Timeline for Level set of convex and plurisubharmonic functions (Ricci curvature and other curvature conditions)
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 2, 2014 at 12:02 | vote | accept | Misha Verbitsky | ||
| Jun 2, 2014 at 6:14 | answer | added | Robert Bryant | timeline score: 4 | |
| Jun 1, 2014 at 20:40 | comment | added | Misha Verbitsky | many thanks! I would be very grateful, the question has some important geometric applications | |
| Jun 1, 2014 at 20:20 | comment | added | Robert Bryant | @MishaVerbitsky: I'm about to board a flight, so I don't have time to input a full answer, but I can say that the Ricci curvature of metric induced on the level sets of a strictly plurisubharmonic function is not always positive. I think it is possible to characterize the induced metrics on level sets that you get this way intrinsically, but I haven't completely worked that out. Maybe when I next get connected to the internet (tomorrow), I'll have that worked out. | |
| Jun 1, 2014 at 17:09 | comment | added | Misha Verbitsky | everything is smooth | |
| Jun 1, 2014 at 16:53 | comment | added | Alexandre Eremenko | Do you have any smoothness condition on your PSH function? Otherwise what is a pull-back? What is curvature? | |
| Jun 1, 2014 at 13:27 | comment | added | Robert Bryant | Moreover, it doesn't actually work in general: For example, consider $f= 1/(z^2-x^2-y^2)$. This is strictly convex on the domain (interior of a cone) where $x^2+y^2 < z^2$, but the function is invariant under the affine action of $\mathrm{SO}(2,1)$, so any induced metrics on the level sets of $f$ have to be invariant under this action as well, and that forces them to be hyperbolic metrics. | |
| Jun 1, 2014 at 13:24 | history | edited | Misha Verbitsky | CC BY-SA 3.0 |
edited body
|
| Jun 1, 2014 at 13:24 | comment | added | Misha Verbitsky | Specifically, I want an argument which might be possibly generalized to the complex case | |
| Jun 1, 2014 at 13:20 | comment | added | Misha Verbitsky | It seems so, but I don't see an easy argument why the metric is induced by the Gauss map | |
| Jun 1, 2014 at 11:31 | history | asked | Misha Verbitsky | CC BY-SA 3.0 |