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6-vertex theorem for closed planar curve which bounds an immersed surface with non-zero genus

Published online by Cambridge University Press:  22 January 2016

Masaaki Umehara
Masaaki Umehara*
Affiliation:
Department of Mathematics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan
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A vertex of a planar curve γ of class is a point which attains a local maximum or minimum of its curvature function. By the definition, the number of vertices are even whenever it is finite. As a generalization of famous four vertex theorem, Pinkall [P] showed that a closed curve γ has at least 4 vertices if it bounds an immersed surface, and he conjectured that γ has at least 4g + 2 vertices when the surface has genus g.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 134 , June 1994 , pp. 75 - 89
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

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[W] Whitney, H., On regular closed curves in the plane, Comp. Math., 4 (1937), 276–284.Google Scholar