On the Complete 2-Dimensional λ-Translators with a Second Fundamental form of Constant Length

Acta Mathematica Scientia volume 40, pages1897–1914(2020)Cite this article

Abstract

In this article we study the two-dimensional complete λ-translators immersed in the Euclidean space ℝ3 and Minkovski space ℝ31 . We obtain two classification theorems: one for two-dimensional complete λ-translators x: M2 → ℝ3 and one for two-dimensional complete space-like λ-translators x: M2 → ℝ31 , with a second fundamental form of constant length.

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References

  1. [1]

    Altschuler S J, Wu L F. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc Var, 1994, 2: 101–111

    MathSciNet  Article  Google Scholar 

  2. [2]

    Chen Q, Qiu H B. Rigidity of self-shrinkers and translating solitons of mean curvature flows. Adv Math, 2016, 294: 517–531

    MathSciNet  Article  Google Scholar 

  3. [3]

    Cheng Q -M, Ogata S. 2-dimensional complete self-shrinkers in ℝ3. Math Z, 2016, 284: 537–542

    MathSciNet  Article  Google Scholar 

  4. [4]

    Cheng Q-M, Ogata S, Wei G X. Rigidity theorems of λ-hypersurfaces. Comm Anal Geom, 2016, 24: 45–58

    MathSciNet  Article  Google Scholar 

  5. [5]

    Cheng Q -M, Wei G X. Complete λ-surfaces in ℝ3. arXiv:1807.06760v1 [math.DG], 2018

  6. [6]

    Cheng Q -M, Wei G X. Complete λ-hypersurfaces of the weighted volume-preserving mean curvature flow. Calc Var, 2018, 57(2): Art 32, DOI https://doi.org/10.1007/s00526-018-1303-4

  7. [7]

    Clutterbuck J, Schnürer O, F Schulze. Stability of translating solutions to mean curvature flow. Calc Var, 2007, 29: 281–293

    MathSciNet  Article  Google Scholar 

  8. [8]

    Gromov M. Isoperimetry of waists and concentration of maps. Geom Func Anal, 2003, 13: 178–215

    MathSciNet  Article  Google Scholar 

  9. [9]

    Halldorsson H P. Helicoidal surfaces rotating/translating under the mean curvature flow. Geom Dedicata, 2013, 162: 45–65

    MathSciNet  Article  Google Scholar 

  10. [10]

    Huisken G, Sinestrari C. Mean curvature flow singularities for mean convex surfaces. Calc Var, 1999, 8: 1–14

    MathSciNet  Article  Google Scholar 

  11. [11]

    Ilmanen T. Elliptic regularization and partial regularity for motion by mean curvature. Mem Amer Math Soc, 1994, 108

  12. [12]

    Li X X, Li X. On the Lagrangian angle and the Kühler angle of immersed surfaces in the complex plane C2. Acta Math Sci, 2019, 39B(6): 1695–1712

    Article  Google Scholar 

  13. [13]

    López R. Invariant surfaces in Euclidean space with a log-linear density. Adv Math, 2018, 339: 285–309

    MathSciNet  Article  Google Scholar 

  14. [14]

    López R. Compact λ-translating solitons with boundary. Mediterranean J Math, 2018, 15(5): Art 196

    Google Scholar 

  15. [15]

    Martín F, Savas-Halilaj A, Smoczyk K. On the topology of translating solitons of the mean curvature flow. Calc Var, 2015, 54: 2853–2882

    MathSciNet  Article  Google Scholar 

  16. [16]

    Minh N, Hieu D T. Ruled minimal surfaces in R3 with density ez. Pacific J Math, 2009, 243: 277–285

    MathSciNet  Article  Google Scholar 

  17. [17]

    Morgan F. Manifolds with density. Notices Amer Math Soc, 2005, 52: 853–858

    MathSciNet  MATH  Google Scholar 

  18. [18]

    Pyo J. Compact translating solitons with non-empty planar boundary. Diff Geom App, 2016, 47: 79–85

    MathSciNet  Article  Google Scholar 

  19. [19]

    Shahriyari L. Translating Graphs by Mean Curvature Flow [D]. The Johns Hopkins University, 2013

  20. [20]

    Smith G. On complete embedded translating solitons of the mean curvature flow that area of finite genus. arXiv:1501.04149 [math.DG], 2015

  21. [21]

    Wang X-J. Convex solutions to the mean curvature flow. Ann Math, 2011, 173: 1185–1239

    MathSciNet  Article  Google Scholar 

  22. [22]

    White B. Subsequent singularities in mean-convex mean curvature flow. Calc Var Pertial Diff Equ, 2015, 54(2): 1457–1468

    MathSciNet  Article  Google Scholar 

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Affiliations

  1. School of Mathematics and Information Sciences, Henan Normal University, Xinxiang, 453007, China

    Xingxiao Li, Ruina Qiao & Yangyang Liu

Authors
  1. Xingxiao Li
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  2. Ruina Qiao
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  3. Yangyang Liu
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Corresponding author

Correspondence to Xingxiao Li.

Additional information

Supported by Foundation of Natural Sciences of China (11671121, 11871197 and 11971153).

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Li, X., Qiao, R. & Liu, Y. On the Complete 2-Dimensional λ-Translators with a Second Fundamental form of Constant Length. Acta Math Sci 40, 1897–1914 (2020). https://doi.org/10.1007/s10473-020-0618-3

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Key words

  • singular solution
  • mean curvature flow
  • second fundamental form
  • λ-translator
  • classification

2010 MR Subject Classification

  • 53C44
  • 53C40