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, p. 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 : 2018-08-07
Classification:  53C44,  35J60
@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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