Classification of limiting shapes for isotropic curve flows.(English)Zbl 1023.53051
Consider a convex curve \(\gamma\) in the plane which is the image of an embedding \(x_0: S^1\to\mathbb{R}^2\). A deformation \(x\) of \(\gamma\) in a way that each point moves in the direction normal to \(\gamma\) with speed equal to a power of the curvature \(\kappa\) is a solution of the equation \[{\partial x\over\partial t}=- {1\over\alpha} \kappa^\alpha{\mathbf n},\] where \(\alpha\neq 0\) and \({\mathbf n}\) is the (outward pointing) unit normal vector. The initial condition is \(x(p,0)= x_0(p)\). The paper provides a complete description of the behaviour of embedded convex curves moving by equations of the above form. In certain cases, depending on the range of \(\alpha\), the description has been proved before.
Reviewer: F.Manhart (Wien)
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 53A04 | Curves in Euclidean and related spaces |
| 35K55 | Nonlinear parabolic equations |
Cite
Full Text:DOI
References:
| [1] | U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175 – 196. ·Zbl 0592.53002 |
| [2] | Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), no. 2, 207 – 230. ·Zbl 0858.53005 |
| [3] | Ben Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, Internat. Math. Res. Notices 20 (1997), 1001 – 1031. ·Zbl 0892.53002 ·doi:10.1155/S1073792897000640 |
| [4] | Ben Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315 – 371. ·Zbl 0931.53030 ·doi:10.1007/s005260050111 |
| [5] | Ben Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), no. 1, 1 – 34. ·Zbl 1028.53072 ·doi:10.2140/pjm.2000.195.1 |
| [6] | -, Nonconvergence and instability in the asymptotic behaviour of curves evolving by curvature, Comm. Anal. Geom. 10 (2002), 409-449. ·Zbl 1029.53079 |
| [7] | Sigurd Angenent, Guillermo Sapiro, and Allen Tannenbaum, On the affine heat equation for non-convex curves, J. Amer. Math. Soc. 11 (1998), no. 3, 601 – 634. ·Zbl 0902.35048 |
| [8] | Bennett Chow and Dong-Ho Tsai, Geometric expansion of convex plane curves, J. Differential Geom. 44 (1996), no. 2, 312 – 330. ·Zbl 0872.58052 |
| [9] | K.-S. Chou and X.-P. Zhu, “The Curve Shortening Problem”, Chapman and Hall/CRC (2000). |
| [10] | Kai-Seng Chou and Xi-Ping Zhu, A convexity theorem for a class of anisotropic flows of plane curves, Indiana Univ. Math. J. 48 (1999), no. 1, 139 – 154. ·Zbl 0979.53074 ·doi: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. ·Zbl 0534.52008 ·doi:10.1215/S0012-7094-83-05052-4 |
| [12] | M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), no. 2, 357 – 364. ·Zbl 0542.53004 ·doi:10.1007/BF01388602 |
| [13] | Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299 – 314. ·Zbl 0708.53045 |
| [14] | Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285 – 314. ·Zbl 0667.53001 |
| [15] | M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69 – 96. ·Zbl 0621.53001 |
| [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. ·Zbl 0846.51010 ·doi:10.1016/1053-8127(94)00130-3 |
| [17] | Gerhard Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), no. 1, 127 – 133. ·Zbl 0931.53032 ·doi:10.4310/AJM.1998.v2.n1.a2 |
| [18] | Jeffrey A. Oaks, Singularities and self-intersections of curves evolving on surfaces, Indiana Univ. Math. J. 43 (1994), no. 3, 959 – 981. ·Zbl 0835.53048 ·doi: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. ·Zbl 0801.53008 ·doi:10.1006/jfan.1994.1004 |
| [20] | Richard Montgomery, Survey of singular geodesics, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 325 – 339. ·Zbl 0868.53020 |
| [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. |
| [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. ·Zbl 0691.35048 ·doi: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. ·Zbl 0970.53039 ·doi:10.4310/AJM.1999.v3.n3.a4 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
