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Halfspace type theorems for self-shrinkers in arbitrary codimension

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In this paper, we generalize some halfspace type theorems for self-shrinkers of codimension 1 to the case of arbitrary codimension.

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References

  1. Arezzo, C., Sun, J.: Self-shrinkers for the mean curvature flow in arbitrary codimension. Math. Z. 274(3–4), 993–1027 (2013)

    Article  MathSciNet  Google Scholar 

  2. Cavalcante, M.P., Espinar, J.M.: Halfspace type theorems for self-shrinkers. Bull. Lond. Math. Soc. 48(2), 242–250 (2016)

    Article  MathSciNet  Google Scholar 

  3. Cheng, Q.-M., Peng, Y.: Complete self-shrinkers of the mean curvature flow. Calc. Var. Partial Differ. Equ. 52(3–4), 497–506 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cheng, X., Zhou, D.: Volume estimate about shrinkers. Proc. Am. Math. Soc. 141(2), 687–696 (2013)

    Article  MathSciNet  Google Scholar 

  5. Colding, T.H., Minicozzi, W.P., II.: A Course in Minimal Surfaces, Graduate Studies in Mathematics, vol. 121. American Mathematical Society, Providence (2011)

    Google Scholar 

  6. Colding, T.H., Minicozzi, W.P., II.: Smooth compactness of self-shrinkers. Comment. Math. Helv. 87(2), 463–475 (2012)

    Article  MathSciNet  Google Scholar 

  7. Colding, T.H., Minicozzi, W.P.: Generic mean curvature flow I generic singularities. Ann. Math. 175(2), 755–833 (2012)

    Article  MathSciNet  Google Scholar 

  8. Daniel, B., Hauswirth, L.: Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group. Proc. Lond. Math. Soc. 98(2), 445–470 (2009)

    Article  MathSciNet  Google Scholar 

  9. Daniel, B., Meeks, W.H., III., Rosenberg, H.: Half-space theorems for minimal surfaces in \({\rm Nil}_3\) and \({\rm Sol}_3,\). J. Differ. Geom. 88(1), 41–59 (2011)

    Article  Google Scholar 

  10. Ding, Q., Xin, Y.L.: Volume growth, eigenvalue and compactness for self-shrinkers. Asian J. Math. 17(3), 443–456 (2013)

    Article  MathSciNet  Google Scholar 

  11. Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 130, 453–471 (1989)

    Article  MathSciNet  Google Scholar 

  12. Gimeno, V., Palmer, V.: Parabolicity, Brownian exit time and properness of solitons of the direct and inverse mean curvature flow. J. Geom. Anal. 31(1), 579–618 (2021)

    Article  MathSciNet  Google Scholar 

  13. Hieu, D.T.: A weighted volume estimate and its application to Bernstein type theorems in Gauss space Colloq. Math. 159(1), 25–28 (2020)

    MathSciNet  Google Scholar 

  14. Hoffman, D., Meeks, W.H., III.: The strong halfspace theorem for minimal surfaces. Invent. Math. 101(2), 373–377 (1990)

    Article  MathSciNet  Google Scholar 

  15. Hurtado, A., Palmer, V., Rosales, C.: Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights. Nonlinear Anal. 192, 111681 (2020)

    Article  MathSciNet  Google Scholar 

  16. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285–299 (1990)

    Article  MathSciNet  Google Scholar 

  17. Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature, In: Greene, R., Yau, S.T. (eds.), Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, Proceedings of Symposium on Pure Mathematics, vol. 54, Amer. Math. Soc, Providence (1990)

  18. Impera, D., Pigola, S., Rimoldi, M.: The Frankel property for self-shrinkers from the viewpoint of elliptic PDE’s. J. Reine Angew. Math. 773, 1–20 (2021)

    Article  MathSciNet  Google Scholar 

  19. Mazet, L.: A general halfspace theorem for constant mean curvature surfaces. Am. J. Math. 135, 801–834 (2013)

    Article  MathSciNet  Google Scholar 

  20. Nelli, B., Sa Earp, R.: A halfspace theorem for mean curvature \(H = 1/2\) surfaces in \(H^2\times {\mathbb{R} }\). J. Math. Anal. Appl. 365(1), 167–170 (2010)

    Article  MathSciNet  Google Scholar 

  21. Pigola, S., Rimoldi, M.: Complete self-shrinkers confined into some regions of the space. Ann. Global Anal. Geom. 45, 47–65 (2014)

    Article  MathSciNet  Google Scholar 

  22. Rodriguez, L., Rosenberg, H.: Half-space theorems for mean curvature one surfaces in hyperbolic space. Proc. Am. Math. Soc. 126(9), 2755–2762 (1998)

    Article  MathSciNet  Google Scholar 

  23. Rosenberg, H., Schultze, F., Spruck, J.: The halfspace property and entire minimal graphs in \(M \times {\mathbb{R} },\). J. Differ. Geom. 95, 321–336 (2013)

    Article  Google Scholar 

  24. Smoczyk, K.: Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 48, 2983–3004 (2005)

    Article  MathSciNet  Google Scholar 

  25. Xin, Y.: Minimal Submanifolds and Related Topics, [MR2035469], 2nd edn. Nankai Tracts in Mathematics, vol. 16, World Scientific Publishing Co. Pte. Ltd., Hackensack (2019)

  26. Vieira, M., Zhou, D.: Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons. J. Geom. Anal. 28(1), 170–189 (2018)

    Article  MathSciNet  Google Scholar 

  27. Wang, L.: A Bernstein type theorem for self-similar shrinkers. Geom. Dedicata 151, 297–303 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the reviewer for carefully reading the paper and for his/her valuable comments and suggestions which helped to improve the manuscript.

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  1. Department of Mathematics, College of Education, Hue University, Hue, Vietnam

    Hieu T. Doan & Duyen T. M. Nguyen

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  1. Hieu T. Doan
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  2. Duyen T. M. Nguyen
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Correspondence to Hieu T. Doan.

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Doan, H.T., Nguyen, D.T.M. Halfspace type theorems for self-shrinkers in arbitrary codimension. Collect. Math. 75, 425–435 (2024). https://doi.org/10.1007/s13348-023-00393-w

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