I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.




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### Recapitulation


for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E\_0 \cup \bigcup\_{k\in\mathbb N} E\_k$ with $\mathcal H^m(E\_0)=0$ and $E\_k\subseteq F\_k$ for some $\mathcal C^1$-manifolds $F\_k$ of dimension $m$, and a non-negativ function $\theta\in L^1\_{\text{loc}}(\mathcal H^m|\_E)$ such that $\mu=\theta \mathcal H^m|\_E$. This is a characterisation of $m$-recitifiable varifolds. 
The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1\_c(\mathbb R^n;\mathbb R^n)$ given by
$$\delta\mu(\eta)=\int div\_\mu\eta\,d\mu,$$
where $ div\_\mu(\eta)(x) = \sum\_{i=1}^n \tau\_i^T(x)\cdot D\eta(x)\cdot \tau\_i(x)$ where $\tau\_i(x)$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T\_xF\_i$ for $x\in E\_i\subseteq F\_i$ as above. So $div\_\mu\eta(x)$ is just the divergence in the manifold $F\_i$, with $x\in E\_i\subseteq F\_i$.


We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that
$$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert\_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1\_c(\Omega'). $$
See for more explanation for example <http://eom.springer.de/G/g130040.htm>.


For a $\mathcal C^2$-manifold $M$ in $\mathbb R^n$ with mean curvature $H\_M$ the first variation is
$$ \delta M(\eta)=-\int\_M H\_M \cdot \eta \,dvol\_M -\int\_{\partial M} \tau\_0 \cdot \eta \,dvol\_{\partial M} \qquad\forall\;\eta\in\mathcal C\_c^1(\mathbb R^n)$$
with the inner normal $\tau\_0\in T\_xM\cap(T\_x\partial M)^\bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H\_M(x)=\sum\_{i=1}^m A\_x(\tau\_i,\tau\_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.



Example for an integral, rectifiable varifold with unbounded first variation
Something is strange here: it seems like for the sawtooth curve (the Lipschitz curve that goes up and down with slope $1$), the first variation is just the sum of $\delta$-measures at the turning points times the unit bisector vectors, so we can have fixed length and arbitrarily large first variation (just make turning points more and more dense), which can be now trivially turned into an example of finite area and infinite first variation: take more and more rigged closed sawtooth curves around infinitely many circles with finite some of radii contained in a compact domain. Am I missing something? 


