I originally was wondering what is a varifold because I wanted to know about the Plateau's problem (if the boundary of curve )is something pathological like a tentacle shape. Hence, I am now seeing the concept that is related to the geometrical measure theory. Anyway, my question that is interesting but difficult to understand while learning is the concept of the blowing-up of a measure in a point, like the title. According to the Toro's lecture, when she proved the Marstrand's theorem( or see Thm1.1 on Marco Inversi's PDF), she refers to the keyword blowing up $\nu$ at $y_0$.( she changed some notations during the lecture). Then what is blowing-up in this sense? To begin with, I, naturally, imagine the blowing-up in the sense of algebraic geometry and 'resolution of singularity' comes to mind. My main question is here : What is the relation between the blowing-up process about the measure in point and blowing-up in algebraical geometrical sense?.
Actually, according to the Marco, he stated what is the meaning of blowing-up in the Remark 1.18.
- Notation : $\mu_{x,r}(A):= \mu_(x+rA)$ where $A \in \mathbb{R}^n$ is a Borel set.
- Blowing-up the measure $\mu$ blowing up at the point $x$ means that we want to study the limiting behaviour of $\mu_{x,r}/r^\alpha $ rα as $r$ goes to $0$. By notiation $\mu_{x,r}(B_1)/r^\alpha = \mu (B_r(x))/r^\alpha$. And as $r$ goes to $0$. the numerator goes to the measure of the point x and the denominator blows up to $\infty$...
When I read Remark1.18, the concept of blowing-up seems to be not related to the algebraic geometry sense.. just literally, the fraction blows up when denominator goes to $0$ . However, when I see the Proposition1.19 and its proof, these might be similar to some extent.
Prop1.19[Tangent measure to a $C^1$ submanifold] Let $\sum$ be $k$-submanifold of $\mathbb{R}^n$ of class $C^1$(without boundary). Letting $\mu:= \mathcal{H}^k\llcorner \sum $, the following hold true :
- for all $x\in \sum$ and for all $r>0$, $\mu_{x,r}/r^k = \mathcal{H}^k\llcorner(\frac{\sum-x}{r}) $
- for all $x\in \sum$ , $\mathcal{H}^k\llcorner(\frac{\sum-x}{r}) \overset{*}{\rightharpoonup} \mathcal{H}^k\llcorner Tan_x \sum $, as $r$ goes $0$
[the proof the proposition of Prop1.19] : ... We denote as $B_\delta^k$ the $k$-dimensional ball centered at the origin in $R^ k$ of radius $\delta$. As for the second statement, we can make the following assumptions:
- $x=0$,
- $Tan_x \sum =\operatorname{Span}(c_1,...c_k)= \mathbb{R}^k $ where $\mathbb{R}^{n}=R^k \times \mathbb{R}^{n-k}$
- there exists $\delta \ge 0$ and a $C^1$ map $\Phi :$B_\delta^{k}$ \to B_\delta ^{n-k}$ such that $\Gamma \cap ( B_\delta^{k} \times B_\delta ^{n-k}) $ is the graph of $\Phi$, that is ...
In other words, the use of the graph $\Gamma$ , and projection map $\pi: \mathbb{R}^{n}=\mathbb{R}^k \times \mathbb{R}^{n-k} \to \mathbb{R}^k $ ) evoke the blow up in the sense of algebraical geometry.
