Classification of limiting shapes for isotropic curve flows
Author:
Ben Andrews
Journal:
J. Amer. Math. Soc. 16 (2003), 443459
MSC (2000):
Primary 53C44; Secondary 35K55, 53A04
Published electronically:
December 11, 2002
MathSciNet review:
1949167
Fulltext PDF Free Access
Abstract 
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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.
 1.
U.
Abresch and J.
Langer, The normalized curve shortening flow and homothetic
solutions, J. Differential Geom. 23 (1986),
no. 2, 175–196. MR 845704
(88d:53001)
 2.
Ben
Andrews, Contraction of convex hypersurfaces by their affine
normal, J. Differential Geom. 43 (1996), no. 2,
207–230. MR 1424425
(97m:58045)
 3.
Ben
Andrews, Monotone quantities and unique limits for evolving convex
hypersurfaces, Internat. Math. Res. Notices 20
(1997), 1001–1031. MR 1486693
(99a:58041), http://dx.doi.org/10.1155/S1073792897000640
 4.
Ben
Andrews, Evolving convex curves, Calc. Var. Partial
Differential Equations 7 (1998), no. 4,
315–371. MR 1660843
(99k:58038), http://dx.doi.org/10.1007/s005260050111
 5.
Ben
Andrews, Motion of hypersurfaces by Gauss curvature, Pacific
J. Math. 195 (2000), no. 1, 1–34. MR 1781612
(2001i:53108), http://dx.doi.org/10.2140/pjm.2000.195.1
 6.
, Nonconvergence and instability in the asymptotic behaviour of curves evolving by curvature, Comm. Anal. Geom. 10 (2002), 409449.
 7.
Sigurd
Angenent, Guillermo
Sapiro, and Allen
Tannenbaum, On the affine heat equation for
nonconvex curves, J. Amer. Math. Soc.
11 (1998), no. 3,
601–634. MR 1491538
(99d:58039), http://dx.doi.org/10.1090/S0894034798002628
 8.
Bennett
Chow and DongHo
Tsai, Geometric expansion of convex plane curves, J.
Differential Geom. 44 (1996), no. 2, 312–330.
MR
1425578 (97m:58041)
 9.
K.S. Chou and X.P. Zhu, ``The Curve Shortening Problem'', Chapman and Hall/CRC (2000).
 10.
KaiSeng
Chou and XiPing
Zhu, A convexity theorem for a class of anisotropic flows of plane
curves, Indiana Univ. Math. J. 48 (1999), no. 1,
139–154. MR 1722196
(2000m:53098), http://dx.doi.org/10.1512/iumj.1999.48.1273
 11.
Michael
E. Gage, An isoperimetric inequality with applications to curve
shortening, Duke Math. J. 50 (1983), no. 4,
1225–1229. MR 726325
(85d:52007), http://dx.doi.org/10.1215/S0012709483050524
 12.
M.
E. Gage, Curve shortening makes convex curves circular,
Invent. Math. 76 (1984), no. 2, 357–364. MR 742856
(85i:52004), http://dx.doi.org/10.1007/BF01388602
 13.
Claus
Gerhardt, Flow of nonconvex hypersurfaces into spheres, J.
Differential Geom. 32 (1990), no. 1, 299–314.
MR
1064876 (91k:53016)
 14.
Matthew
A. Grayson, The heat equation shrinks embedded plane curves to
round points, J. Differential Geom. 26 (1987),
no. 2, 285–314. MR 906392
(89b:53005)
 15.
M.
Gage and R.
S. Hamilton, The heat equation shrinking convex plane curves,
J. Differential Geom. 23 (1986), no. 1, 69–96.
MR 840401
(87m:53003)
 16.
Richard
S. Hamilton, Isoperimetric estimates for the curve shrinking flow
in the plane, Modern methods in complex analysis (Princeton, NJ, 1992)
Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ,
1995, pp. 201–222. MR 1369140
(96k:58043)
 17.
Gerhard
Huisken, A distance comparison principle for evolving curves,
Asian J. Math. 2 (1998), no. 1, 127–133. MR 1656553
(99m:58052)
 18.
