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STABLE SURFACES IN EUCLIDEAN THREE SPACE

NICOLAAS H. KUIPER
Mathematica Scandinavica
Vol. 36, No. 1 (May 2, 1975), pp. 83-96
Published by: Mathematica Scandinavica
Stable URL: http://www.jstor.org/stable/24490684
Page Count: 14
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STABLE SURFACES IN EUCLIDEAN THREE SPACE
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Abstract

This paper consists of two related parts. In A we present smooth maps of the real projective plane P with the non euclidean metric σ, into euclidean spaces such that we can read various interesting properties from the image. We mention and indicate some proofs of known facts. This part is expository. In B we consider C∞-stable (in the sense of R. Thom) maps of surfaces in E3. We call these "stable surfaces" for short. The Gauss curvature as a measure (∫ K dσ) then exists although the scalar Gauss curvature K may explode at the C∞-stable singularities. The infimum of the total absolute curvature (2π)–1 ∫ |K dσ| of a compact surface M equals 4 – Χ(M). This infimum can be reached for any surface in the class of stable maps, but not for all surfaces in the class of immersions, as we know. Stable surfaces of minimal total absolute curvature (tight) are given for the exceptions: the projective plane with 0 or 1 handles and the Klein-bottle. Recall that tight (closed) surfaces in EN are also characterized as those that are divided into at most two (connected) parts by any (hyper-)plane.

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