Self-similar solutions of fully nonlinear curvature flows
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 317-333.

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.

Published online:
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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McCoy, James Alexander. Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 317-333. http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0/

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