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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.
Alessandroni, Roberta 1 ; Sinestrari, Carlo 1
@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},
pages = {541--571},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {3},
year = {2010},
mrnumber = {2722655},
zbl = {1248.53047},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_541_0/}
}
TY - JOUR AU - Alessandroni, Roberta AU - Sinestrari, Carlo TI - Evolution of hypersurfaces by powers of the scalar curvature JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 541 EP - 571 VL - 9 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_541_0/ LA - en ID - ASNSP_2010_5_9_3_541_0 ER -
%0 Journal Article %A Alessandroni, Roberta %A Sinestrari, Carlo %T Evolution of hypersurfaces by powers of the scalar curvature %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 541-571 %V 9 %N 3 %I Scuola Normale Superiore, Pisa %U http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_541_0/ %G en %F 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, Série 5, Tome 9 (2010) no. 3, pp. 541-571. http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_541_0/