Geodesic
Classification of limiting shapes for isotropic curve flows
Andrews, Ben1
1 Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia
Journal of the American Mathematical Society, Tome 16 (2003) no. 2, pp. 443-459

Voir la notice de l'article provenant de la source American Mathematical Society

A complete classification is given of curves in the plane which contract homothetically when evolved according to a power of their curvature. Applications are given to the limiting behaviour of the flows in various situations.
DOI : 10.1090/S0894-0347-02-00415-0

Andrews, Ben 1

1 Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia 
@article{10_1090_S0894_0347_02_00415_0,
     author = {Andrews, Ben},
     title = {Classification of limiting shapes for isotropic curve flows},
     journal = {Journal of the American Mathematical Society},
     pages = {443--459},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2003},
     doi = {10.1090/S0894-0347-02-00415-0},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00415-0/}
}
TY  - JOUR
AU  - Andrews, Ben
TI  - Classification of limiting shapes for isotropic curve flows
JO  - Journal of the American Mathematical Society
PY  - 2003
SP  - 443
EP  - 459
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00415-0/
DO  - 10.1090/S0894-0347-02-00415-0
ID  - 10_1090_S0894_0347_02_00415_0
ER  - 
%0 Journal Article
%A Andrews, Ben
%T Classification of limiting shapes for isotropic curve flows
%J Journal of the American Mathematical Society
%D 2003
%P 443-459
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00415-0/
%R 10.1090/S0894-0347-02-00415-0
%F 10_1090_S0894_0347_02_00415_0
Andrews, Ben. Classification of limiting shapes for isotropic curve flows. Journal of the American Mathematical Society, Tome 16 (2003) no. 2, pp. 443-459. doi: 10.1090/S0894-0347-02-00415-0

[1] Abresch, U., Langer, J. The normalized curve shortening flow and homothetic solutions J. Differential Geom. 1986 175 196

[2] Andrews, Ben Contraction of convex hypersurfaces by their affine normal J. Differential Geom. 1996 207 230

[3] Andrews, Ben Monotone quantities and unique limits for evolving convex hypersurfaces Internat. Math. Res. Notices 1997 1001 1031

[4] Andrews, Ben Evolving convex curves Calc. Var. Partial Differential Equations 1998 315 371

[5] Andrews, Ben Motion of hypersurfaces by Gauss curvature Pacific J. Math. 2000 1 34

[6] Angenent, Sigurd, Sapiro, Guillermo, Tannenbaum, Allen On the affine heat equation for non-convex curves J. Amer. Math. Soc. 1998 601 634

[7] Chow, Bennett, Tsai, Dong-Ho Geometric expansion of convex plane curves J. Differential Geom. 1996 312 330

[8] Chou, Kai-Seng, Zhu, Xi-Ping A convexity theorem for a class of anisotropic flows of plane curves Indiana Univ. Math. J. 1999 139 154

[9] Gage, Michael E. An isoperimetric inequality with applications to curve shortening Duke Math. J. 1983 1225 1229

[10] Gage, M. E. Curve shortening makes convex curves circular Invent. Math. 1984 357 364

[11] Gerhardt, Claus Flow of nonconvex hypersurfaces into spheres J. Differential Geom. 1990 299 314

[12] Grayson, Matthew A. The heat equation shrinks embedded plane curves to round points J. Differential Geom. 1987 285 314

[13] Gage, M., Hamilton, R. S. The heat equation shrinking convex plane curves J. Differential Geom. 1986 69 96

[14] Hamilton, Richard S. Isoperimetric estimates for the curve shrinking flow in the plane 1995 201 222

[15] Huisken, Gerhard A distance comparison principle for evolving curves Asian J. Math. 1998 127 133

[16] Oaks, Jeffrey A. Singularities and self-intersections of curves evolving on surfaces Indiana Univ. Math. J. 1994 959 981

[17] Sapiro, Guillermo, Tannenbaum, Allen On affine plane curve evolution J. Funct. Anal. 1994 79 120

[18] Montgomery, Richard Survey of singular geodesics 1996 325 339

[19] Urbas, John I. E. Correction to: “An expansion of convex hypersurfaces” [J. Differential Geom. 33 (1991), no. 1, 91–125 J. Differential Geom. 1992 763 765

[20] Urbas, John I. E. On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures Math. Z. 1990 355 372

[21] Urbas, John Convex curves moving homothetically by negative powers of their curvature Asian J. Math. 1999 635 656

Cité par Sources :