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Self-similar solutions of fully nonlinear curvature flows

Journal Article


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Abstract


  • We consider closed hypersurfaces which shrink self-similarly under

    a natural class of fully nonlinear curvature flows. For those flows in our class

    with speeds homogeneous of degree 1 and either convex or concave, we show

    that the only such hypersurfaces are shrinking spheres. In the setting of convex

    hypersurfaces, we show under a weaker second derivative condition on the speed

    that again only shrinking spheres are possible. For surfaces this result is extended

    in some cases by a different method to speeds of homogeneity greater than 1. Finally

    we show that self-similar hypersurfaces with sufficiently pinched principal

    curvatures, depending on the flow speed, are again necessarily spheres.

Publication Date


  • 2011

Citation


  • McCoy, J. Alexander. (2011). Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale Superiore di Pisa: Classe di Scienze, 10 (5), 317-333.

Scopus Eid


  • 2-s2.0-83655196943

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1863&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/847

Number Of Pages


  • 16

Start Page


  • 317

End Page


  • 333

Volume


  • 10

Issue


  • 5

Place Of Publication


  • http://annaliscienze.sns.it/index.php?page=Home&PHPSESSID=3f396383f456e0db756bdb7c535ba6a7

Abstract


  • We consider closed hypersurfaces which shrink self-similarly under

    a natural class of fully nonlinear curvature flows. For those flows in our class

    with speeds homogeneous of degree 1 and either convex or concave, we show

    that the only such hypersurfaces are shrinking spheres. In the setting of convex

    hypersurfaces, we show under a weaker second derivative condition on the speed

    that again only shrinking spheres are possible. For surfaces this result is extended

    in some cases by a different method to speeds of homogeneity greater than 1. Finally

    we show that self-similar hypersurfaces with sufficiently pinched principal

    curvatures, depending on the flow speed, are again necessarily spheres.

Publication Date


  • 2011

Citation


  • McCoy, J. Alexander. (2011). Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale Superiore di Pisa: Classe di Scienze, 10 (5), 317-333.

Scopus Eid


  • 2-s2.0-83655196943

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1863&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/847

Number Of Pages


  • 16

Start Page


  • 317

End Page


  • 333

Volume


  • 10

Issue


  • 5

Place Of Publication


  • http://annaliscienze.sns.it/index.php?page=Home&PHPSESSID=3f396383f456e0db756bdb7c535ba6a7