We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.
@article{ASNSP_2010_5_9_3_541_0,
author = {Alessandroni, Roberta and Sinestrari, Carlo},
title = {Evolution of hypersurfaces by powers of the scalar curvature},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {3},
year = {2010},
pages = {541-571},
zbl = {1248.53047},
mrnumber = {2722655},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_3_541_0}
}
Alessandroni, Roberta; Sinestrari, Carlo. Evolution of hypersurfaces by powers of the scalar curvature. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 3, pp. 541-571. http://www.numdam.org/item/ASNSP_2010_5_9_3_541_0/
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