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Convexity estimates for hypersurfaces moving by convex curvature functions

Journal Article


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Abstract


  • We consider the evolution of compact hypersurfaces by fully non-linear, parabolic

    curvature

    ows for which the normal speed is given by a smooth, convex, degree one homoge-

    neous function of the principal curvatures. We prove that solution hypersurfaces on which the

    speed is initially positive become weakly convex at a singularity of the

    ow. The result extends

    the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature

    ow to a large

    class of speeds, and leads to an analogous description of `type-II' singularities. We remark that

    many of the speeds considered are positive on larger cones than the positive mean half-space,

    so that the result in those cases also applies to non-mean-convex initial data.

Authors


  •   Andrews, Ben H. (external author)
  •   Langford, Mat (external author)
  •   McCoy, James A.

Publication Date


  • 2014

Citation


  • Andrews, B., Langford, M. & McCoy, J. (2014). Convexity estimates for hypersurfaces moving by convex curvature functions. Analysis and PDE, 7 (2), 407-433.

Scopus Eid


  • 2-s2.0-84903173584

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=4469&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3452

Has Global Citation Frequency


Number Of Pages


  • 26

Start Page


  • 407

End Page


  • 433

Volume


  • 7

Issue


  • 2

Place Of Publication


  • United States

Abstract


  • We consider the evolution of compact hypersurfaces by fully non-linear, parabolic

    curvature

    ows for which the normal speed is given by a smooth, convex, degree one homoge-

    neous function of the principal curvatures. We prove that solution hypersurfaces on which the

    speed is initially positive become weakly convex at a singularity of the

    ow. The result extends

    the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature

    ow to a large

    class of speeds, and leads to an analogous description of `type-II' singularities. We remark that

    many of the speeds considered are positive on larger cones than the positive mean half-space,

    so that the result in those cases also applies to non-mean-convex initial data.

Authors


  •   Andrews, Ben H. (external author)
  •   Langford, Mat (external author)
  •   McCoy, James A.

Publication Date


  • 2014

Citation


  • Andrews, B., Langford, M. & McCoy, J. (2014). Convexity estimates for hypersurfaces moving by convex curvature functions. Analysis and PDE, 7 (2), 407-433.

Scopus Eid


  • 2-s2.0-84903173584

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=4469&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3452

Has Global Citation Frequency


Number Of Pages


  • 26

Start Page


  • 407

End Page


  • 433

Volume


  • 7

Issue


  • 2

Place Of Publication


  • United States