Complete self-shrinkers of the mean curvature flow

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• 作者：Qing-Ming Cheng (1)
  Yejuan Peng (2)
  
  1. Department of Applied Mathematics ; Faculty of Sciences ; Fukuoka University ; Fukuoka ; 814-0180 ; Japan
  2. Department of Mathematics ; Henan Normal University ; Xinxiang ; 453007 ; Henan ; People鈥檚 Republic of China
  
• 关键词：53C44 ; 53C40
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：52
• 期：3-4
• 页码：497-506
• 全文大小：175 KB
• 参考文献：1. Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Diff. Geom. 23, 175鈥?96 (1986)
  2. Angenent, S.: Shrinking doughnuts. In Nonlinear Diffusion Equations and Their Equilibrium States, vol 7. Boston-Basel-Berlin: Birkha眉ser, pp. 21鈥?8 (1992)
  3. Bryant, R.: Minimal surfaces of constant curvature in \(S^n\) . Trans. Am. Math. Soc. 290, 259鈥?71 (1985)
  4. Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Differ. Geom. 1, 111鈥?25 (1967)
  5. Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Differ. Equ. 46, 879鈥?89 (2013) CrossRef
  6. Chang, S.: On minimal hypersurfaces with constant scalar curvatures in \(S^4\) . J. Differ. Geom. 37, 523鈥?34 (1993)
  7. Cheng, Q. -M., Peng, Y.: Estimates for eigenvalues of \(\cal L\) operator on self-shrinkers. Commun. Contemp. Math. 15(6), 1350011 (2013). doi:10.1142/S0219199713500119
  8. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333鈥?54 (1975) CrossRef
  9. Cheng, X., Zhou, D.: Volume estimate about shrinkers. Proc. Am. Math. Soc. 141, 687鈥?96 (2013). arXiv:1106.4950
  10. Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I; Generic singularities. Ann. Math. 175, 755鈥?33 (2012) CrossRef
  11. Ding, Q., Xin, Y.L.: Volume growth, eigenvalue and compactness for self-shrinkers. Asian J. Math. 17(3), 391鈥?82 (2013) CrossRef
  12. Q. Ding and Y. L. Xin, The rigidity theorems of self shrinkers, to appear in, Trans. Am. Math. Soc., arXiv:1105.4962v1 (2011)
  13. Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 130, 453鈥?71 (1989) CrossRef
  14. Huisken, G.: Flow by mean curvature convex surfaces into spheres. J. Differ. Geom. 20, 237鈥?66 (1984)
  15. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285鈥?99 (1990)
  16. Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, : Proc. Sympos. Pure Math., 54, Part 1, Am. Math. Soc. Providence, 1993), pp. 175鈥?91 (1990)
  17. Kenmotsu, K.: On minimal immersions of \(\mathbb{R}^2\) into \(S^N\) . J. Math. Soc. Jpn. 28, 182鈥?91 (1976) CrossRef
  18. Kleene, S., M酶ller, N.M.: Self-shrinkers with a rotation symmetry, to appear in Trans. Am. Math. Soc., arXiv:1008.1609v2 (2012)
  19. Lawson, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. 89, 187鈥?97 (1969) CrossRef
  20. Le, N.Q., 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), 633鈥?59 (2011)
  21. Li, H., Wei, Y.: Classification and rigidity of self-shrinkers in the mean curvature flow. to appear in J. Math. Soc. Japan., arXiv:1201.4623 (2012)
  22. Smoczyk, K.: Self-Shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Notices 48, 2983鈥?004 (2005) CrossRef
  23. Wallach, N.R.: Extension of locally defined minimal immersions of spheres into spheres. Arch. Math. 21, 210鈥?13 (1970) CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for \(\mathcal {L}\) -operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on polynomial volume growth, which is assumed in Cao and Li [5]. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on polynomial volume growth. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.