Timeline for Finding constant curvature metrics on surfaces for the case of positive Euler characteristic
Current License: CC BY-SA 2.5
6 events
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| Nov 18, 2010 at 16:29 | vote | accept | Alejandro Betancourt | ||
| Nov 18, 2010 at 7:11 | comment | added | Ian Agol | @macbeth: you're right, I hadn't looked at the references closely before answering the question. So this suggestion probably needs some work to complete, given your constraints. I think one could approach this by first making a conformal change of the metric around $x,y, z$ to look like a hyperbolic cusp (which is possible, e.g. using isothermal coordinates), getting a new metric $g'$ on $S^2-\\{x,y,z\\}$. Then one could search for solutions $u$ to the scalar curvature equation which are asymptotically constant near $x,y,z$. Such boundary conditions may enable the techniques to go through. | |
| Nov 17, 2010 at 2:23 | comment | added | macbeth | This idea of dealing with $S^2-\{x,y,z\}$ instead of $S^2$ is interesting. But can you clarify how to prove "directly" (i.e., without complex analysis or Ricci flow) that any metric on $S^2-\{x,y,z\}$ is conformal to a hyperbolic one? As far as I can tell, it doesn't immediately follow from the references (Berger, Taylor) cited by Betan -- they solve the constant-curvature equation for compact hyperbolic-type surfaces only. | |
| Nov 16, 2010 at 17:25 | history | edited | Ian Agol | CC BY-SA 2.5 |
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| Nov 16, 2010 at 7:27 | history | edited | Ian Agol | CC BY-SA 2.5 |
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| Nov 16, 2010 at 7:02 | history | answered | Ian Agol | CC BY-SA 2.5 |