Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T06:21:44.331Z Has data issue: false hasContentIssue false
  • English
  • Français

SELF-SHRINKERS WITH SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH

Part of: Global differential geometry

Published online by Cambridge University Press:  02 March 2017

QIANG GUANG
QIANG GUANG*
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA email guang@math.ucsb.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a new and simple proof of a result of Ding and Xin, which states that any smooth complete self-shrinker in $\mathbb{R}^{3}$ with the second fundamental form of constant length must be a generalised cylinder $\mathbb{S}^{k}\times \mathbb{R}^{2-k}$ for some $k\leq 2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

Keywords

self-shrinkersecond fundamental formmean curvature flow

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 326 - 332
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Abresch, U. and Langer, J., ‘The normalized curve shortening flow and homothetic solutions’, J. Differential Geom. 23(2) (1986), 175–196.Google Scholar
Angenent, S. B., ‘Shrinking doughnuts’, in: Nonlinear Diffusion Equations and their Equilibrium States, 3 (Gregynog, 1989), Progress in Nonlinear Differential Equations and their Applications, 7 (Birkhäuser, Boston, 1992), 21–38.CrossRefGoogle Scholar
Brendle, S., ‘Embedded self-similar shrinkers of genus 0’, Ann. of Math. (2) 183(2) (2016), 715–728.Google Scholar
Cao, H. and Li, H., ‘A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension’, Calc. Var. Partial Differential Equations 46(3–4) (2013), 879–889.Google Scholar
Cheng, X. and Zhou, D., ‘Volume estimate about shrinkers’, Proc. Amer. Math. Soc. 141(2) (2013), 687–696.Google Scholar
Chopp, D., ‘Computation of self-similar solutions for mean curvature flow’, Experiment. Math. 3(1) (1994), 1–15.Google Scholar
Colding, T. H., Ilmanen, T. and Minicozzi, W. P. II, ‘Rigidity of generic singularities of mean curvature flow’, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 363–382.Google Scholar
Colding, T. H. and Minicozzi, W. P. II, ‘Generic mean curvature flow I: Generic singularities’, Ann. of Math. (2) 175(2) (2012), 755–833.Google Scholar
Colding, T. H. and Minicozzi, W. P. II, ‘Smooth compactness of self-shrinkers’, Comment. Math. Helv. 87(2) (2012), 463–475.Google Scholar
Ding, Q. and Xin, Y. L., ‘Volume growth, eigenvalue and compactness for self-shrinkers’, Asian J. Math. 17(3) (2013), 443–456.Google Scholar
Ding, Q. and Xin, Y. L., ‘The rigidity theorems of self-shrinkers’, Trans. Amer. Math. Soc. 366(10) (2014), 5067–5085.Google Scholar
Huisken, G., ‘Asymptotic behavior for singularities of the mean curvature flow’, J. Differential Geom. 31(1) (1990), 285–299.Google Scholar
Huisken, G., ‘Local and global behaviour of hypersurfaces moving by mean curvature’, in: Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proceedings of Symposia in Pure Mathematics, 54 (American Mathematical Society, Providence, RI, 1993), 175–191.Google Scholar
Ilmanen, T., ‘Singularities of mean curvature flow of surfaces’, Preprint, 1997.Google Scholar
Kleene, S. and Møller, N. M., ‘Self-shrinkers with a rotational symmetry’, Trans. Amer. Math. Soc. 366(8) (2014), 3943–3963.Google Scholar
Le, N. Q. and Sesum, N., ‘Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers’, Comm. Anal. Geom. 19(4) (2011), 633–659.Google Scholar
Møller, N. M., ‘Closed self-shrinking surfaces in $\mathbf{R}^{3}$ via the torus’, Preprint, 2011,arXiv:1111.7318.Google Scholar
Nguyen, X. H., ‘Construction of complete embedded self-similar surfaces under mean curvature flow. I’, Trans. Amer. Math. Soc. 361(4) (2009), 1683–1701.CrossRefGoogle Scholar
White, B., ‘Partial regularity of mean-convex hypersurfaces flowing by mean curvature’, Int. Math. Res. Not. IMRN 4 (1994), 185–192.CrossRefGoogle Scholar