Contracting convex immersed closed plane curves with slow speed of curvature
Authors:
YuChu Lin, ChiCheung Poon and DongHo Tsai
Journal:
Trans. Amer. Math. Soc. 364 (2012), 57355763
MSC (2010):
Primary 53C44, 35K15, 35K55
Published electronically:
June 20, 2012
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The authors study the contraction of a convex immersed plane curve with speed , where is a constant, and show that, if the blowup rate of the curvature is of type one, it will converge to a homothetic selfsimilar solution. They also discuss a special symmetric case of type two blowup and show that it converges to a translational selfsimilar solution. In the case of curve shortening flow (i.e., when ), this translational selfsimilar solution is the familiar ``Grim Reaper'' (a terminology due to M. Grayson).
 [AL]
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)
 [AN1]
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
 [AN2]
Ben
Andrews, Harnack inequalities for evolving hypersurfaces,
Math. Z. 217 (1994), no. 2, 179–197. MR 1296393
(95j:58178), http://dx.doi.org/10.1007/BF02571941
 [AN3]
Ben
Andrews, Classification of limiting shapes for
isotropic curve flows, J. Amer. Math. Soc.
16 (2003), no. 2,
443–459 (electronic). MR 1949167
(2004a:53083), http://dx.doi.org/10.1090/S0894034702004150
 [AN4]
Ben
Andrews, Nonconvergence and instability in the asymptotic
behaviour of curves evolving by curvature, Comm. Anal. Geom.
10 (2002), no. 2, 409–449. MR 1900758
(2003e:53086)
 [ANG]
Sigurd
Angenent, On the formation of singularities in the curve shortening
flow, J. Differential Geom. 33 (1991), no. 3,
601–633. MR 1100205
(92c:58016)
 [AV]
S.
B. Angenent and J.
J. L. Velázquez, Asymptotic shape of cusp singularities in
curve shortening, Duke Math. J. 77 (1995),
no. 1, 71–110. MR 1317628
(95k:58036), http://dx.doi.org/10.1215/S0012709495077047
 [C]
Bennett
Chow, Geometric aspects of Aleksandrov reflection and gradient
estimates for parabolic equations, Comm. Anal. Geom.
5 (1997), no. 2, 389–409. MR 1483984
(98k:53045)
 [CLT]
Bennett
Chow, LiiPerng
Liou, and DongHo
Tsai, Expansion of embedded curves with turning angle greater than
𝜋, Invent. Math. 123 (1996), no. 3,
415–429. MR 1383955
(97c:58025), http://dx.doi.org/10.1007/s002220050034
 [CM]
XuYan
Chen and Hiroshi
Matano, Convergence, asymptotic periodicity, and finitepoint
blowup in onedimensional semilinear heat equations, J. Differential
Equations 78 (1989), no. 1, 160–190. MR 986159
(90e:35018), http://dx.doi.org/10.1016/00220396(89)900818
 [CPE]
Carmen
Cortázar, Manuel
del Pino, and Manuel
Elgueta, On the blowup set for
𝑢_{𝑡}=Δ𝑢^{𝑚}+𝑢^{𝑚},
𝑚>1, Indiana Univ. Math. J. 47 (1998),
no. 2, 541–561. MR 1647932
(99h:35085), http://dx.doi.org/10.1512/iumj.1998.47.1399
 [CT]
Bennett
Chow and DongHo
Tsai, Geometric expansion of convex plane curves, J.
Differential Geom. 44 (1996), no. 2, 312–330.
MR
1425578 (97m:58041)
 [CZ]
KaiSeng
Chou and XiPing
Zhu, The curve shortening problem, Chapman & Hall/CRC,
Boca Raton, FL, 2001. MR 1888641
(2003e:53088)
 [FS]
Eduard
Feireisl and Frédérique
Simondon, Convergence for degenerate parabolic equations, J.
Differential Equations 152 (1999), no. 2,
439–466. MR 1674569
(2000d:35123), http://dx.doi.org/10.1006/jdeq.1998.3545
 [FM]
Avner
Friedman and Bryce
McLeod, Blowup of solutions of nonlinear degenerate parabolic
equations, Arch. Rational Mech. Anal. 96 (1986),
no. 1, 55–80. MR 853975
(87j:35051), http://dx.doi.org/10.1007/BF00251413
 [GA]
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
 [GH]
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)
 [GR]
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)
 [LPT]
TaiChia
Lin, ChiCheung
Poon, and DongHo
Tsai, Expanding convex immersed closed plane curves, Calc.
