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2010 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 ${\bf M}(V):=\mu_V(U)$. The support of a varifold is defined by ${\rm spt}(V):={\rm spt}(\mu_V)$.

We can also define almost everywhere the tangent plane to a varifold by setting $T_xV:=T_xM$ at each point $x\in {\rm spt}(V)$ where the tangent plane to $M$ exists (see Rectifiable set, Proposition 7).

The following are standard construction of Geometric Measure Theory:

  • Restriction of a varifold to a set $A \subset \mathbb \R^n$ defined by

$$V A:=(A\cap M,\theta|_A)$$

  • The push-forward of a varifold via a smooth map defined by

$$f_\#(V):=(f(M), \tilde{\theta}), \quad \text{where }\tilde{\theta}(y):=\sum_{x\in f^{-1}(y)}\theta(x).$$

First Variation and Stationariety

Consider a vector field $X\in C^1_c(U; \mathbb R^n)$, then we can define a one parameter family of diffeomorphism $\phi_t(x)=\phi(x,t)\colon U\times \mathbb R \to U$ as the unique solution of the system $$\left\{ \begin{array}{cc} \frac{\partial\phi}{\partial t}= X(\phi)\\ \phi(x,0)=x \end{array} \right. $$

Now the first variation of the varifold is given by $$\delta V(X):=\frac{d}{dt}\Big|_{t=0}{\rm M}(\phi_\# V)$$ which corresponds essentially to the first variation of the Area Functionals.

Definition 2 A rectifiable varifold $V$ is called a stationary varifold if $$\delta V(X)=0 \quad \text{for every }X \in C^1_c(U; \mathbb R^n).$$

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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Luca.Spolaor/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luca.Spolaor/sandbox&oldid=27916