Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The extension and convergence of mean curvature flow in higher codimension


Authors: Kefeng Liu, Hongwei Xu, Fei Ye and Entao Zhao
Journal: Trans. Amer. Math. Soc. 370 (2018), 2231-2262
MSC (2010): Primary 53C44, 53C40
DOI: https://doi.org/10.1090/tran/7281
Published electronically: November 1, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the convergence of the mean curvature flow of closed submanifolds in $ \mathbb{R}^{n+q}$. We show that if the initial submanifold satisfies some suitable integral curvature conditions, then along the mean curvature flow it will shrink to a round point in finite time.


References [Enhancements On Off] (What's this?)

  • [1] Ben Andrews and Charles Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differential Geom. 85 (2010), no. 3, 357-395. MR 2739807
  • [2] Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 485012
  • [3] Bang-yen Chen, On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof, Math. Ann. 194 (1971), 19-26. MR 0291994, https://doi.org/10.1007/BF01351818
  • [4] Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749-786. MR 1100211
  • [5] Jingyi Chen and Weiyong He, A note on singular time of mean curvature flow, Math. Z. 266 (2010), no. 4, 921-931. MR 2729297, https://doi.org/10.1007/s00209-009-0604-x
  • [6] Xianzhe Dai, Guofang Wei, and Rugang Ye, Smoothing Riemannian metrics with Ricci curvature bounds, Manuscripta Math. 90 (1996), no. 1, 49-61. MR 1387754, https://doi.org/10.1007/BF02568293
  • [7] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635-681. MR 1100206
  • [8] David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727. MR 0365424, https://doi.org/10.1002/cpa.3160270601
  • [9] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266. MR 772132
  • [10] Gerhard Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463-480. MR 837523, https://doi.org/10.1007/BF01388742
  • [11] Gerhard Huisken, Deforming hypersurfaces of the sphere by their mean curvature, Math. Z. 195 (1987), no. 2, 205-219. MR 892052, https://doi.org/10.1007/BF01166458
  • [12] Nam Q. Le and Natasa Sesum, On the extension of the mean curvature flow, Math. Z. 267 (2011), no. 3-4, 583-604. MR 2776050, https://doi.org/10.1007/s00209-009-0637-1
  • [13] 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 0344978, https://doi.org/10.1002/cpa.3160260305
  • [14] W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900-904. MR 0078836
  • [15] Nataša Šešum, Curvature tensor under the Ricci flow, Amer. J. Math. 127 (2005), no. 6, 1315-1324. MR 2183526
  • [16] Katsuhiro Shiohama and Hong Wei Xu, Rigidity and sphere theorems for submanifolds, I, Kyushu J. Math. 48 (1994), no. 2, 291-306, MR 1294532; II, Kyushu J. Math. 54 (2000), no. 2, 221-232, MR 1762795,
  • [17] Katsuhiro Shiohama and Hongwei Xu, The topological sphere theorem for complete submanifolds, Compositio Math. 107 (1997), no. 2, 221-232. MR 1458750, https://doi.org/10.1023/A:1000189116072
  • [18] Knut Smoczyk, Longtime existence of the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 20 (2004), no. 1, 25-46. MR 2047144, https://doi.org/10.1007/s00526-003-0226-9
  • [19] Knut Smoczyk and Mu-Tao Wang, Mean curvature flows of Lagrangians submanifolds with convex potentials, J. Differential Geom. 62 (2002), no. 2, 243-257. MR 1988504
  • [20] Peter Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539-546. MR 2410779, https://doi.org/10.4171/CMH/135
  • [21] Bing Wang, On the conditions to extend Ricci flow, Int. Math. Res. Not. IMRN 8 (2008), Art. ID rnn012, 30. MR 2428146, https://doi.org/10.1093/imrn/rnn012
  • [22] Mu-Tao Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J. Differential Geom. 57 (2001), no. 2, 301-338. MR 1879229
  • [23] Mu-Tao Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. Math. 148 (2002), no. 3, 525-543. MR 1908059, https://doi.org/10.1007/s002220100201
  • [24] Mu-Tao Wang, Lectures on mean curvature flows in higher codimensions, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 525-543. MR 2483374
  • [25] Hong Wei Xu, $ L_{n/2}$-pinching theorems for submanifolds with parallel mean curvature in a sphere, J. Math. Soc. Japan 46 (1994), no. 3, 503-515. MR 1276835, https://doi.org/10.2969/jmsj/04630503
  • [26] Hong-Wei Xu and Juan-Ru Gu, An optimal differentiable sphere theorem for complete manifolds, Math. Res. Lett. 17 (2010), no. 6, 1111-1124. MR 2729635, https://doi.org/10.4310/MRL.2010.v17.n6.a10
  • [27] Hong-Wei Xu, Fei Ye, and En-Tao Zhao, Extend mean curvature flow with finite integral curvature, Asian J. Math. 15 (2011), no. 4, 549-556. MR 2853649, https://doi.org/10.4310/AJM.2011.v15.n4.a4
  • [28] HongWei Xu, Fei Ye, and EnTao Zhao, The extension for mean curvature flow with finite integral curvature in Riemannian manifolds, Sci. China Math. 54 (2011), no. 10, 2195-2204. MR 2838131, https://doi.org/10.1007/s11425-011-4244-3
  • [29] Hong-Wei Xu and En-Tao Zhao, Topological and differentiable sphere theorems for complete submanifolds, Comm. Anal. Geom. 17 (2009), no. 3, 565-585. MR 2550209, https://doi.org/10.4310/CAG.2009.v17.n3.a6
  • [30] Xi-Ping Zhu, Lectures on mean curvature flows, AMS/IP Studies in Advanced Mathematics, vol. 32, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1931534

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 53C40

Retrieve articles in all journals with MSC (2010): 53C44, 53C40


Additional Information

Kefeng Liu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China – and – Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California, 90095-1555
Email: kefeng@zju.edu.cn, liu@math.ucla.edu

Hongwei Xu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: xuhw@zju.edu.cn

Fei Ye
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: yf@cms.zju.edu.cn, flemmingye@163.com

Entao Zhao
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: zhaoet@zju.edu.cn

DOI: https://doi.org/10.1090/tran/7281
Keywords: Mean curvature flow, submanifold, maximal existence time, convergence theorem, integral curvature
Received by editor(s): September 23, 2016
Received by editor(s) in revised form: January 23, 2017
Published electronically: November 1, 2017
Additional Notes: This research was supported by the National Natural Science Foundation of China, Grant Nos. 11531012, 11371315, 11201416.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society