I want to prove that $D$, standard unit ball in ${\bf R}^2$ with $|\ |$, with a metric $\| \ \|_1$ is a limit of Riemannian manifolds $X_i$. Here problem is to find $X_i$
(If necessary, all metrics in here is strictly intrinsic)
Proof : $Y:=\{ (x,y)| x$ or $y$ is an integer$\}$ has a metric $d=\|\ \|_1$ : $$d((a,b),(x,y))=|a-x| + |b-y|$$
Then $$ \lim_i \ \bigg(Y,\frac{1}{i} d\bigg)=({\bf R}^2,\|\ \|_1)$$
( This is followed from asymptotic cone : Asymptotic cone of a metric space $(X,d)$ is $\lim_{t>0,\ t\rightarrow 0} \ (X,td)$ )
Definition of limit of metric space : $(X_i=X,d_i)$ is a metric space Then $(X,d)$ is a limit if $$ {\rm sup}_{x,\ x'\in X} \ |d_i(x,x')-d(x,x')| \rightarrow 0 $$
But $Y$ is a not Riemannian manifolds (Not smooth) But from this we consider $g_n$ in ${\bf R}^2$ : On $Y$, $g_n$ conincides to standard inner product $g$ and $g_n$ is strictly larger then $g$ on ${\bf R}^2-Y $
That is consider a function : If $d_n$ is a metric from $ng$, define $$ f_n(p):=d_{ng}(p,Y)^{2n} $$
If $F_n\geq 0$ with $|f_n-F_n| < \frac{1}{n}$ is smooth, then define $$ g_n = \frac{g+F_ng}{n^2} $$
Here I have a question : $g_n$ is smooth ? (That is, it is a Riemannian metric)
And we must show that $d_{g_n}\rightarrow \|\ \|_1$ By construction, in $g_n$, one coordinate of shortest path from $(0,0)$ to $(1,1)$ is close to an integer and its length is close to $\frac{2}{n}$.
Here $(x,y)\in D$ is corresponded to $(nx,ny)$ in $X_n={\bf R}^2$. I believe that $X_\infty$ is $({\bf R}^2,\|\ \|_1)$. So why do we consider $D$ not ${\bf R}^2$ ?
My idea is right ? Thank you in anticipation (reference : A course in metric geometry - D. Burago, Y. Burago, and S. Ivanov 248p. exe 7.1.3)