Geometric measure theory formulas on rectifiable metric spaces

2007 ◽  
pp. 103-136 ◽  
Author(s):  
Maria Karmanova
Keyword(s):  
Metric Spaces ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Geometric Measure

RECTIFIABILITY OF MEASURES WITH LOCALLY UNIFORM CUBE DENSITY

2003 ◽  
Vol 86 (1) ◽  
pp. 153-249 ◽  
Author(s):  
ANDREW LORENT
Keyword(s):  
Euclidean Space ◽  
Isometric Immersion ◽  
Metric Spaces ◽  
Model Problem ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Initial Condition ◽  
Radon Measures ◽  
Geometric Measure ◽  
Euclidean Unit Ball

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question.In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space.2000 Mathematical Subject Classification: 28A75.

scholarly journals Some fine properties of currents and applications to distributional Jacobians

2002 ◽  
Vol 132 (4) ◽  
pp. 815-842 ◽  
Author(s):  
Camillo De Lellis
Keyword(s):  
Metric Spaces ◽  
Special Functions ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Finite Mass ◽  
Geometric Measure ◽  
Closure Theorem

We study fine properties of currents in the framework of geometric measure theory on metric spaces developed by Ambrosio and Kirchheim, and we prove a rectifiability criterion for flat currents of finite mass. We apply these tools to study the structure of the distributional Jacobians of functions in the space BnV, defined by Jerrard and Soner. We define the subspace of special functions of bounded higher variation and we prove a closure theorem.

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