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A Bernstein type theorem for self-similar shrinkers

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Abstract

In this paper, we prove that smooth self-shrinkers in \({\mathbb R^{n+1}}\), that are entire graphs, are hyperplanes. Previously, Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this paper is that no growth assumption at infinity is needed.

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References

  1. Angenent, S.B.: Shrinking doughnuts. In: Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Gregynog, 1989), Progress in Nonlinear Differential Equations & Their Applications, vol. 7, pp. 21–38. Birkhuser Boston, Boston, MA (1992)

  2. Colding, T.H., Minicozzi II, W.P. : Generic Mean Curvature Flow I; Generic Singularities. Preprint available at arXiv:0908.3788

  3. Colding, T.H., Minicozzi II, W.P. : Smooth compactness of self-shrinkers. Comment. Math. Helv. (to appear)

  4. Colding T.H., Minicozzi II W.P.: Minimal Surfaces, vol. 4 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York (1999)

    Google Scholar 

  5. Colding T.H., Minicozzi II W.P.: Sharp estimates for mean curvature flow of graphs. J. Reine Angew. Math. 574, 187–195 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ecker K.: Regularity Theory for Mean Curvature Flow, vol. 57 of Progress in Nonlinear Differential Equations and Their Applications. Birkhuser Boston, Inc, Boston, MA (2004)

    Google Scholar 

  7. Ecker K., Huisken G.: Mean curvature evolution of entire graphs. Ann. Math. (2) 130(3), 453–471 (1989)

    Article  MathSciNet  Google Scholar 

  8. Ecker K., Huisken G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105(3), 547–569 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bombieri E., De Giorgi E., Miranda M.: Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Ration. Mech. Anal. 32, 255–267 (1969) (Italian)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. In: Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proceedings of the Symposia in Pure Mathematics, vol. 54. Part 1, pp. 175–191. Am. Math. Soc. Providence, RI (1993)

  13. Schoen R., Simon L., Yau S.T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134(3–4), 275–288 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Simons J.: Minimal varieties in Riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Lu Wang

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  1. Lu Wang
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Correspondence to Lu Wang.

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Wang, L. A Bernstein type theorem for self-similar shrinkers. Geom Dedicata 151, 297–303 (2011). https://doi.org/10.1007/s10711-010-9535-2

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  • DOI: https://doi.org/10.1007/s10711-010-9535-2

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