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Proceedings of the International Congress of Mathematicians pp 481–490Cite as

Birkhäuser

Constant Mean Curvature Surfaces in Euclidean Spaces

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Abstract

A variant of the isoperimetric problem is to classify and study the hypersurfaces in the Euclidean space \({\mathbb{E}^{n + 1}}\) that have critical area subject to the requirement that they enclose a fixed volume. In physical terms this is equivalent to having a soap film in equilibrium under its surface tension and a uniform gas pressure applied to one of its sides; hence, such surfaces are often called soap bubbles. The geometric condition for such a surface is that its mean curvature H is a nonzero constant. The precise value of the constant is not important because it can be changed to any desired value by a homothetic expansion. We will be using the abbreviation “CMC surface” to mean “complete smooth hypersurface properly immersed in \({\mathbb{E}^{n + 1}}\) with H ≡ 1”. Notice that the above definitions do not require embeddedness.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, RI, 02912, USA

    Nikolaos Kapouleas

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  1. Nikolaos Kapouleas
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Editor information

Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

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© 1995 Birkhäser Verlag, Basel, Switzerland

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Kapouleas, N. (1995). Constant Mean Curvature Surfaces in Euclidean Spaces. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_41

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  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_41

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

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