Timeline for Einstein metrics on connected sums
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 26, 2021 at 21:20 | vote | accept | Piotr Chrusciel | ||
| Jul 15, 2021 at 20:35 | answer | added | Robert Bryant | timeline score: 9 | |
| Jul 13, 2021 at 15:43 | comment | added | Robert Bryant | @PiotrChrusciel: Well, one can take a 'conformally Einstein' connected sum in the above sense of any $n$-dimensional compact space form with the round $S^n$, of course, but one cannot take a 'conformally Einstein' connected sum in the above sense of any two $n$-dimensional compact space forms unless at least one of them is the round $S^n$. One can also construct other trivial, uninteresting examples in the above sense from the $n$-sphere endowed with any Einstein metric (not just the conformally flat ones). | |
| Jul 6, 2021 at 8:43 | comment | added | Piotr Chrusciel | Thank your for your answer, I should have indeed added: which is not a connected sum of two spheres. Any other examples? I would have thought that cutting a compact negatively curved space form along a totally geodesic sphere could perhaps do what I want, but I don't know if such spheres exist in dimension higher than two. | |
| Jul 5, 2021 at 14:17 | comment | added | Robert Bryant | @BenMcKay: Well, after one has replaced 'the connected sum' with 'a connected sum' (i.e., specified the removed disks and the precise method of attaching a 'neck' with its own metric), it makes sense to talk about an Einstein metric on such a connected sum that is conformal to the original metrics outside the neck. That's one way of making sense of it. In this sense, one could take $(M_i,g_i)$ to be $(S^n,g_{\mathrm{can}})$ and perform a connected sum that would still be a conformally flat metric on $S^n$, which would support a (family) of Einstein metrics in that conformal class. | |
| Jul 5, 2021 at 10:08 | comment | added | Ben McKay | I don't understand. There is no open subset of $M_1\# M_2$ which is diffeomorphic to $M_1$, so how do I figure out whether the metric on the connected sum is conformal ``on'' $M_1$ to $g_1$? | |
| Jul 5, 2021 at 9:55 | history | asked | Piotr Chrusciel | CC BY-SA 4.0 |