Timeline for Nonexistence of sphere with one conical point [reference request]
Current License: CC BY-SA 4.0
6 events
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| Oct 20, 2022 at 20:46 | comment | added | Alexandre Eremenko | In the papers about metrics with conic singularities, the case of 1 singularity is considered trivial, and it is usually only briefly mentioned that such metrics on the sphere do not exist. (In the paper that I cited it is not even mentioned). There is a complete description of possible sets of angles at the conic singularities for metrics on the sphere, MR3556430, MR4105912. The statement is of course a formal consequence of the results of these papers. | |
| Oct 20, 2022 at 14:16 | vote | accept | Tom Sharpe | ||
| Oct 20, 2022 at 14:16 | comment | added | Tom Sharpe | The proof you gave is perfectly satisfactory; however, I was hoping to be able to simply quote a result (which I can now also do, thanks to your further explanation!) | |
| Oct 20, 2022 at 14:05 | comment | added | Alexandre Eremenko | Yes, you understand correctly. Just introduce the second singularity with angle $2\pi$ and apply Troyanov's result, if you are not satisfied with the proof I gave. | |
| Oct 20, 2022 at 14:00 | comment | added | Tom Sharpe | I can't see where in the paper you reference Troyanov actually discusses the case of one conical point. Or were you suggesting to deduce it from the case of two conical points? Something like, a sphere with one conical point can be viewed as a sphere with two conical points, one having conical angle $2\pi$. Then, by Troyanov's result, the other must have conical angle $2\pi$? | |
| Oct 20, 2022 at 13:32 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |