Constant mean curvature surfaces and Heun's differential equations

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Mota, Eduardo
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University College Cork
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This thesis is concerned with the problem of constructing surfaces of constant mean curvature with irregular ends by using the class of Heun’s Differential Equations. More specifically, we are interested in obtaining immersion of punctured Riemann spheres into three dimensional Euclidean space with constant mean curvature. These immersions can be described by a Weierstrass representation in terms of holomorphic loop Lie algebra valued 1-forms. We describe how to encode each of the differential equations in Heun’s family in the Weierstrass representation. Next, we investigate monodromy problems for each of the cases in order to ensure periodicity of all the resulting immersions. This allows us to find four families of surfaces with constant mean curvature and irregular ends. These families can be described as trinoids, cylinders, perturbed Delaunay surfaces and planes. Finally, we study some symmetry properties of these groups of surfaces.
Differential geometry , Theory of surfaces , Mean curvature , Ordinary differential equations , Singularities
Mota, E. 2019. Constant mean curvature surfaces and Heun's differential equations. PhD Thesis, University College Cork.
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