User:Luca.Spolaor/sandbox

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 $V$ 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.

To a varifold we can naturally associate a measure defined and denoted by $$\mu_V(A):=\int_{M\cap A} \theta \,d\mathcal H^m,\quad \text{for every Borel set } A\subset \mathbb R^n.$$ The mass of the varifold is defined by ${\bf M}(V):=\mu_V(U)$.

We can also define almost everywhere the tangent plane to a varifold by setting $T_xV:=T_xM$ for each point where the tangent plane to $M$ exists (see Rectifiable set, Proposition 7).

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