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Estimates of Gromov's box distance

Author: Kei Funano
Journal: Proc. Amer. Math. Soc. 136 (2008), 2911-2920
MSC (2000): Primary 28E99, 53C23
Published electronically: April 11, 2008
MathSciNet review: 2399058
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Abstract: In 1999, M. Gromov introduced the box distance function $ \mathop{\underline{\square}_{\lambda}}\nolimits$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate $ \mathop{\underline{\square}_{\lambda}}\nolimits (\mathbb{S}^n,\mathbb{S}^m)$ and $ \mathop{\underline{\square}_{\lambda}}\nolimits(\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 $ \mathop{\underline{\square}_{\lambda}}\nolimits\big(SO(n), SO(m)\big)$ from below, where $ SO(n)$ is the special orthogonal group.

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Kei Funano
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Keywords: mm-space, box distance function, observable distance function
Received by editor(s): June 18, 2007
Published electronically: April 11, 2008
Additional Notes: This work was partially supported by research fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Dedicated: This paper is dedicated to our advisors.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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