Timeline for Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 8, 2021 at 18:53 | comment | added | abx | Fine, no problem. As for how I came up with my answer, I think it is quite common to prove real-analyticity of a map by proving that it is the restriction of a holomorphic map. Complex analyticity behaves much better that the real one. | |
| May 8, 2021 at 11:31 | comment | added | aglearner | Dear abx, I feel like the answer that you gave solves the question, but I can't accept it until I fully understand it. This might take time, since I need to learn how to make the calculation of integrability. For a time being I just added 50 bounty to your answer (instead of accepting). I wonder, whether the type of reasoning that you gave applies to some other examples of moduli spaces in similar circumstances. I am curious also how you came up with this answer. | |
| May 8, 2021 at 11:24 | history | bounty ended | aglearner | ||
| May 3, 2021 at 19:08 | comment | added | abx | This seems likely, but I am sorry I don't have time right now to do the computation. | |
| May 3, 2021 at 8:57 | comment | added | aglearner | Dear abx, I am thinking about your answer. For the linear algebra part, I have the space of all Riemannian metrics on $\mathbb R^2$. This can be identified with the positive cone $z^2>x^2+y^2$. The projectivisation of the cone is the hyperbolic plane $H^2$ in the Klein model. A path $tg_1+(1-t)g_0$ in the cone gives a geodesic in the Klein model. Clearly, the real analytic map $[0,1]\to H^2$ that sends $[0,1]$ to a geodesic has a natural complexification (for the natural complex structure on $H^2$). Is this equivalent to the complexification you had in mind? | |
| May 2, 2021 at 16:35 | comment | added | abx | @John Pardon: I have added some explanation. I agree that details should be checked, but this is an MO post, not a proof from a textbook. | |
| May 2, 2021 at 16:34 | history | edited | abx | CC BY-SA 4.0 |
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| May 2, 2021 at 13:49 | comment | added | John Pardon | I think some further explanation is warranted to justify the assertion that $V\to M$ is holomorphic. | |
| May 2, 2021 at 6:26 | history | edited | abx | CC BY-SA 4.0 |
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| May 2, 2021 at 4:38 | history | answered | abx | CC BY-SA 4.0 |