Timeline for Diffeomorphism group of the 2-sphere with $C^0$ topology
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 27, 2013 at 2:41 | comment | added | Igor Belegradek | The diffeomorphism $\phi$ solves Beltrami equation in which the dilatation $\mu$ continuously depends on the metric. Ahlfors (or maybe Ahlfors-Bers?) proved that the solution of Beltrami equation depends continuously on $\mu$. | |
| Sep 26, 2013 at 18:29 | comment | added | Lev Soukhanov | Actually, I cannot convince myself why the $\phi$ depends continously on the metric - I wanted to use inverse function theorem for it... | |
| Sep 26, 2013 at 15:05 | comment | added | Igor Belegradek | I agree with what you say about contractibility in $C^0$ topology (thanks for this). The inverse function theorem seems problematic but looks like it is not needed: every metric on $S^2$ is written uniquely as $\phi^*(e^u g_1)$ where $\phi$ is a self-diffeomorphism of $S^2$ fixing $0,1, \infty$, $u$ is a smooth function on $S^2$ and $g_1$ is the curvature $1$ metric, and the map $\phi^*(e^u g_1)\to \phi$ is probably a fiber bundle with contractible total space and fibers. If so, the base is contractible. | |
| Sep 23, 2013 at 18:18 | comment | added | Lev Soukhanov | What I wanted to say is that the space of riemmanian metrics is contractable in any topology, $C^0$ or $C^\infty$. Maybe, the problem with my argument is in the check that inverse function theorem can be applied (i.e. the morphism from diffeomorphisms to the almost-complex structures is differentiable as the morphism of Banach spaces with $C^0$ topology?) | |
| Sep 11, 2013 at 2:52 | comment | added | Igor Belegradek | I am aware of all this (and the reference is Earle-Eeels, 1969 JDG), but I do not see why the argument works in $C^0$ topology, in particular I do not understand your "obviously won't make this space uncontractable". | |
| Sep 11, 2013 at 2:10 | history | answered | Lev Soukhanov | CC BY-SA 3.0 |