some special Dirichlet type boundary conditions. With help from Dr. Eugenio Aulisa and Mr. Bhagya Athukorallage, I have provided some numerical computations for the profiles of these surfaces, using COMSOL Multiphysics. Willmore surfaces of revolution have lots of application in the real world, such as elastic biological membranes. At the microbiological level, a model of such Willmore surfaces of revolution is provided by the beta barrels arising from secondary structures in proteins (beta sheets configured as a rotationally symmetric model).