Metric geometry

I may have made a stupid mistake, but if not, we could construct a metric by pullback a metric on a suitable linear normalized space $latex H$ which we carefully constructed. Let we define the generators of free group $latex F_2$ by $latex a,b$. Step 1. Constructed the linear normalized space $latex H$. the space $latex H$ was spanned by basis $latex \Lambda=\Lambda_a \coprod \Lambda_b$, $latex \Lambda_a, \Lambda_b$ are defined by look at the Cayley graph of $latex F_2$, there is a lot of vertic…

Gromov’s idea applicate to parabolic equation. Key point: 1.rescaling+renormalization. 2.analysis it on every scale.

There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $latex \mathbb{S}_n \to \mathbb{R}^{n+1}$. Is it possible for us to compute the Gromov-Hausdorff distance $latex d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $latex \mathbb{S}_n$ and $latex \mathbb{S}_m$, $latex m\neq n$? For example if we want to calculate $latex d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathb…