Existence of stationary tangent cones

Question

My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42

So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)>0$ and $\lim_{r\searrow 0} r^{1-n} \|\delta V\|(B_r(x))=0$ $(\ast)$ then one can use the standard compactness theorem to select a sequence $\lambda_j\searrow 0$, s.t. $\eta_{x,\lambda_j\#}V$ converges to a limit Varifold $C$. $\eta$ the typical blow-up function.

My question: Why is $C$ stationary in $\mathbb{R}^{n+k}$?? (i.e. $\delta C(X)=0 \; \forall X\in C^1_c(\mathbb{R}^{n+k},\mathbb{R}^{n+k})$)

I think it has something to do with $(\ast)$, but how goes the exact argument??

Answer 1

You can easily check the following things:

1) if $ V_{i} $ and $ V $ are varifolds in an open set $ U $ and $ V_{i} \to V $ then $\| \delta V \|(G)\leq\liminf_{i \to 0}\| \delta V_{i} \|(G) $ whenever $ G \subseteq U $ is open.

2) if $ V $ is a $ k $ dimensional varifold in $ U $ and $ 0 < r < \infty $ then $ \| \delta(\mu_{r,\sharp}V) \| = r^{k-1}\mu_{r,\sharp}\| \delta V \| $, where $ \mu_{r}(x) = rx $.

Therefore if $ C \in TanVar(V,0) $ and $ \Theta^{k-1}(\| \delta V \|,0) =0 $ then there exists a sequence $ r_{i} \to \infty $ such that $ C = \lim_{i \to 0 } \mu_{r_{i},\sharp}V $ and

$ \| \delta C \|(B_{R} ) \leq \liminf_{i \to \infty}r_{i}^{k-1}\| \delta V \|( B_{R/r_{i}}) =0 $ whenever $ 0 < R < \infty $.