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

  • Masaaki Umehara (a1)

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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.

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References

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[COT] Cairns, G., Õzdemir, M. and Tjaden, E. H., A counterexample to a conjecture of U. pinkall, Topology, 31 (1992), 557558.
[J] Jackson, S. B., Vertices of plane curves, Bull. Amer. Math. Soc., 50 (1944), 564578.
[KB] Kauffiman, L. H. and Banchoff, T. F., Immersions and Mod-2 quadratic forms, Amer. Math. Monthly, 84 (1977), 168185.
[P] Pinkall, U., On the four-vertex theorem, Aequat, Math., 34 (1987), 221230.
[W] Whitney, H., On regular closed curves in the plane, Comp. Math., 4 (1937), 276284.
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6-vertex theorem for closed planar curve which bounds an immersed surface with non-zero genus

  • Masaaki Umehara (a1)

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