Abstract
A concentrated (ξ, m) almost monotone measure inR n is a Radon measure Φ satisfying the two following conditions: (1) Θm(Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R n the ratioexp [ξ(r)]r−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R n and a λ-Lipschitz function f from x+W into x+W‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allard, W.K. On the first variation of a varifold,Ann. of Math.,95(2), 417–491, (1972).
Almgren, F.J.The theory of varifolds, Mimeographed notes, (1965).
Almgren, F.J. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Memoirs of the A.M.S., no. 165, American Math. Soc., (1976).
Almgren, F.J., Taylor, J.E., and Wang, L. Curvature-driven flows: a variational approach,SIAM J. Control and Opt.,31(2), 387–437, (1993).
Federer, H.Geometric Measure Theory, Springer-Verlag, (1969).
Huovinen, P. and Kirchheim, B. Personal communication, (1998).
Kowalski, O. and Preiss, D. Besicovitch-type properties of measures and submanifolds,J. Reine Angew. Math.,379, 115–151, (1987).
Marstrand, J.M. Hausdorff two-dimensional measure in 3 space,Proc. London Math. Soc.,11(3), 91–108, (1961).
Mattila, P. Hausdorffm regular and rectifiable sets inn space,Trans. Am. Math. Soc.,205, 263–274, (1975).
Mattila, P.Geometry of Sets and Measures in Euclidean Spaces-Fractals and Rectifiability, Cambridge University Press, (1995).
O’Neil, T. A measure with a large set of tangent measures,Proc. Am. Math. Soc.,123(7), 2217–2220, (1995).
Preiss, D. Geometry of measures inR n: distribution, rectifiability, and densities,Ann. of Math.,125 (2), 537–643, (1987).
Reifenberg, R.E. Solutions of the plateau problem for m-dimensional surfaces of varying topological typeActa Math.,104, 1–92, (1960).
Taylor, J.E. The structure of singularities in soap-bubble-like and soap-film-like minimal surfacesAnn. of Math.,103(2), 489–539, (1976).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Pauw, T. Nearly flat almost monotone measures are big pieces of lipschitz graphs. J Geom Anal 12, 29–61 (2002). https://doi.org/10.1007/BF02930859
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02930859