An immersed 𝑆² self-shrinker

Drugan, Gregory

doi:10.1090/S0002-9947-2014-06051-0

An immersed $S^2$ self-shrinker
HTML articles powered by AMS MathViewer

by Gregory Drugan PDF

Trans. Amer. Math. Soc. 367 (2015), 3139-3159 Request permission

Abstract:

We construct an immersed and non-embedded $S^2$ self-shrinker.

References

U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704, DOI 10.4310/jdg/1214440025
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
M. S. Baouendi and C. Goulaouic, Singular nonlinear Cauchy problems, J. Differential Equations 22 (1976), no. 2, 268–291. MR 435564, DOI 10.1016/0022-0396(76)90028-0
David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1–15. MR 1302814, DOI 10.1080/10586458.1994.10504572
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, DOI 10.4007/annals.2012.175.2.7
G. Drugan, Embedded $S^2$ self-shrinkers with rotational symmetry, preprint. Available at http://www.math.washington.edu/~drugan/papers.
Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2024995, DOI 10.1007/978-0-8176-8210-1
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, DOI 10.1002/cpa.3160400106
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
N. Kapouleas, S. J. Kleene, N. M. Møller, Mean curvature self-shrinkers of high genus: Non-compact examples, preprint. Available at arXiv:1106.5454.
Stephen Kleene and Niels Martin Møller, Self-shrinkers with a rotational symmetry, Trans. Amer. Math. Soc. 366 (2014), no. 8, 3943–3963. MR 3206448, DOI 10.1090/S0002-9947-2014-05721-8
N. M. Møller, Closed self-shrinking surfaces in $R^3$ via the torus, preprint. Available at arXiv:1111.7318.
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, DOI 10.1090/S0002-9947-08-04748-X
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
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, DOI 10.1215/00127094-2795108
Lu Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297–303. MR 2780753, DOI 10.1007/s10711-010-9535-2