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Self-similar solutions of fully nonlinear curvature flows

Journal Article


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. �� 2009.

Publication Date


  • 2011

Citation


  • McCoy, J. A. (2011). Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale - Classe di Scienze, 10(2), 317-333.

Scopus Eid


  • 2-s2.0-83655196943

Start Page


  • 317

End Page


  • 333

Volume


  • 10

Issue


  • 2

Place Of Publication


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. �� 2009.

Publication Date


  • 2011

Citation


  • McCoy, J. A. (2011). Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale - Classe di Scienze, 10(2), 317-333.

Scopus Eid


  • 2-s2.0-83655196943

Start Page


  • 317

End Page


  • 333

Volume


  • 10

Issue


  • 2

Place Of Publication