Timeline for Obstructions for a metric to be conformally equivalent to a product metric
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 11, 2014 at 17:44 | comment | added | Alexandre Eremenko | @Will Sawin: thanks, this is much simpler than my proposal:-) I wanted the metric to be complete. | |
| Dec 29, 2013 at 16:14 | comment | added | Alexandre Eremenko | The references are given in my answer. A plane domain with a metric defines an open simply connected Riemann surface. Every such Riemann surface is conformally equivalent to the disc or the plane. | |
| Dec 29, 2013 at 16:09 | comment | added | Ali Taghavi | @AlexandreEremenko By geometric version of Riemann mapping theorem I mean : If U is a simply connected subset of the plane with a metric, then there is a bioholomorphic map which sends U to either $\mathbb{R}^{2}$ or the unit disk $\mathbb{D}$ which the resulting metric is conformally equivalent to the euclidean or hyperbolic metric. I think your answer is bases on this statement. Yes? After your statment I realized that I heard this from some one, many years ago, but I do not know the reference. Did you used this stement in your answerr? And what is a reference for proof ofthis statement? | |
| Dec 29, 2013 at 4:03 | comment | added | Will Sawin | e.g. the flat metric on the unit square. | |
| Dec 29, 2013 at 3:12 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Dec 29, 2013 at 1:55 | comment | added | Alexandre Eremenko | But you can FIND a product metric in the plane which is conformally equivalent to the hyperbolic metric. And this is enough. | |
| Dec 28, 2013 at 20:57 | comment | added | Deane Yang | This is probably a silly question, but I don't see why the hyperbolic metric is globally conformal to a product metric. It's globally conformal to the flat metric on the unit disk, which is locally but not globally a product metric. | |
| Dec 28, 2013 at 19:48 | comment | added | Alexandre Eremenko | What is the "geometric version" of the Riemann mapping theorem? | |
| Dec 28, 2013 at 19:46 | comment | added | Ali Taghavi | what is a reference for a proof of geometric version of Riemann mapping theorem? | |
| Dec 28, 2013 at 15:48 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Dec 28, 2013 at 6:59 | vote | accept | Ali Taghavi | ||
| Dec 30, 2013 at 17:20 | |||||
| Dec 28, 2013 at 6:59 | comment | added | Ali Taghavi | Thank you for the answer to my first question. I give a new question about the first part. | |
| Dec 28, 2013 at 6:51 | comment | added | Ali Taghavi | Thank you very much for the answer.Is it the geometric version of riemann mapping theorem? what is a reference to prove this. I would appreciate if you give an answer or a reference about the remaining part of my question.For Example is it true to say that every metric on torous is conformally equivalent to a product metric. What is an example of a metric on $M\times N$ which is not conformal equivalent to a product metric | |
| Dec 28, 2013 at 6:45 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |