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http://dx.doi.org/10.4134/BKMS.2007.44.1.031

RECTIFIABILITY PROPERTIES OF VARIFOLDS IN l3∞  

Zhao, Peibiao (Department of Applied Mathematics Nanjing University of Science and Technology)
Yang, Xiaoping (Department of Applied Mathematics Nanjing University of Science and Technology)
Publication Information
Bulletin of the Korean Mathematical Society / v.44, no.1, 2007 , pp. 31-45 More about this Journal
Abstract
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.
Keywords
Varifolds; tangent measures; rectifiable sets; rectifiable measures;
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1 A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann. 98 (1928), no. 1, 422-464   DOI
2 A. S. Besicovitc, On the fundamental geometrical properties of linearly measurable plane sets of points (II), Math. Ann. 115 (1938), no. 1, 296-329   DOI
3 G. David and S. Semmes, Singular integrals and rectifiable sets in $R^n$: Beyond Lipschitz graphs. Ast risque 193, Soc. Math. France, 1991
4 J. M. Marstrand, Hausdorff two-dimensional measure in 3-space, Proc. London Math. Soc. (3) 11 (1961), 91-108   DOI
5 P. Mattila, Hausdorffm regular and rectifiable sets in n-space, Trans. Amer. Math. Soc. 205 (1975), 263-274   DOI   ScienceOn
6 E. F. Morse, Density ratios and $(\phi,1)$ rectifiability in n-space, Trans. Amer. Math. Soc. 69 (1950), 324-334   DOI   ScienceOn
7 P. Morters and D. Preiss, Tangent measure distributions of fractal measures, Math. Ann. 312 (1998), no. 1, 53-93   DOI
8 T. D. Pauw, Nearly flat almost monotone measures are big pieces of Lipschitz graphs, J. Geom. Anal. 12 (2002), no. 1, 29-61   DOI   ScienceOn
9 H. Federer, The $(\varphi,\kappa)$ rectifiable subsets of n-space, Amer. Math. Soc. 62 (1947), 114-192   DOI   ScienceOn
10 H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissensch-aften, Band 153 Springer-Verlag New York Inc., New York, 1969
11 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
12 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
13 P. B. Zhao and X. P. Yang, Marstrand Theorem for Cube in $R^d$ with respect to Varifolds, in preparation
14 D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), no. 3, 537-643   DOI
15 I. Rubinstein and L. Rubinstein, Partial Differential Equations in Classical Mathenatical Physics, Cambridge University Press, 1998
16 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
17 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
18 P. B. Zhao and X. P. Yang, Geometric Analysis of Tangent Merasures, Chinese Annals of mathematics 26 (2005), no. 2, 151-164
19 W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491   DOI
20 L. Ambrosio, M. Gobbino, and D. Pallara, Approximation problems for curvature vari-folds, J. Geom. Anal. 8 (1998), no. 1, 1-19   DOI
21 L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1-80   DOI
22 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
23 A. Lorent, Rectifiability of measures with locally uniform cube density, Proc. London. Math. Soc. 86 (2003), no. 1, 153-249   DOI
24 L. Ambrosio and B. Kirchhei, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527-555   DOI
25 K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986
26 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   DOI   ScienceOn