Abstract:We generalize Colding and Minicozzi’s work (2012) on the stability of hypersurface self-shrinkers to higher co-dimension. The first and second variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general co-dimension is found. We also prove that $\mathbb R^n$ is the only stable product self-shrinker and show that the closed embedded Lagrangian self-shrinkers constructed by Anciaux are unstable.
- 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
- Henri Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\Bbb C^n$, Geom. Dedicata 120 (2006), 37–48. MR 2252892, DOI 10.1007/s10711-006-9082-z
- 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
- S. Angenent, T. Ilmanen, and D. L. Chopp, A computed example of nonuniqueness of mean curvature flow in $\mathbf R^3$, Comm. Partial Differential Equations 20 (1995), no. 11-12, 1937–1958. MR 1361726, DOI 10.1080/03605309508821158
- B. Andrews, H. Li and Y. Wei, $F$-stability for self-shrinking solutions to mean curvature flow, arXiv:1204.5010.
- 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
- Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
- Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
- Gerhard Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR 1216584, DOI 10.1090/pspum/054.1/1216584
- T. Ilmanen, Singularities of mean curvature flow of surfaces, preprint, 1995, http://www.math.ethz.ch/ ilmanen/papers/pub.html.
- Dominic Joyce, Yng-Ing Lee, and Mao-Pei Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84 (2010), no. 1, 127–161. MR 2629511
- Yng-Ing Lee and Mu-Tao Wang, Hamiltonian stationary cones and self-similar solutions in higher dimension, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1491–1503. MR 2563738, DOI 10.1090/S0002-9947-09-04795-3
- Knut Smoczyk, Self-shrinkers of the mean curvature flow in arbitrary codimension, Int. Math. Res. Not. 48 (2005), 2983–3004. MR 2189784, DOI 10.1155/IMRN.2005.2983
- Yng-Ing Lee
- Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan – and – National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan
- Email: firstname.lastname@example.org
- Yang-Kai Lue
- Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan
- Address at time of publication: Department of Mathematics, National Taiwan Normal University, Hsinchu, Taiwan
- Email: email@example.com
- Received by editor(s): July 18, 2012
- Published electronically: November 24, 2014
- © 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), 2411-2435
- MSC (2010): Primary 53C44, 35C06
- DOI: https://doi.org/10.1090/S0002-9947-2014-05969-2
- MathSciNet review: 3301868