• Zhao, Peibiao (Department of Applied Mathematics Nanjing University of Science and Technology) ;
  • Yang, Xiaoping (Department of Applied Mathematics Nanjing University of Science and Technology)
  • Published : 2007.02.28


We prove the following theorem: Given a Varifold V in $l^{3}_{\infty}$ with the property that 0 < $lim_{r}_{\rightarrow}_{o}\;\frac{{\mu}v(C_{r}(x))}{r^{2}}\;<\;{\infty}\;for\;{\mu}v\;a.e.x\;{\in}$ SptV, then V is rectifiable.


Varifolds;tangent measures;rectifiable sets;rectifiable measures


  1. W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491
  2. L. Ambrosio, M. Gobbino, and D. Pallara, Approximation problems for curvature vari-folds, J. Geom. Anal. 8 (1998), no. 1, 1-19
  3. L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1-80
  4. A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann. 98 (1928), no. 1, 422-464
  5. A. S. Besicovitc, On the fundamental geometrical properties of linearly measurable plane sets of points (II), Math. Ann. 115 (1938), no. 1, 296-329
  6. G. David and S. Semmes, Singular integrals and rectifiable sets in $R^n$: Beyond Lipschitz graphs. Ast risque 193, Soc. Math. France, 1991
  7. H. Federer, The $(\varphi,\kappa)$ rectifiable subsets of n-space, Amer. Math. Soc. 62 (1947), 114-192
  8. H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissensch-aften, Band 153 Springer-Verlag New York Inc., New York, 1969
  9. F. H. Lin, Varifold type theory for Sobolev mappings, First International Congress of Chinese Mathematicians (Beijing, 1998), 423-430, AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001
  10. F. H. Lin and X. P. Yang, Geometric measure theory|an introduction, Advanced Math-ematics (Beijing/Boston), 1. Science Press, Beijing; International Press, Boston, MA, 2002
  11. J. M. Marstrand, Hausdorff two-dimensional measure in 3-space, Proc. London Math. Soc. (3) 11 (1961), 91-108
  12. P. Mattila, Hausdorffm regular and rectifiable sets in n-space, Trans. Amer. Math. Soc. 205 (1975), 263-274
  13. E. F. Morse, Density ratios and $(\phi,1)$ rectifiability in n-space, Trans. Amer. Math. Soc. 69 (1950), 324-334
  14. P. Morters and D. Preiss, Tangent measure distributions of fractal measures, Math. Ann. 312 (1998), no. 1, 53-93
  15. T. D. Pauw, Nearly flat almost monotone measures are big pieces of Lipschitz graphs, J. Geom. Anal. 12 (2002), no. 1, 29-61
  16. D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), no. 3, 537-643
  17. I. Rubinstein and L. Rubinstein, Partial Differential Equations in Classical Mathenatical Physics, Cambridge University Press, 1998
  18. L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Math-ematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983
  19. P. B. Zhao and X. P. Yang, Some remarks on currents in metric spaces, Southeast Asian Bulletin of Mathematics, 29 (2005), no. 5, 1011-1021
  20. P. B. Zhao and X. P. Yang, Geometric Analysis of Tangent Merasures, Chinese Annals of mathematics 26 (2005), no. 2, 151-164
  21. P. B. Zhao and X. P. Yang, Marstrand Theorem for Cube in $R^d$ with respect to Varifolds, in preparation
  22. L. Ambrosio and B. Kirchhei, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527-555
  23. K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986
  24. A. Lorent, A Marstrand type theorem for measures with cube density in general dimension, Math. Proc. Camb. Phil. Soc. 137 (2004), no. 3, 657-696
  25. P. Mattila, Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cam-bridge, 1995
  26. A. Lorent, Rectifiability of measures with locally uniform cube density, Proc. London. Math. Soc. 86 (2003), no. 1, 153-249