Constant Mean Curvature Surfaces in Euclidean Spaces

Nikolaos Kapouleas
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Nikolaos Kapouleas: Brown University, Department of Mathematics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 481-490 from Springer

Abstract: Abstract A variant of the isoperimetric problem is to classify and study the hypersurfaces in the Euclidean space $${\mathbb{E}^{n + 1}}$$ 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 $${\mathbb{E}^{n + 1}}$$ E n + 1 with H ≡ 1”. Notice that the above definitions do not require embeddedness.

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_41

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