Estimates of Gromov’s box distance

Abstract:

In 1999, M. Gromov introduced the box distance function $\underline {\square }_\lambda$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate $\underline {\square }_\lambda (\mathbb {S}^n,\mathbb {S}^m)$ and $\underline {\square }_\lambda (\mathbb {C}P^n, \mathbb {C}P^m)$, where $\mathbb {S}^n$ is the $n$-dimensional unit sphere in $\mathbb {R}^{n+1}$ and $\mathbb {C}P^n$ is the $n$-dimensional complex projective space equipped with the Fubini-Study metric. In particular, we give the complete answer to an exercise of Gromov’s green book. We also estimate $\underline {\square }_\lambda \big (\operatorname {SO}(n), \operatorname {SO}(m)\big )$ from below, where $\operatorname {SO}(n)$ is the special orthogonal group.