Limit shape of the cross-section of shrinking doughnuts

Naoyuki ISHIMURA

doi:10.2969/jmsj/04530569

Naoyuki ISHIMURA

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JOURNAL FREE ACCESS

1993 Volume 45 Issue 3 Pages 569-582

DOI https://doi.org/10.2969/jmsj/04530569

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Published: 1993 Received: October 16, 1991 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) K. Ahara and N. Ishimura, On the mean curvature flow of “thin” doughnuts, to appear in Lecture Notes Numer. Appl. Anal., see also Preprint series UTYO-MATH, 91-14, Univ. of Tokyo, 1991.
2) S.J. Altschuler, S.B. Angenent and Y. Giga, Mean curvature flow through singularities for surfaces of rotation, preprint (1991).
3) S.B. Angenent, Parabolic equations for curves on surfaces, Part I, Ann. of Math., 132 (1990), 451-483.
4) S.B. Angenent, Parabolic equations for curves on surfaces, Part II, Ann. of Math., 133 (1991), 171-215.
5) S.B. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633.
6) S.B. Angenent, Shrinking doughnuts, in “Nonliner Diffusion Equation and Their Equilibrium States, 3,” (eds. N.G. Lloyd, W.-M. Ni, L.A. Peletier and J. Serrin), Birkhäuser, Boston, 1992, pp. 21-38.
7) Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
8) L.C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681.
9) M.E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J., 50 (1983), 1225-1229.
10) M.E. Gage, Curve shortening makes convex curves circular, Invent. Math., 76 (1984), 357-364.
11) M.E. Gage and R.S. Hamilton, The shrinking of convex plane curves by the heat equation, J. Differential Geom., 23 (1986), 69-96.
12) Y. Giga and R. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
13) Y. Giga and R. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.
14) M.A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
15) M.A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J., 58 (1989), 555-558.
16) M.A. Grayson, The shape of a figure-eight under the curve shortening flow, Invent. Math., 96 (1989), 177-180.
17) R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.
18) G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
19) G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differential Geom., (1990), 285-299.
20) H.M. Soner and P.E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, preprint (1992).
21) M. Struwe, On the evolution of harmonic maps in high dimensions, J. Differential Geom., (1988), 485-502.

Right : [1] K. Ahara and N. Ishimura, On the mean curvature flow of “thin” doughnuts, to appear in Lecture Notes Numer. Appl. Anal., see also Preprint series UTYO-MATH, 91-14, Univ. of Tokyo, 1991.
[2] S. J. Altschuler, S. B. Angenent and Y. Giga, Mean curvature flow through singularities for surfaces of rotation, preprint (1991).
[3] S. B. Angenent, Parabolic equations for curves on surfaces, Part I, Ann. of Math., 132 (1990), 451-483.
[4] S. B. Angenent, Parabolic equations for curves on surfaces, Part II, Ann. of Math., 133 (1991), 171-215.
[5] S. B. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633.
[6] S. B. Angenent, Shrinking doughnuts, in “Nonliner Diffusion Equation and Their Equilibrium States, 3,” (eds. N. G. Lloyd, W. -M. Ni, L. A. Peletier and J. Serrin), Birkhäuser, Boston, 1992, pp. 21-38.
[7] Y. -G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
[8] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681.
[9] M. E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J., 50 (1983), 1225-1229.
[10] M. E. Gage, Curve shortening makes convex curves circular, Invent. Math., 76 (1984), 357-364.
[11] M. E. Gage and R. S. Hamilton, The shrinking of convex plane curves by the heat equation, J. Differential Geom., 23 (1986), 69-96.
[12] Y. Giga and R. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
[13] Y. Giga and R. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.
[14] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
[15] M. A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J., 58 (1989), 555-558.
[16] M. A. Grayson, The shape of a figure-eight under the curve shortening flow, Invent. Math., 96 (1989), 177-180.
[17] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.
[18] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
[19] G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differential Geom., (1990), 285-299.
[20] H. M. Soner and P. E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, preprint (1992).
[21] M. Struwe, On the evolution of harmonic maps in high dimensions, J. Differential Geom., (1988), 485-502.

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