On the affine heat equation for non-convex curves.(English)Zbl 0902.35048

Recently G. Sapiro and A. Tannenbaum [J. Funct. Anal. 119, 79-120 (1994; Zbl 0801.53008)] considered the evolution of convex plane curves moving with velocity given by their affine normal. Their result is that any smooth closed convex curve moving under this flow remains smooth and convex until it shrinks to a point. Moreover, the limiting shape is an ellipse rather than a circle as in the usual curve shortening flow studied by M. Gage and R. S. Hamilton [J. Differ. Geom. 23, 69-96 (1986; Zbl 0621.53001)] because this is an affine invariant flow, which makes it particularly interesting for applications to computer vision and image processing.
Here the authors continue this study by considering this affine invariant flow for non-convex curves. The first difficulty that needs to be addressed is finding a suitable extension of the flow to non-convex curves, because the basic invariants of affine differential geometry are defined only for convex curves. A suitable extension to non-convex curves is obtained by taking the normal velocity vector to be $$\kappa^{1/3}{\mathcal N}$$, where $$\kappa$$ is the curvature and $${\mathcal N}$$ is the usual Euclidean normal to the curve. The basic result is that under this flow any simple closed curve shrinks to a point, and moreover, the total curvature of the curve tends to $$2\pi$$.
L. Alvarez, F. Guichard, P. L. Lions and J. M. Morel [C. R. Acad. Sci., Paris, Sér. I 315, 135-138 (1992; Zbl 0792.68196) and 315, 265-268 (1992; Zbl 0805.68134)] have considered an equivalent flow from the point of view of viscosity solutions. In addition, a higher dimensional affine invariant flow for convex hypersurfaces has been studied by B. Andrews [J. Differ. Geom. 43, 207-230 (1996; Zbl 0858.53005)].
Reviewer: J.Urbas (Bonn)

MathOverflow Questions:

Properties of the affine curve-shortening flow

MSC:

 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 53A15 Affine differential geometry 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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References:

 [1] Luis Álvarez, Frédéric Guichard, Pierre-Louis Lions, and Jean-Michel Morel, Axiomes et équations fondamentales du traitement d’images (analyse multiéchelle et EDP), C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 2, 135 – 138 (French, with English and French summaries). · Zbl 0792.68196 [2] Luis Alvarez, Frédéric Guichard, Pierre-Louis Lions, and Jean-Michel Morel, Axiomatisation et nouveaux opérateurs de la morphologie mathématique, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 3, 265 – 268 (French, with English and French summaries). · Zbl 0805.68134 [3] Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), no. 2, 207 – 230. · Zbl 0858.53005 [4] Sigurd Angenent, Parabolic equations for curves on surfaces. I. Curves with \?-integrable curvature, Ann. of Math. (2) 132 (1990), no. 3, 451 – 483. · Zbl 0789.58070 [5] Sigurd Angenent, Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math. (2) 133 (1991), no. 1, 171 – 215. · Zbl 0749.58054 [6] Sigurd Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), no. 3, 601 – 633. · Zbl 0731.53002 [7] Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79 – 96. · Zbl 0644.35050 [8] W. Blaschke, Vorlesungen über Differentialgeometrie II, Verlag Von Julius Springer, Berlin, 1923. · JFM 49.0499.01 [9] Bu Chin Su, Affine differential geometry, Science Press Beijing, Beijing; Gordon & Breach Science Publishers, New York, 1983. [10] Jean A. Dieudonné and James B. Carrell, Invariant theory, old and new, Academic Press, New York-London, 1971. · Zbl 0258.14011 [11] Xu-Yan Chen and Hiroshi Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), no. 1, 160 – 190. · Zbl 0692.35013 [12] C. L. Epstein and Michael Gage, The curve shortening flow, Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986) Math. Sci. Res. Inst. Publ., vol. 7, Springer, New York, 1987, pp. 15 – 59. · Zbl 0645.53028 [13] Michael E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225 – 1229. · Zbl 0534.52008 [14] M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), no. 2, 357 – 364. · Zbl 0542.53004 [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] 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 [17] Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71 – 111. · Zbl 0686.53036 [18] Heinrich W. Guggenheimer, Differential geometry, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. Heinrich W. Guggenheimer, Differential geometry, Dover Publications, Inc., New York, 1977. Corrected reprint of the 1963 edition; Dover Books on Advanced Mathematics. · Zbl 0472.51008 [19] Benjamin B. Kimia, Allen Tannenbaum, and Steven W. Zucker, On the evolution of curves via a function of curvature. I. The classical case, J. Math. Anal. Appl. 163 (1992), no. 2, 438 – 458. · Zbl 0771.53003 [20] B. B. Kimia, A. Tannenbaum, and S. W. Zucker, “Shapes, shocks, and deformations,” Int. J. Computer Vision 15 (1995), 189-224. [21] Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401 – 441. · Zbl 0496.35011 [22] Peter J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003 [23] P. J. Olver, “Differential invariants,” to appear in Acta Appl. Math. · Zbl 0842.53013 [24] P. Olver, G. Sapiro, and A. Tannenbaum, “Differential invariant signatures and flows in computer vision: A symmetry group approach,” Geometric Driven Diffusion, edited by Bart ter har Romeny, Kluwer, 1994. · Zbl 0863.53008 [25] Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum, Classification and uniqueness of invariant geometric flows, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 339 – 344 (English, with English and French summaries). · Zbl 0863.53008 [26] Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum, Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), no. 1, 176 – 194. · Zbl 0874.35044 [27] P. Olver, G. Sapiro, and A. Tannenbaum, “Affine invariant edge maps and active contours,” to appear in CVIU. · Zbl 1005.53010 [28] Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12 – 49. · Zbl 0659.65132 [29] Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. · Zbl 0549.35002 [30] Guillermo Sapiro and Allen Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), no. 1, 79 – 120. · Zbl 0801.53008 [31] G. Sapiro and A. Tannenbaum, “Affine invariant scale-space,” Int. J. Computer Vision 11, pp. 25-44, 1993. [32] Guillermo Sapiro and Allen Tannenbaum, On invariant curve evolution and image analysis, Indiana Univ. Math. J. 42 (1993), no. 3, 985 – 1009. · Zbl 0793.53002 [33] G. Sapiro and A. Tannenbaum, “Area and length preserving geometric invariant scale-spaces,” IEEE Trans. Pattern Analysis and Machine Intelligence 17 (1995), 1066-1070. [34] Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. · Zbl 0439.53001 [35] Brian White, Some recent developments in differential geometry, Math. Intelligencer 11 (1989), no. 4, 41 – 47. · Zbl 0701.53004
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