Metric geometry

Title: Constructing a Metric by Pullback on a Linear Normalized Space In this article, we aim to construct a metric by pulling back a metric on a suitable linear normalized space $H$ that we carefully constructed. We begin by defining the generators of the free group $F_2$ as $a$ and $b$. Step 1: Constructing the Linear Normalized Space $H We construct the linear normalized space $H$, which is spanned by the basis $\Lambda = \Lambda_a \cup \Lambda_b$. Here, $\Lambda_a$ and $\Lambda_b$ are define…

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 $\mathbb{S}n o \mathbb{R}^{n+1}$. Is it possible for us to compute the Gromov-Hausdorff distance $d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $meq n$? For example, if we want to calculate $d_{G-H}(\mathbb{S}2,\mathbb{S}3)=\inf{M,f,g}d{M}(\mathbb{S}_2,\mathbb{S}_3)$, where $M$ ranges over…