Question :
There is a sequence of Riemannian metrics on 3-dimensional sphere s.t. they converges to four dimensional ball in Gromov-Hausdorff $d_{GH}$ sense. How can we prove this ? (cf. 105p. in [1])
Background
This question is out from the following theorem : If $(X,g_i)$ is a sequence of closed Riemannian manifolds of dimension $>3$, then $(X,g_i)\rightarrow (X_0,d)$ where $d$ is a path metric iff there is a continuous map $X\rightarrow X_0$ s.t. it induces a sujective map on fundamental groups
Examples :
Note that Riemannian manifolds $(S^2,g_i)$ does not converge to $n$-dimensional ball for $n=2,\ 3$.
There is a sequence of three dimensional spheres converging to three dimensional ball (cf. [2]).
Proof : If $D$ is unit disc in $\mathbb{E}^3$, then $N$ is $1/i$-net for $D$ s.t. $N$ does not contain origin. Hence each $x\in N$ has a foot $x_f$ in $\partial D$. If $(xx_f)$ is an open $1/i^2$-tubular neighborhood of a segment $[xx_f]$, then $X:=D-\bigcup_{x\in N}\ (xx_f)$ is homeomorphic to unit disc. And $S:=X\bigcup_{\partial X}X$ is 3-dimensional sphere. Here $d_{GH} (D,X),\ d_{GH}(X,S)$ are small.
We have a sequence of two dimensional closed surfaces converging to a three dimensional ball.
Proof : Consider $1/i$-net $N$, a set of finite points, in $D^3$ so that we construct a graph $G$ by connecting the points. Hence ${\rm incl} :G\rightarrow D^3$ is $1/i$-isometry. So the boundary of $1/i$-tubular neighborhood of ${\rm incl}\ (G)$ is desired.
[1] Metric structures for Riemannian and Non-Riemannian spaces
[2] A course of metric geometry - D. Burago, Y. Burago, and S. Ivanov
References :
(1) Gromov-Hausdorff convergence to non-manifolds - Moore
(2) Approximation topological metrics by Riemannian metrics - Ferry and Okun