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Award ID contains: 1711400

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  1. ; ( , Discrete & Computational Geometry)
    null (Ed.)
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  2. ; ( , Calculus of Variations and Partial Differential Equations)
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  3. ( , Notices of the American Mathematical Society)
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  4. ( , Geometriae dedicata)
    Accepted Manuscript Full Text Available
  5. ( , Journal of differential geometry)
    We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical 4 vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects. 
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