Timeline for Flat metric on compact surface minus a point
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2023 at 12:01 | answer | added | gaoqiang | timeline score: 0 | |
| Dec 16, 2019 at 0:39 | vote | accept | Gianni del Fiore | ||
| Dec 14, 2019 at 21:23 | answer | added | Robert Bryant | timeline score: 18 | |
| Dec 14, 2019 at 20:30 | comment | added | Martin de Borbon | No. Take a flat metric with a single cone angle. These are smooth (in complex coordinates) because the cone angle is an integer multiple of 2 pi. You can get these metrics either from a holomorphic one form with a single zero, or branching over a torus with only one critical point. | |
| Dec 14, 2019 at 15:04 | comment | added | José Figueroa-O'Farrill | @GiannidelFiore Ah, sorry, I misread the question. I too will leave the comment up as a warning :) | |
| Dec 14, 2019 at 15:02 | comment | added | Gianni del Fiore | @JoséFigueroa-O'Farrill: Note that the flat metric induced on the 2-sphere minus the north pole by the stereographic projection does not extend to a $\left(0,2\right)$-tensor on the whole sphere. | |
| Dec 14, 2019 at 14:07 | comment | added | Alexandre Eremenko | For every compact Riemann surface $S$ and a point $p\in S$ there is a harmonic function $u$ on $S\backslash p$. | |
| Dec 14, 2019 at 13:26 | comment | added | José Figueroa-O'Farrill | Take the 2-sphere and remove the north pole. This is stereographically diffeomorphic to the plane, which admits a flat metric. Transport that flat metric back to the sphere. It is flat away from the north pole. But the sphere has nonzero Euler characteristic. | |
| Dec 14, 2019 at 5:56 | comment | added | Michael Albanese | I see. I will leave my comment up as a warning. | |
| Dec 14, 2019 at 5:48 | comment | added | Gianni del Fiore | @MichaelAlbanese: note that the $\left(0,2\right)$-tensor may fail to be positive definite at $p$ | |
| Dec 14, 2019 at 5:44 | comment | added | Michael Albanese | $\|\operatorname{Rm}_g\|^2$ is a continuous function, so if it is zero in the complement of a point, it is zero everywhere. So if the restriction of $g$ to the complement of a point is flat, then $g$ is flat. The only compact surfaces which admit flat metrics are the torus and the Klein bottle, both of which have Euler characteristic zero. | |
| Dec 14, 2019 at 5:43 | history | edited | Gianni del Fiore | CC BY-SA 4.0 |
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| Dec 14, 2019 at 5:12 | history | asked | Gianni del Fiore | CC BY-SA 4.0 |