Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Immersed self-shrinkers


Authors: Gregory Drugan and Stephen J. Kleene
Journal: Trans. Amer. Math. Soc. 369 (2017), 7213-7250
MSC (2010): Primary 53C44, 53C42
DOI: https://doi.org/10.1090/tran/6907
Published electronically: June 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct infinitely many complete, immersed self-shrinkers with rotational symmetry for each of the following topological types: the sphere, the plane, the cylinder, and the torus.


References [Enhancements On Off] (What's this?)

  • [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
  • [2] F. J. Almgren Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math. (2) 84 (1966), 277-292. MR 0200816
  • [3] Sigurd B. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21-38. MR 1167827
  • [4] M. S. Baouendi and C. Goulaouic, Singular nonlinear Cauchy problems, J. Differential Equations 22 (1976), no. 2, 268-291. MR 0435564
  • [5] Simon Brendle, Embedded self-similar shrinkers of genus 0, Ann. of Math. (2) 183 (2016), no. 2, 715-728. MR 3450486, https://doi.org/10.4007/annals.2016.183.2.6
  • [6] David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1-15. MR 1302814
  • [7] Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755-833. MR 2993752, https://doi.org/10.4007/annals.2012.175.2.7
  • [8] Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. MR 0394451
  • [9] G. Drugan, Embedded $ S^2$ self-shrinkers with rotational symmetry, preprint.
  • [10] Gregory Drugan, An immersed $ S^2$ self-shrinker, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3139-3159. MR 3314804, https://doi.org/10.1090/S0002-9947-2014-06051-0
  • [11] Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2024995
  • [12] Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453-471. MR 1025164, https://doi.org/10.2307/1971452
  • [13] C. L. Epstein and M. I. Weinstein, A stable manifold theorem for the curve shortening equation, Comm. Pure Appl. Math. 40 (1987), no. 1, 119-139. MR 865360, https://doi.org/10.1002/cpa.3160400106
  • [14] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69-96. MR 840401
  • [15] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • [16] Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285-314. MR 906392
  • [17] Hoeskuldur P. Halldorsson, Self-similar solutions to the curve shortening flow, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5285-5309. MR 2931330, https://doi.org/10.1090/S0002-9947-2012-05632-7
  • [18] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • [19] Heinz Hopf, Differential geometry in the large, 2nd ed., Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1989. Notes taken by Peter Lax and John W. Gray; With a preface by S. S. Chern; With a preface by K. Voss. MR 1013786
  • [20] Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285-299. MR 1030675
  • [21] N. Kapouleas, S. J. Kleene, and N. M. Møller, Mean curvature self-shrinkers of high genus: Non-compact examples, preprint. Available at arXiv:1106.5454.
  • [22] Stephen Kleene and Niels Martin Møller, Self-shrinkers with a rotational symmetry, Trans. Amer. Math. Soc. 366 (2014), no. 8, 3943-3963. MR 3206448, https://doi.org/10.1090/S0002-9947-2014-05721-8
  • [23] N. M. Møller, Closed self-shrinking surfaces in $ R^3$ via the torus, preprint. Available at arXiv:1111.7318.
  • [24] Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. I, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1683-1701. MR 2465812, https://doi.org/10.1090/S0002-9947-08-04748-X
  • [25] Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. II, Adv. Differential Equations 15 (2010), no. 5-6, 503-530. MR 2643233
  • [26] Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow, Part III, Duke Math. J. 163 (2014), no. 11, 2023-2056. MR 3263027, https://doi.org/10.1215/00127094-2795108
  • [27] Lu Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297-303. MR 2780753, https://doi.org/10.1007/s10711-010-9535-2

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 53C42

Retrieve articles in all journals with MSC (2010): 53C44, 53C42


Additional Information

Gregory Drugan
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: drugan@math.washington.edu

Stephen J. Kleene
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: skleene@math.mit.edu

DOI: https://doi.org/10.1090/tran/6907
Keywords: Mean curvature flow, self-shrinker
Received by editor(s): June 22, 2013
Received by editor(s) in revised form: November 12, 2015, December 28, 2015, and January 4, 2016
Published electronically: June 27, 2017
Additional Notes: The first author was partially supported by NSF RTG 0838212.
The second author was partially supported by NSF DMS 1004646.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society