Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, that c shrinks through embedded sccs to a "round" point iff it bounds disks in S^2 of unequal area, and the "side" it shrinks to (which given the shrink can be unambiguously be identified as the "inside" or the "outside") is the one of smaller area. If the two sides have the same area, I would similarly guess that the curve shrinks to an equator. Does anyone know if these guesses are correct? 



curvature flow for loops in S^2
Actually, this result seems to be due to Mike Gage, and is so attributed by Grayson:
@article {MR1046497,
 AUTHOR = {Gage, Michael E.},
 TITLE = {Curve shortening on surfaces},
 JOURNAL = {Ann. Sci. \'Ecole Norm. Sup. (4)},
 FJOURNAL = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure.
 Quatri`eme S\'erie},
 VOLUME = {23},
 YEAR = {1990},
 NUMBER = {2},
 PAGES = {229--256},
 ISSN = {0012-9593},
 CODEN = {ASENAH},
 MRCLASS = {53C22 (35K55 58G11)},
 MRNUMBER = {1046497 (91a:53072)},
MRREVIEWER = {Dennis M. DeTurck},
 URL = {<http://www.numdam.org/item?id=ASENS_1990_4_23_2_229_0>},
}


