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Non-unique conical and non-conical tangents to rectifiable stationary varifolds in \(\mathbb R^4\)

  • Jan KolářEmail author
Article

Abstract

We construct a rectifiable stationary 2-varifold in \({\mathbb {R}}^4\) with non-conical, and hence non-unique, tangent varifold at a point. This answers a question of Simon (Lectures on geometric measure theory, p 243, 1983) and provides a new example for a related question of Allard (Ann Math (2) 95(3):417–491, 1972, p 460). There is also a (rectifiable) stationary 2-varifold in \({\mathbb {R}}^{4}\) that has more than one conical tangent varifold at a point.

Mathematics Subject Classification

28A75 49Q20 35B65 

References

  1. 1.
    Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95(3), 417–491 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allard, W.K., Almgren, F.J.: The structure of stationary one dimensional varifolds with positive density. Invent. Math. 34(2), 83–97 (1976). doi: 10.1007/BF01425476 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allard, W.K., Almgren Jr, F.J.: On the radial behavior of minimal surfaces and the uniqueness of their tangent cones. Ann. Math. 113, 215–265 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bellettini, C.: Tangent cones to positive-(1, 1) De Rham currents. J. Reine Angew. MathGoogle Scholar
  5. 5.
    Bellettini, C.: Uniqueness of tangent cones to positive-\((p, p)\) integral cycles. Duke Math. J. 163, 705–732 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bellettini, C.: Semi-calibrated 2-currents are pseudoholomorphic, with applications. Bull. Lond. Math. Soc. 46(4), 881–888 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blel, M.: Sur le cône tangent à un courant positif fermé. J. Math. Pures Appl. 72, 517–536 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brakke, K.A.: The motion of a surface by its mean curvature. In: Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)Google Scholar
  9. 9.
    Brendle, S.: Embedded minimal tori in \(S^3\) and the Lawson conjecture. arXiv:1203.6597
  10. 10.
    Chang, S.X.-D.: Two-dimensional area minimizing integral currents are classical minimal surfaces. J. Am. Math. Soc. 1(4), 699–778 (1988)zbMATHGoogle Scholar
  11. 11.
    Černý, R., Kolář, J., Rokyta, M.: Concentrated monotone measures with non-unique tangential behavior in \(R^3\). Czechoslov. Math. J. 61(4), 1141–1167 (2011)Google Scholar
  12. 12.
    Černý, R., Kolář, J., Rokyta, M.: Monotone measures with bad tangential behavior in the plane. Comment. Math. Univ. Carol. 52(3), 317–339 (2011)zbMATHGoogle Scholar
  13. 13.
    De Lellis, C., Spadaro, E.N.: Q-valued functions revisited. Mem. Am. Math. Soc. 211(991), vi+79 (2011). ISBN 978-0-8218-4914-9. arXiv:0803.0060v4
  14. 14.
    Federer, H.: Geometric Measure Theory. Springer, New York (1969)zbMATHGoogle Scholar
  15. 15.
    Harvey, R., Lawson Jr, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hutchinson, J.E., Meier, M.: A remark on the nonuniqueness of tangent cones. Proc. Am. Math. Soc. 97(1), 184–185 (1986)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kiselman, C.O.: Tangents of plurisubharmonic functions. In: International Symposium in Memory of Hua Loo Keng (August, 1988), vol. II, pp. 157–167. Science Press and Springer, New York (1991)Google Scholar
  18. 18.
    Kolář, J.: Non-regular tangential behaviour of a monotone measure. Bull. Lond. Math. Soc. 38, 657–666 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Lawlor, G.: The angle criterion. Invent. math. 95, 437–446 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    O’Neil, T.C.: Geometric measure theory, online at Encyclopedia of Mathematics. http://www.encyclopediaofmath.org/index.php?title=Geometric_measure_theory&oldid=28204. (First appeared in Supplement III of the Encyclopedia of Mathematics. Kluwer Academic Publishers, Kluwer, 2002)
  21. 21.
    Osserman, R.: Minimal varieties. Bull. Am. Math. Soc. 75(6), 1092–1120 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pumberger, D., Riviére, T.: Uniqueness of tangent cones for semi-calibrated 2-cycles. Duke Math. J. 152(3), 441–480 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, vol. 3. Australian National University, Canberra (1983)Google Scholar
  24. 24.
    Simon, L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. Second Ser. 118(3), 525–571 (1983)CrossRefzbMATHGoogle Scholar
  25. 25.
    Simon, L.: Cylindrical tangent cones and the singular set of minimal submanifolds. J. Differ. Geom. 38, 585–652 (1993)zbMATHGoogle Scholar
  26. 26.
    Simon, L.: Uniqueness of some cylindrical tangent cones. Commun. Anal. Geom. 2(1), 1–33 (1994)zbMATHGoogle Scholar
  27. 27.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Taylor, J.E.: The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. Math. (2) 103(3), 489–539 (1976)CrossRefzbMATHGoogle Scholar
  29. 29.
    White, B.: Tangent cones to two-dimensional area-minimizing integral currents are unique. Duke Math. J. 50(1), 143–160 (1983). doi: 10.1215/S0012-7094-83-05005-6
  30. 30.
    White, B.: The mathematics of F. J. Almgren, Jr. J. Geom. Anal 8(5), 681–702 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of MathematicsCzech Academy of SciencesPrague 1Czech Republic

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