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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Isoperimetric inequalities for immersed closed spherical curves


Author: Joel L. Weiner
Journal: Proc. Amer. Math. Soc. 120 (1994), 501-506
MSC: Primary 53A04; Secondary 53C42
MathSciNet review: 1163337
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Abstract: Let $ \alpha :{S^1} \to {S^2}$ be a $ {C^2}$ immersion with length $ L$ and total curvature $ K$. If $ \alpha $ is regularly homotopic to a circle traversed once then $ {L^2} + {K^2} \geqslant 4{\pi ^2}$ with equality if and only if $ \alpha $ is a circle traversed once. If $ \alpha $ has nonnegative geodesic curvature and multiple points then $ L + K \geqslant 4\pi $ with equality if and only if $ \alpha $ is a great circle traversed twice.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1163337-4
PII: S 0002-9939(1994)1163337-4
Keywords: Immersed spherical curve, isoperimetric inequality, length, total curvature
Article copyright: © Copyright 1994 American Mathematical Society