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Contracting convex hypersurfaces by curvature

Journal Article


Abstract


  • We consider compact convex hypersurfaces contracting by functions of their curvature.

    Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve

    to remain smooth and uniformly convex, and contract to points after finite time. The same

    holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the

    final time is spherical.We provide a surprisingly large family of flows for which such results

    fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex,

    and smooth ones may develop curvature singularities; even where this does not occur, nonuniformly

    convex regions and singular parts in the initial hypersurface may persist, including

    flat sides, ridges of infinite curvature, or ‘cylindrical’ regions where some of the principal

    curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in

    such cases the hypersurface may even collapse to a line segment or higher-dimensional disc

    rather than to a point. We provide sufficient conditions for these various disasters to occur,

Authors


  •   Andrews, Ben H. (external author)
  •   McCoy, James A.
  •   Zheng, Yu (external author)

Publication Date


  • 2013

Citation


  • Andrews, B., McCoy, J. & Zheng, Y. (2013). Contracting convex hypersurfaces by curvature. Calculus of Variations and Partial Differential Equations, 47 (3-4), 611-665.

Scopus Eid


  • 2-s2.0-84879507926

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1154

Number Of Pages


  • 54

Start Page


  • 611

End Page


  • 665

Volume


  • 47

Issue


  • 3-4

Abstract


  • We consider compact convex hypersurfaces contracting by functions of their curvature.

    Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve

    to remain smooth and uniformly convex, and contract to points after finite time. The same

    holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the

    final time is spherical.We provide a surprisingly large family of flows for which such results

    fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex,

    and smooth ones may develop curvature singularities; even where this does not occur, nonuniformly

    convex regions and singular parts in the initial hypersurface may persist, including

    flat sides, ridges of infinite curvature, or ‘cylindrical’ regions where some of the principal

    curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in

    such cases the hypersurface may even collapse to a line segment or higher-dimensional disc

    rather than to a point. We provide sufficient conditions for these various disasters to occur,

Authors


  •   Andrews, Ben H. (external author)
  •   McCoy, James A.
  •   Zheng, Yu (external author)

Publication Date


  • 2013

Citation


  • Andrews, B., McCoy, J. & Zheng, Y. (2013). Contracting convex hypersurfaces by curvature. Calculus of Variations and Partial Differential Equations, 47 (3-4), 611-665.

Scopus Eid


  • 2-s2.0-84879507926

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1154

Number Of Pages


  • 54

Start Page


  • 611

End Page


  • 665

Volume


  • 47

Issue


  • 3-4