Timeline for Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces
Current License: CC BY-SA 4.0
19 events
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| S May 8, 2021 at 11:24 | history | bounty ended | aglearner | ||
| S May 8, 2021 at 11:24 | history | notice removed | aglearner | ||
| S May 6, 2021 at 22:21 | history | bounty started | aglearner | ||
| S May 6, 2021 at 22:21 | history | notice added | aglearner | Reward existing answer | |
| May 2, 2021 at 4:38 | answer | added | abx | timeline score: 4 | |
| May 1, 2021 at 22:59 | history | edited | aglearner | CC BY-SA 4.0 |
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| May 1, 2021 at 22:56 | comment | added | aglearner | @GrishaPapayanov You can phrase it in the way you stated. I had in mind the fact that the moduli space of Riemann surfaces has a natural complex holomorphic structure, and any complex holomorphic manifold has a natural real analytic structure. But if you prefer Fenchel-Nielsen coordinates, you can use them. They give the same real analytic structure on the moduli space | |
| May 1, 2021 at 22:50 | comment | added | Grisha Papayanov | @aglearner What is this natural analytic structure? Am I right that you are asking whether the path in question (more precisely, its lifting to the Teichmueller space or at least one of its liftings) is analytical in Fenchel-Nielsen coordinates or do you mean something else? | |
| May 1, 2021 at 22:30 | history | edited | aglearner | CC BY-SA 4.0 |
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| May 1, 2021 at 22:23 | comment | added | aglearner | Grisha, "$J$ corresponding to $g$ is just rotation by 90 degrees" that's very true. Now, take $g_t=g_1t+g_0(1-t)$ and take the corresponding path of Riemann surfaces $(S,J_t)$, $t\in (0,1)$. This is a path in the moduli space of Riemann surfaces. For what is the Moduli space of Riemann surfaces you can look here: www-personal.umich.edu/~alexmw/CourseNotes.pdf This space has a natural analytic structure | |
| May 1, 2021 at 22:11 | comment | added | Grisha Papayanov | $J$ corresponding to $g$ is just rotation by 90 degrees. What do you mean by an analytical path in a moduli space, and by moduli space itself? | |
| May 1, 2021 at 20:46 | comment | added | Andy Putman | I haven't done the calculation, but the Riemann surface structure associated to a metric is obtained by finding isothermal coordinates, and thus by solving the Beltrami equation. So what you're asking should be equivalent to asking how the solution to the Beltrami equation depends on the initial conditions. This is definitely known (and should e.g. be in Ahlfor's "Lectures on quasiconformal maps). | |
| May 1, 2021 at 19:23 | history | edited | aglearner | CC BY-SA 4.0 |
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| May 1, 2021 at 19:17 | comment | added | aglearner | It seems to me that $J_t$ that you choose is not the one that I choose? For me $J_t$ is the unique one that preserves $g_t$. And for you? (I can't get it from your first comment, unfortunately) | |
| May 1, 2021 at 19:13 | history | edited | aglearner | CC BY-SA 4.0 |
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| May 1, 2021 at 19:05 | history | edited | aglearner | CC BY-SA 4.0 |
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| May 1, 2021 at 15:12 | comment | added | aglearner | I am a bit confused. $\omega$ should be the area form of $g$ for equality $\omega(X,JY)=g(X,Y)$ to hold. So I don't think you need to fix $\omega$ in advance (the area forms of $g_t$ are all different). However, indeed $J_x$ will depend analytically on $t$. So this observation makes the analyticity of $\gamma$ look more plausible indeed... I wonder how to complete this argument to make it a full proof (I guess this is would require something from PDEs?). | |
| May 1, 2021 at 14:57 | comment | added | abx | Fix a symplectic form $\omega $ on $S$. For each metric $g$, there is a unique complex structure $J$ on $S$ such that $\omega (X,JY)=g(X,Y)$; at each point $x\in S$ $J_x$ depends analytically on $g_x$. So the $J$ associated to $tg_1+(1-t)g_0$ depends analytically on $t$, and $\gamma$ is analytic. | |
| May 1, 2021 at 14:09 | history | asked | aglearner | CC BY-SA 4.0 |