Geodesic
Self-similar solutions of fully nonlinear curvature flows
McCoy, James Alexander1
1 Institute for Mathematics and its Applications School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW 2522, Australia
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 317-333

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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.

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Classification : 53C44, 35J60

McCoy, James Alexander 1

1 Institute for Mathematics and its Applications School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW 2522, Australia 
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     title = {Self-similar solutions of fully nonlinear curvature flows},
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McCoy, James Alexander. Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 317-333. http://geodesic.mathdoc.fr/item/ASNSP_2011_5_10_2_317_0/