## 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

## Additional Information

**Gregory Drugan**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- MR Author ID: 1097133
- Email: drugan@math.washington.edu, drugan@uoregon.edu
- Received by editor(s): June 6, 2012
- Received by editor(s) in revised form: November 10, 2012
- Published electronically: December 18, 2014
- Additional Notes: This work was partially supported by NSF RTG [DMS-0838212].
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**367**(2015), 3139-3159 - MSC (2010): Primary 53C44, 53C42
- DOI: https://doi.org/10.1090/S0002-9947-2014-06051-0
- MathSciNet review: 3314804