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.
@article{ASNSP_2011_5_10_2_317_0, author = {McCoy, James Alexander}, title = {Self-similar solutions of fully nonlinear curvature flows}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {2}, year = {2011}, pages = {317-333}, zbl = {1234.53018}, mrnumber = {2856150}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0} }
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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