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'''Definition 1'''
 
'''Definition 1'''
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Let $m\leq n$. A m-[[Rectifiable varifold]] is a couple $(M, \theta)$, where $M$ is a m-dimensional [[Rectifiable set]] in $\mathbb R^n$ and $\theta$ is a $\mathcal H^m$ measurable function defined on $M$, called the density function. A varifold is calle integral rectifiable if $\theta$ is integer valued.
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Let $U \subset \mathbb R^n. A [[Rectifiable varifold]] of dimension $m$ in $U$ is a couple $(M, \theta)$, where $M\subset U$ is an m-dimensional [[Rectifiable set]] and $\theta\colon M \to \mathbb R_+$ is a $\mathcal H^m$ measurable function, called density function. A varifold is called integral rectifiable if $\theta$ is integer valued.
  
  

Revision as of 09:58, 11 September 2012

2020 Mathematics Subject Classification: Primary: 49Q15 [MSN][ZBL]


Rectifiable varifolds are a generalization of rectifiable sets in the sense that they allow for a density function to be defined on the set. They are also strictly connected to rectifiable currents, in fact to such a current one can always associate a varifold by putting aside the orientation.

Definitions

Definition 1 Let $U \subset \mathbb R^n. A [[Rectifiable varifold]] of dimension $m$ in $U$ is a couple $(M, \theta)$, where $M\subset U$ is an m-dimensional [[Rectifiable set]] and $\theta\colon M \to \mathbb R_+$ is a $\mathcal H^m$ measurable function, called density function. A varifold is called integral rectifiable if $\theta$ is integer valued.


First Variation and Stationariety

Allard's Regularity Theorem

References

[Sim] Leon Simon, "Lectures on Geometric Measure Theory". Proceedings of the centre for Mathematical Analysis. Australian National University, Canberra, 1983. MR0756417Zbl 0546.49019
[FX] Lin Fanghua, Yang Xiaoping, "Geometric Measure Theory-An Introduction". Advanced Mathematics Vol.1. International Press, Boston, 2002. MR2030862Zbl 1074.49011
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