Timeline for compatification of $\mathbb{R}^2$ under a conformal metric
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 8, 2020 at 14:39 | comment | added | student | I see. Thank you very much! | |
| Jul 8, 2020 at 14:39 | vote | accept | student | ||
| Jul 6, 2020 at 19:54 | comment | added | Alexandre Eremenko | Choose the chart $1/z$ and the condition is that your $e^u$ written in this chart is smooth at $0$. | |
| Jul 6, 2020 at 14:26 | comment | added | student | Sorry I didn't clarify in my question. I know that if the criteria is satisfied, then $\mathbb{R}\cup \{\infty\}$ is a complete metric space, under the metric $g=e^{2u}\delta$. Also, I know that if $\int_{\mathbb{R}^2}e^{2u}dx<\infty$, then the criteria is satisfied. My true question is that, how to make sure such metric is still a Riemannian metric at $\infty$? Or how to choose a local chart in a neighborhood of $\infty$ such that it is a smooth extension of the original metric defined only on $\mathbb{R}^2$? | |
| Jul 6, 2020 at 12:56 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 617 characters in body
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| Jul 6, 2020 at 12:50 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |