Timeline for Compact surface with genus$\geq 2$ with Killing field
Current License: CC BY-SA 3.0
5 events
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| Feb 21, 2013 at 14:51 | comment | added | Joe | Thanks again for the clarification. I should have written "when the curvature of metric is degenerate appropriately at higher order" | |
| Feb 21, 2013 at 12:35 | comment | added | Robert Bryant | @Joe: Well, if the metric actually did degenerate, you certainly could have poles for solutions to the Killing equation. For example, if $M$ is a Riemann surface of genus $g>1$ and $\zeta$ is a nonzero holomorphic $1$-form on $M$, then the pseudometric $ds^2 = \zeta\circ\bar\zeta$ has a meromorphic Killing field $X=\mathrm{Re}(Z)$ where $Z$ is the meromorphic vector field that satisfies $\zeta(Z)=1$; $X$ will have 'poles' where $\zeta$ has zeros. As long as the metric is nondegenerate though, you're OK (as my parenthetical remark and your expansion of it in your comment both indicate). | |
| Feb 21, 2013 at 10:33 | comment | added | Joe | Thanks for the answer! The coefficients of a Killing field $v$ and its first derivative satisfy a closed system of DE's of the form $dv=\omega v$, and since $\omega$, which determines the rate of change of $v$, is bounded in a neighborhood of the puncture, $v$ must be bounded. I mistakenly thought that there might exist a Killing field with a pole-type singularity at a point, when the metric is degenerate appropriately at higher order. | |
| Feb 21, 2013 at 10:01 | vote | accept | Joe | ||
| Feb 21, 2013 at 0:44 | history | answered | Robert Bryant | CC BY-SA 3.0 |