Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On a classification theorem for self-shrinkers


Author: Michele Rimoldi
Journal: Proc. Amer. Math. Soc. 142 (2014), 3605-3613
MSC (2010): Primary 53C44, 53C21
DOI: https://doi.org/10.1090/S0002-9939-2014-12074-0
Published electronically: June 12, 2014
MathSciNet review: 3238436
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by Colding and Minicozzi, by replacing the assumption on polynomial volume growth with a weighted $ L^2$ condition on the norm of the second fundamental form. Our approach adopts the viewpoint of weighted manifolds and also permits us to recover and to extend some other recent classification and gap results for self-shrinkers.


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 (88d:53001)
  • [2] Huai-Dong Cao and Haizhong Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (2013), no. 3-4, 879-889. MR 3018176, https://doi.org/10.1007/s00526-012-0508-1
  • [3] Q.-M. Cheng and Y. Peng, Complete self-shrinkers of the mean curvature flow, Calc. Var. Partial Differential Equations, to appear, DOI 10.1007/s00526-014-0720-2, http://link.springer.com/article/10.1007%2Fs00526-014-0720-2
  • [4] Xu Cheng and Detang Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc. 141 (2013), no. 2, 687-696. MR 2996973, https://doi.org/10.1090/S0002-9939-2012-11922-7
  • [5] 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
  • [6] Q. Ding and Y.L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443-456. MR 3119795
  • [7] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. MR 664497 (84a:53050)
  • [8] 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 (94c:58037)
  • [9] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266. MR 772132 (86j:53097)
  • [10] Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285-299. MR 1030675 (90m:53016)
  • [11] H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187-197. MR 0238229 (38 #6505)
  • [12] Nam Q. Le and Natasa Sesum, Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers, Comm. Anal. Geom. 19 (2011), no. 4, 633-659. MR 2880211
  • [13] Stefano Pigola, Marco Rigoli, Michele Rimoldi, and Alberto G. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 4, 757-799. MR 2932893
  • [14] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Vanishing theorems on Riemannian manifolds, and geometric applications, J. Funct. Anal. 229 (2005), no. 2, 424-461. MR 2182595 (2006k:53055), https://doi.org/10.1016/j.jfa.2005.05.007
  • [15] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Vanishing and finiteness results in geometric analysis, A generalization of the Bochner technique, Progress in Mathematics, vol. 266, Birkhäuser Verlag, Basel, 2008. MR 2401291 (2009m:58001)
  • [16] Stefano Pigola and Giona Veronelli, Remarks on $ L^p$-vanishing results in geometric analysis, Internat. J. Math. 23 (2012), no. 1, 1250008, 18 pp.. MR 2888937, https://doi.org/10.1142/S0129167X11007513
  • [17] Zhongmin Qian, On conservation of probability and the Feller property, Ann. Probab. 24 (1996), no. 1, 280-292. MR 1387636 (97d:58206), https://doi.org/10.1214/aop/1042644717
  • [18] Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. MR 2577473 (2011a:53064)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C44, 53C21

Retrieve articles in all journals with MSC (2010): 53C44, 53C21


Additional Information

Michele Rimoldi
Affiliation: Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via Valleggio 11, I-22100 Como, Italy
Address at time of publication: Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, via Cozzi, 55, I-20125 Milano, Italy
Email: michele.rimoldi@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12074-0
Keywords: Self--shrinkers, classification, weighted manifolds
Received by editor(s): July 10, 2012
Received by editor(s) in revised form: October 17, 2012
Published electronically: June 12, 2014
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society