|
Classification of limiting shapes for isotropic curve flows
Author(s):
Ben
Andrews
Journal:
J. Amer. Math. Soc.
16
(2003),
443-459.
MSC (2000):
Primary 53C44;
Secondary 35K55, 53A04
Posted:
December 11, 2002
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
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.
References:
-
- 1.
- U. Abresch and J. Langer, The normalised curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), 175-196. MR 88d:53001
- 2.
- B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), 207-230. MR 97m:58045
- 3.
- -, Monotone quantities and unique limits for evolving convex hypersurfaces, Internat. Math. Res. Notices 20 (1997), 1001-1031. MR 99a:58041
- 4.
- -, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no 3, 315-371. MR 99k:58038
- 5.
- -, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), 1-34. MR 2001i:53108
- 6.
- -, Nonconvergence and instability in the asymptotic behaviour of curves evolving by curvature, Comm. Anal. Geom. 10 (2002), 409-449.
- 7.
- S. Angenent, G. Sapiro and A. Tannenbaum, On the affine heat equation for non-convex curves, J. Amer. Math. Soc. 11 (1998), 601-634. MR 99d:58039
- 8.
- B. Chow and H.-D. Tsai, Geometric expansion of convex plane curves, J. Differential Geom. 44 (1996), 312-330. MR 97m:58041
- 9.
- K.-S. Chou and X.-P. Zhu, ``The Curve Shortening Problem'', Chapman and Hall/CRC (2000).
- 10.
- -, A convexity theorem for a class of anisotropic flows of plane curves, Indiana Univ. Math. J. 48 (1999), 139-154. MR 2000m:53098
- 11.
- M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), 1225-1229. MR 85d:52007
- 12.
- -, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), 357-364. MR 85i:52004
- 13.
- C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), 299-314. MR 91k:53016
- 14.
- M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285-314. MR 89b:53005
- 15.
- M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69-96. MR 87m:53003
- 16.
- R. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Ann. of Math. Stud. 137 (1995), 201-222. MR 96k:58043
- 17.
- G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127-133. MR 99m:58052
- 18.
- J. Oaks, Singularities and self-intersections of curves evolving on surfaces, Indiana Univ. Math. J. 43 (1994), 959-981. MR 95k:58039
- 19.
- G. Sapiro and A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79-120. MR 94m:58049
- 20.
- D.-H. Tsai, Geometric expansion of star-shaped plane curves, Comm. Anal. Geom. 4 (1996), 459-480. MR 97m:58042
- 21.
- J. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), 91-125; Correction to, ibid. 35, 763-765. MR 93b:58142
- 22.
- -, On the expansion of star-shaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), 355-372. MR 92c:53037
- 23.
- -, Convex curves moving homothetically by negative powers of their curvature, Asian J. Math. 3 (1999), 635-656. MR 2001m:53119
Similar Articles:
Retrieve articles in Journal of the American Mathematical Society
with MSC
(2000):
53C44,
35K55, 53A04
Retrieve articles in all Journals with MSC
(2000):
53C44,
35K55, 53A04
Additional Information:
Ben
Andrews
Affiliation:
Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia
Email:
andrews@maths.anu.edu.au
DOI:
10.1090/S0894-0347-02-00415-0
PII:
S 0894-0347(02)00415-0
Received by editor(s):
November 4, 2002
Posted:
December 11, 2002
Additional Notes:
Research supported by a grant from the Australian Research Council
Copyright of article:
Copyright
2002,
American Mathematical Society
|
Comments: webmaster@ams.org