Timeline for Does nonexpanding map between manifolds decrease volume?
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| S Sep 28, 2016 at 12:41 | history | bounty ended | CommunityBot | ||
| S Sep 28, 2016 at 12:41 | history | notice removed | CommunityBot | ||
| S Sep 20, 2016 at 11:30 | history | bounty started | Asaf Shachar | ||
| S Sep 20, 2016 at 11:30 | history | notice added | Asaf Shachar | Draw attention | |
| Sep 13, 2016 at 16:52 | history | edited | Asaf Shachar | CC BY-SA 3.0 |
Added pointers to relevant examples given so far. Clarified some conditions concerning the question.
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| Sep 9, 2016 at 20:56 | comment | added | Neal | (This is less general than what you are asking, however.) | |
| Sep 9, 2016 at 20:55 | comment | added | Neal | I think the answer is "yes" in the case that $f$ is actually a diffeomorphism. Compare the metric $g$ on $M$ to the pullback $h$ by $f$ of the metric on $N$. By the hypothesis that $f$ is nonexpanding, the pointwise eigenvalues of $h$ with respect to $g$ are all no greater than $1$. The hypothesis on distance implies the $h$-geodesic $\gamma$ connecting $p,q$ has shorter $h$-length than $g$-length, so by looking at the arc-length integral we should see at least one pointwise eigenvalue of $h$ wrt $g$ is strictly less than $1$. The volume inequality should follow by comparing volume forms. | |
| Sep 9, 2016 at 19:02 | answer | added | Nik Weaver | timeline score: 3 | |
| Sep 9, 2016 at 15:32 | comment | added | Asaf Shachar | @Neal: Actually, even though I am not assuming smoothness, $f$ is in fact differentiable almost everywhere (via Rademacher's theorem). It is not hard to show that at each point of differentiability $p \in M$, the singular values of the differential $df_p:T_pM \to T_{f(p)}N$ are not greater than one. | |
| Sep 9, 2016 at 14:58 | comment | added | Asaf Shachar | Yes, I do not assume the map is smooth. | |
| Sep 9, 2016 at 14:56 | comment | added | Neal | Do you continue to not assume $f$ is smooth? | |
| Sep 9, 2016 at 14:33 | history | asked | Asaf Shachar | CC BY-SA 3.0 |