Timeline for A counterexample for Sard's theorem in $C^1$ regularity
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2019 at 6:33 | history | edited | Pietro Majer | CC BY-SA 4.0 |
deleted 355 characters in body
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| Apr 13, 2019 at 16:41 | comment | added | Pietro Majer | (I edited and added details and rectified the unclear sentence about infinite variation. Thank you!) | |
| Apr 13, 2019 at 16:35 | history | edited | Pietro Majer | CC BY-SA 4.0 |
details added
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| Apr 13, 2019 at 12:49 | history | edited | Pietro Majer | CC BY-SA 4.0 |
m
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| Apr 13, 2019 at 11:43 | comment | added | Mizar | Ah yes, now it's clear! So at each iteration you are reparametrizing by arclength, right? If you don't do that, having infinite variation does not give $|\phi(x)-\phi(y)|=o(|x-y|)$ (this fails if you don't reparametrize by arclength, as I was saying in my previous comment). | |
| Jun 8, 2018 at 16:24 | comment | added | Mizar | $\kappa$ having infinite variation is not sufficient to get the estimate on $\phi$: build "one third" of the Koch snowflake starting from the trivial map $\gamma_0:[0,1]\to\mathbb R^2$ (with $\gamma_0(t)=(t,0)$), subdividing $[0,1]$ into three thirds and replacing the straight segment with a tent on the middle interval (thus obtaining $\gamma_1$), and so on. The limiting map $f_\infty:[0,1]\to\mathbb R^2$ will have $f_\infty(0)=(0,0)$ and $f_\infty(3^{-k})=(3^{-k},0)$! Maybe you are reparametrizing by arclength before taking the limit? (I don't know how to deal with $f_\infty$ in that case...) | |
| Mar 16, 2018 at 20:50 | comment | added | Piotr Hajlasz | For generalizations of this example see mathscinet.ams.org/mathscinet-getitem?mr=1991757 | |
| Dec 27, 2016 at 23:13 | comment | added | BigM | Very interesting. I actually had not heard about Whitney's extension theorem (Tras. AMS 1934) before. It is a very neat analogue to Tietze-Urysohn extension theorem. | |
| Dec 27, 2016 at 22:30 | history | edited | Johannes Hahn | CC BY-SA 3.0 |
grammar++;
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| Dec 27, 2016 at 20:02 | history | answered | Pietro Majer | CC BY-SA 3.0 |