The rigidity theorems of self-shrinkers

Ding, Qi; Xin, Y. L.

doi:10.1090/S0002-9947-2014-05901-1

The rigidity theorems of self-shrinkers
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by Qi Ding and Y. L. Xin PDF

Trans. Amer. Math. Soc. 366 (2014), 5067-5085 Request permission

Abstract:

By using certain techniques developed in minimal submanifold theory, we study the rigidity problem for self-shrinkers in the present paper. We prove rigidity results for the squared norm of the second fundamental form of self-shrinkers, either under pointwise conditions or under integral conditions.

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
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, DOI 10.1007/s00526-012-0508-1
Qing Chen and Sen Lin Xu, Rigidity of compact minimal submanifolds in a unit sphere, Geom. Dedicata 45 (1993), no. 1, 83–88. MR 1199729, DOI 10.1007/BF01667404
S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR 0273546
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
Tobias H. Colding and William P. Minicozzi II, Smooth compactness of self-shrinkers, Comment. Math. Helv. 87 (2012), no. 2, 463–475. MR 2914856, DOI 10.4171/CMH/260
Qi Ding and Y. L. Xin, On Chern’s problem for rigidity of minimal hypersurfaces in the spheres, Adv. Math. 227 (2011), no. 1, 131–145. MR 2782189, DOI 10.1016/j.aim.2011.01.018
Qi Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443–456. MR 3119795, DOI 10.4310/AJM.2013.v17.n3.a3
Klaus Ecker, Logarithmic Sobolev inequalities on submanifolds of Euclidean space, J. Reine Angew. Math. 522 (2000), 105–118. MR 1758578, DOI 10.1515/crll.2000.033
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
H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
Li An-Min and Li Jimin, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. (Basel) 58 (1992), no. 6, 582–594. MR 1161925, DOI 10.1007/BF01193528
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, DOI 10.4310/CAG.2011.v19.n4.a1
J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
Lei Ni, Gap theorems for minimal submanifolds in $\textbf {R}^{n+1}$, Comm. Anal. Geom. 9 (2001), no. 3, 641–656. MR 1895136, DOI 10.4310/CAG.2001.v9.n3.a2
Chia-Kuei Peng and Chuu-Lian Terng, Minimal hypersurfaces of spheres with constant scalar curvature, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 177–198. MR 795235
Chia Kuei Peng and Chuu-Lian Terng, The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1983), no. 1, 105–113. MR 722930, DOI 10.1007/BF01458707
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
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
Yuanlong Xin, Mean curvature flow with convex Gauss image, Chinese Ann. Math. Ser. B 29 (2008), no. 2, 121–134. MR 2392328, DOI 10.1007/s11401-007-0212-1
Yuanlong Xin, Minimal submanifolds and related topics, Nankai Tracts in Mathematics, vol. 8, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 2035469, DOI 10.1142/9789812564382
Shing Tung Yau, Submanifolds with constant mean curvature. I, II, Amer. J. Math. 96 (1974), 346–366; ibid. 97 (1975), 76–100. MR 370443, DOI 10.2307/2373638