Timeline for Recovering the length metric from Hausdorff measure
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 6, 2021 at 16:39 | comment | added | Anton Petrunin | A simpler way: it is well known that any noncontracting map from compact metric space to itself is an isometry --- apply it to the partial inverse of $f\colon X \to X$. | |
| May 6, 2021 at 16:26 | comment | added | Anton Petrunin | Oh, it seems that you expect me reading your post to the end :) | |
| May 6, 2021 at 11:07 | comment | added | Moishe Kohan | @AntonPetrunin: As I said, it works for $X=Y$. As for your example, I do not understand it. | |
| May 6, 2021 at 5:02 | comment | added | Anton Petrunin | Take the standard sphere $X$. Shrink its equator by factor 2; denote the obtained space by $Y$. The induced map $X\to Y$ is measure-preserving and short. So, something wrong with your argument. So you need to assume more about spaces. For Alexandrov spaces it was done by Nan Li arxiv.org/abs/1110.5498 | |
| May 5, 2021 at 13:30 | comment | added | Moishe Kohan | @JialongDeng: No, you need also the assumption that the map is measure-preserving and that it is a map from a manifold to itself. | |
| May 5, 2021 at 12:22 | comment | added | Jialong Deng | @ Moishe: Does it mean that a 1-Lipschitz map between Riemannian $n$-manifolds is an isometric map by the argument? | |
| May 4, 2021 at 13:10 | history | edited | Moishe Kohan | CC BY-SA 4.0 |
added 231 characters in body
|
| May 4, 2021 at 13:08 | comment | added | Moishe Kohan | @JohannesHahn: You are right: The argument was originally written in the case of self-maps, $X=Y$ and I did not think through the general case. I will correct... | |
| May 4, 2021 at 11:49 | comment | added | Johannes Hahn | $f^2(\Delta_r)$ need not be a proper subset for non-isometries. Extreme counterexample: Constant maps. You need to use the volume-condition or manifold-ness somewhere to infer that. | |
| May 3, 2021 at 16:11 | history | edited | Moishe Kohan | CC BY-SA 4.0 |
added 295 characters in body
|
| May 3, 2021 at 15:07 | history | answered | Moishe Kohan | CC BY-SA 4.0 |