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

### 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.

`$D(\eta\tau_i)(x)$`

stands for here? Also, it would be great if you could tell what it all means for regular smooth manifolds (if I understand it right, we just want to show that certain quantitative estimate fails, i.e., we may have small area and big "first variation" (whatever that means)). $\endgroup$ – fedja Jan 28 '10 at 16:10