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Constant Mean Curvature Embedded Surfaces

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Constant Mean Curvature Surfaces with Boundary

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

The interest in this chapter is what type of symmetries imposes a given a space closed curve \(\varGamma\subset \mathbb{R}^{3}\) on the shape of a compact cmc surface spanning Γ. In the context of embedded surfaces, this problem may be partially answered using the reflection technique of Alexandrov. With this method we will give conditions whether the symmetries of the boundary are inherited to the surface, in particular, when the boundary is a plane curve. We employ the method of reflection by means of planes orthogonal to the plane containing the boundary, proving that, under some conditions, if the boundary is a circle the surface is rotational. In this context, we also apply the reflection method with planes parallel to the boundary plane obtaining results that prove that the surface is a graph. Finally, we use the Alexandrov method for embedded cmc surfaces whose boundary lies in a sphere and we give conditions that ensure that the surface lies on one side of the sphere.

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Authors and Affiliations

  1. Department of Geometry and Topology, University of Granada, Granada, Spain

    Rafael López

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  1. Rafael López
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López, R. (2013). Constant Mean Curvature Embedded Surfaces. In: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39626-7_4

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