Jeffrey
A. Oaks, Singularities and selfintersections of curves evolving on
surfaces, Indiana Univ. Math. J. 43 (1994),
no. 3, 959–981. MR 1305955
(95k:58039), http://dx.doi.org/10.1512/iumj.1994.43.43042
 19.
Guillermo
Sapiro and Allen
Tannenbaum, On affine plane curve evolution, J. Funct. Anal.
119 (1994), no. 1, 79–120. MR 1255274
(94m:58049), http://dx.doi.org/10.1006/jfan.1994.1004
 20.
Richard
Montgomery, Survey of singular geodesics, SubRiemannian
geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996,
pp. 325–339. MR 1421824
(97m:58042)
 21.
John
I. E. Urbas, Correction to: “An expansion of convex
hypersurfaces” [J.\ Differential Geom.\ {33} (1991), no.\ 1,
91–125; MR1085136 (91j:58155)], J. Differential Geom.
35 (1992), no. 3, 763–765. MR 1163459
(93b:58142)
 22.
John
I. E. Urbas, On the expansion of starshaped hypersurfaces by
symmetric functions of their principal curvatures, Math. Z.
205 (1990), no. 3, 355–372. MR 1082861
(92c:53037), http://dx.doi.org/10.1007/BF02571249
 23.
John
Urbas, Convex curves moving homothetically by negative powers of
their curvature, Asian J. Math. 3 (1999), no. 3,
635–656. MR 1793674
(2001m:53119)
 1.
 U. Abresch and J. Langer, The normalised curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), 175196. MR 88d:53001
 2.
 B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), 207230. MR 97m:58045
 3.
 , Monotone quantities and unique limits for evolving convex hypersurfaces, Internat. Math. Res. Notices 20 (1997), 10011031. MR 99a:58041
 4.
 , Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no 3, 315371. MR 99k:58038
 5.
 , Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), 134. MR 2001i:53108
 6.
 , Nonconvergence and instability in the asymptotic behaviour of curves evolving by curvature, Comm. Anal. Geom. 10 (2002), 409449.
 7.
 S. Angenent, G. Sapiro and A. Tannenbaum, On the affine heat equation for nonconvex curves, J. Amer. Math. Soc. 11 (1998), 601634. MR 99d:58039
 8.
 B. Chow and H.D. Tsai, Geometric expansion of convex plane curves, J. Differential Geom. 44 (1996), 312330. 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), 139154. MR 2000m:53098
 11.
 M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), 12251229. MR 85d:52007
 12.
 , Curve shortening makes convex curves circular, Invent. Math. 76 (1984), 357364. MR 85i:52004
 13.
 C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), 299314. MR 91k:53016
 14.
 M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285314. MR 89b:53005
 15.
 M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 6996. MR 87m:53003
 16.
 R. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Ann. of Math. Stud. 137 (1995), 201222. MR 96k:58043
 17.
 G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127133. MR 99m:58052
 18.
 J. Oaks, Singularities and selfintersections of curves evolving on surfaces, Indiana Univ. Math. J. 43 (1994), 959981. MR 95k:58039
 19.
 G. Sapiro and A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79120. MR 94m:58049
 20.
 D.H. Tsai, Geometric expansion of starshaped plane curves, Comm. Anal. Geom. 4 (1996), 459480. MR 97m:58042
 21.
 J. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), 91125; Correction to, ibid. 35, 763765. MR 93b:58142
 22.
 , On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), 355372. MR 92c:53037
 23.
 , Convex curves moving homothetically by negative powers of their curvature, Asian J. Math. 3 (1999), 635656. MR 2001m:53119
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Additional Information
Ben Andrews
Affiliation:
Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia
Email:
andrews@maths.anu.edu.au
DOI:
http://dx.doi.org/10.1090/S0894034702004150
PII:
S 08940347(02)004150
Received by editor(s):
November 4, 2002
Published electronically:
December 11, 2002
Additional Notes:
Research supported by a grant from the Australian Research Council
Article copyright:
© Copyright 2002
American Mathematical Society