Var. Partial Differential Equations 34 (2009), no. 2,
153–178. MR 2448648
(2010m:35191), http://dx.doi.org/10.1007/s0052600801807
 [LT1]
YuChu
Lin and DongHo
Tsai, On a simple maximum principle technique applied to equations
on the circle, J. Differential Equations 245 (2008),
no. 2, 377–391. MR 2428003
(2010b:35252), http://dx.doi.org/10.1016/j.jde.2008.04.007
 [LT2]
YuChu
Lin and DongHo
Tsai, Evolving a convex closed curve to another one via a
lengthpreserving linear flow, J. Differential Equations
247 (2009), no. 9, 2620–2636. MR 2568066
(2010j:53135), http://dx.doi.org/10.1016/j.jde.2009.07.024
 [NT]
ChiaHsing
Nien and DongHo
Tsai, Convex curves moving translationally in the plane, J.
Differential Equations 225 (2006), no. 2,
605–623. MR 2225802
(2007a:35070), http://dx.doi.org/10.1016/j.jde.2006.03.005
 [PT]
ChiCheung
Poon and DongHo
Tsai, Contracting convex immersed closed plane curves with fast
speed of curvature, Comm. Anal. Geom. 18 (2010),
no. 1, 23–75. MR 2660457
(2011g:53146), http://dx.doi.org/10.4310/CAG.2010.v18.n1.a2
 [T1]
DongHo
Tsai, Blowup and convergence of expanding immersed convex plane
curves, Comm. Anal. Geom. 8 (2000), no. 4,
761–794. MR 1792373
(2001k:53133)
 [T2]
DongHo
Tsai, Behavior of the gradient for solutions of parabolic equations
on the circle, Calc. Var. Partial Differential Equations
23 (2005), no. 3, 251–270. MR 2142063
(2006d:35116), http://dx.doi.org/10.1007/s0052600402981
 [T3]
DongHo
Tsai, Blowup behavior of an equation arising from planecurves
expansion, Differential Integral Equations 17 (2004),
no. 78, 849–872. MR 2075410
(2005e:35120)
 [T4]
DongHo
Tsai, Asymptotic closeness to limiting shapes for expanding
embedded plane curves, Invent. Math. 162 (2005),
no. 3, 473–492. MR 2198219
(2006j:53099), http://dx.doi.org/10.1007/s0022200504499
 [TSO]
Kaising
Tso, Deforming a hypersurface by its GaussKronecker
curvature, Comm. Pure Appl. Math. 38 (1985),
no. 6, 867–882. MR 812353
(87e:53009), http://dx.doi.org/10.1002/cpa.3160380615
 [U1]
John
I. E. Urbas, An expansion of convex hypersurfaces, J.
Differential Geom. 33 (1991), no. 1, 91–125. MR 1085136
(91j:58155)
 [U2]
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)
 [AL]
 Abresch, U., Langer, J. (1986) The normalized curve shortening flow and homothetic solutions, J. Diff. Geom., 23, 175196. MR 845704 (88d:53001)
 [AN1]
 Andrews, B. (1998) Evolving convex curves, Cal. of Var. & PDE., 7, no. 4, 315371. MR 1660843 (99k:58038)
 [AN2]
 Andrews, B. (1994) Harnack inequality for evolving hypersurfaces, Math. Zeit., 217, 179197. MR 1296393 (95j:58178)
 [AN3]
 Andrews, B. (2003) Classification of limiting shapes for isotropic curve flows, J. of the AMS, 16, no. 2, 443459. MR 1949167 (2004a:53083)
 [AN4]
 Andrews, B. (2002) Nonconvergence and instability in the asymptotic behavior of curves evolving by curvature, Comm. Anal. & Geom., vol. 10, no. 2, 409449. MR 1900758 (2003e:53086)
 [ANG]
 Angenent, S. (1991) On the formation of singularities in the curve shortening flow, J. Diff. Geom., 33, 601633. MR 1100205 (92c:58016)
 [AV]
 Angenent, S., Velázquez, J. J. L. (1995) Asymptotic shape of cusp singularities in curve shortening, Duke Math. J., vol. 77, no. 1, 71110. MR 1317628 (95k:58036)
 [C]
 Chow, B. (1997) Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. Geom., vol. 5, no. 2, 389409. MR 1483984 (98k:53045)
 [CLT]
 Chow, B., Liou, L. P., Tsai, D. H. (1996) Expansion of embedded curves with turning angle greater than, Invent. Math., 123, 415429. MR 1383955 (97c:58025)
 [CM]
 Chen, X. Y., Matano, H. (1989) Convergence, asymptotic periodicity, and finitepoint blowup in onedimensional semilinear heat equations, J. Diff. Eq., 78, 160190. MR 986159 (90e:35018)
 [CPE]
