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Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

Journal Article


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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, non-uniformly 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, and by avoiding these

    arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become

    smooth and uniformly convex and contract to points.

Publication Date


  • 2012

Citation


  • Andrews, B. & McCoy, J. (2012). Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Transactions of the American Mathematical Society, 364 (7), 3427-3447.

Scopus Eid


  • 2-s2.0-84859075865

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9265&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1930

Has Global Citation Frequency


Number Of Pages


  • 20

Start Page


  • 3427

End Page


  • 3447

Volume


  • 364

Issue


  • 7

Place Of Publication


  • United States

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, non-uniformly 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, and by avoiding these

    arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become

    smooth and uniformly convex and contract to points.

Publication Date


  • 2012

Citation


  • Andrews, B. & McCoy, J. (2012). Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Transactions of the American Mathematical Society, 364 (7), 3427-3447.

Scopus Eid


  • 2-s2.0-84859075865

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9265&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1930

Has Global Citation Frequency


Number Of Pages


  • 20

Start Page


  • 3427

End Page


  • 3447

Volume


  • 364

Issue


  • 7

Place Of Publication


  • United States