Timeline for Every homotopy class contains at least a harmonic representative
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 19, 2020 at 17:47 | comment | added | Robert Bryant | @EduardoLonga: I don't know an example off the top of my head, but I suspect that existence of a harmonic mapping in a given homotopy class of maps $f:M^3\to S^2$ fails in many cases, just because the regularity theory for a nonlinear PDE gets harder as the dimension of the domain goes up. Maybe Andy knows something more specific about this. A good place to start is the work of Schoen and Uhlenbeck on regularity of harmonic maps. | |
| Apr 19, 2020 at 16:35 | comment | added | Eduardo Longa | The domain of dimension $3$ and the codomain being the round $2$-sphere. | |
| Apr 19, 2020 at 16:31 | comment | added | Robert Bryant | @EduardoLonga: Do you mean dimension 3 for the range or the domain, or both? | |
| Apr 19, 2020 at 15:42 | comment | added | Eduardo Longa | What does this imply for the case of dimension $3$? | |
| Apr 19, 2020 at 14:04 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a correction in response to Andy's comment.
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| Apr 19, 2020 at 14:01 | comment | added | Robert Bryant | @AndySanders: Thanks for the reference! I knew it had been known a long time, but I had forgot where I learned it. | |
| Apr 19, 2020 at 13:40 | comment | added | Andy Sanders | I think the result goes back to Wood, a discussion can be found here core.ac.uk/download/pdf/82593933.pdf. | |
| Apr 19, 2020 at 11:46 | history | answered | Robert Bryant | CC BY-SA 4.0 |