 Cortazar, C., Del Pino, M., Elgueta, M. (1998) On the blowup set for , Indiana Univ. Math. J., 47, p. 541561. MR 1647932 (99h:35085)
 [CT]
 Chow, B., Tsai, D. H. (1996) Geometric expansion of convex plane curves, J. Diff. Geom., 44, 312330. MR 1425578 (97m:58041)
 [CZ]
 Chou, K. S., Zhu, X. P. (2000) The Curve Shortening Problem, Chapman and Hall/ CRC. MR 1888641 (2003e:53088)
 [FS]
 Feireisl, E. and Simondon, F. (1999) Convergence for degenerate parabolic equations, J. Diff. Eq., vol. 152, 439466. MR 1674569 (2000d:35123)
 [FM]
 Friedman, A., McLeod (1986) Blowup of solutions of nonlinear degenerate parabolic equations, Arch. Rational Mech. Anal., 96, 5580. MR 853975 (87j:35051)
 [GA]
 Gage, M. (1984) Curve shortening makes convex curves circular, Invent. Math., 76, 357364. MR 742856 (85i:52004)
 [GH]
 Gage, M., Hamilton, R. S. (1986) The heat equation shrinking convex plane curves, J. Diff. Geom., 23, 6996. MR 840401 (87m:53003)
 [GR]
 Grayson, M. (1987) The heat equation shrinks embedded plane curves to round points, J. Diff. Geom., 26, 285314. MR 906392 (89b:53005)
 [LPT]
 Lin, T. C., Poon, C. C., Tsai, D. H. (2009) Expanding convex immersed closed plane curves, Cal. of Var. & PDEs., 34, 153178. MR 2448648 (2010m:35191)
 [LT1]
 Lin, Y. C., Tsai, D. H. (2008) On a simple maximum principle technique applied to equations on the circle, J. Diff. Eqs., 245, 377391. MR 2428003 (2010b:35252)
 [LT2]
 Lin, Y. C., Tsai, D. H. (2009) Evolving a convex closed curve to another one via a lengthpreserving linear flow, J. Diff. Eqs., 247, 26202636. MR 2568066 (2010j:53135)
 [NT]
 Nien, C. H., Tsai, D. H. (2006) Convex curves moving translationally in the plane, J. of Diff. Eq., 225, 605623. MR 2225802 (2007a:35070)
 [PT]
 Poon, C. C., Tsai, D. H. (2010) Contracting convex immersed closed plane curves with fast speed of curvature, Comm. Anal. & Geom., vol. 18, no. 1, 2375. MR 2660457 (2011g:53146)
 [T1]
 Tsai, D. H. (2000) Blowup and convergence of expanding immersed convex plane curves, Comm. Anal. & Geom., vol. 8, no. 4, 761794. MR 1792373 (2001k:53133)
 [T2]
 Tsai, D. H. (2005) Behavior of the gradient for solutions of parabolic equations on the circle, Cal. of Var. & PDE., vol. 23, 251270. MR 2142063 (2006d:35116)
 [T3]
 Tsai, D. H. (2004) Blowup behavior of an equation arising from plane curves expansion, Diff. and Integ. Eq., vol. 17, no. 78, 849872. MR 2075410 (2005e:35120)
 [T4]
 Tsai, D. H. (2005) Asymptotic closeness to limiting shapes for expanding embedded plane curves, Invent. Math., 162, 473492. MR 2198219 (2006j:53099)
 [TSO]
 K. Tso, (1985) Deforming a hypersurface by its GaussKronecker curvature, Comm. Pure and Appl. Math., 38, 867882. MR 812353 (87e:53009)
 [U1]
 Urbas, J. (1991) An expansion of convex hypersurfaces, J. Diff. Geom., 33, 91125. MR 1085136 (91j:58155)
 [U2]
 Urbas, J. (1999) Convex curves moving homothetically by negative powers of their curvature, Asian J. Math., vol. 3, no. 3, 635658. MR 1793674 (2001m:53119)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
53C44,
35K15,
35K55
Retrieve articles in all journals
with MSC (2010):
53C44,
35K15,
35K55
Additional Information
YuChu Lin
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Email:
yclin@math.ncku.edu.tw
ChiCheung Poon
Affiliation:
Department of Mathematics, National Chung Cheng University, Chiayi 621, Taiwan
Email:
ccpoon@math.ccu.edu.tw
DongHo Tsai
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
Email:
dhtsai@math.nthu.edu.tw
DOI:
http://dx.doi.org/10.1090/S00029947201205611X
PII:
S 00029947(2012)05611X
Received by editor(s):
October 12, 2010
Published electronically:
June 20, 2012
Additional Notes:
The third author’s research was supported by the NCTS and the NSC of Taiwan under grant number 962115M007010MY3.
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
