Timeline for First eigenvalue of the Laplacian on the traceless-transverse 2-forms
Current License: CC BY-SA 4.0
12 events
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| Nov 8, 2021 at 14:05 | comment | added | emiliocba | I have just added the simplest case at the end: cyclic subgroups. I hope it helps, but I don't think I will be able to do it for the rest of the subgroups of $SO(4)$ acting freely on $S^3$. Good luck! | |
| Nov 8, 2021 at 14:04 | history | edited | emiliocba | CC BY-SA 4.0 |
An example added at the end.
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| Nov 6, 2021 at 1:05 | comment | added | Zhiqiang | Yes. A more precise description of all such $\Gamma$ would be helpful. I am particularly interested in the case of whether $\Gamma$ is conjugate in $O(4)$ to a subgroup of $U(2)$. | |
| Nov 5, 2021 at 8:28 | comment | added | emiliocba | Oh, I see. Now everything fits! In any case, the smallest eigenvalue of $\Delta_L$ and $\Delta$ are attained at the same places, so the guide in my answer still holds. | |
| Nov 5, 2021 at 8:23 | history | edited | emiliocba | CC BY-SA 4.0 |
Added a coment about the relation of the Rough Laplacian and the Lichnerowicz Laplacian
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| Nov 4, 2021 at 12:48 | history | edited | emiliocba | CC BY-SA 4.0 |
A mistake fixed in the last paragraph. A new paragraph at the end adding information.
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| Nov 4, 2021 at 9:57 | comment | added | Zhiqiang | You mean the first eigenvalue of the Lichnerowicz Laplacian is $12$, which is equivalent to the first eigenvalue of the rough Laplacian is $6$ since $\Delta_L h=\Delta h-6h$ in the case of $3$-dimensional spherical space forms. | |
| Nov 4, 2021 at 8:40 | comment | added | emiliocba | Well, to be honest, I don't fully understand that paper. However, I found a precise reference for the fact that the first eigenvalue of the Lichnerowicz Laplacian on the unit 3-sphere is $12$ (Klaus Kröncke's thesis), and I added to my answer. By the way, this reference is a great source for the kind of problems that, I think, you are considering (according to your other questions about the spectrum of the Laplace-Beltrami operator, it seems you are interested on some kind of stability of 3-dimensional spherical spaceforms). | |
| Nov 4, 2021 at 8:35 | history | edited | emiliocba | CC BY-SA 4.0 |
A references to Klaus Kröncke's thesis was added
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| Nov 4, 2021 at 0:36 | comment | added | Zhiqiang | Please check out the paper "Symmetric-tensor eigenspectrum of the Laplacian on n-spheres" by Rubin and Ordonez. It contains the table of the spectra on symmetric two-forms. | |
| Nov 3, 2021 at 22:00 | history | edited | emiliocba | CC BY-SA 4.0 |
Several modifications in the last three paragraphs
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| Nov 3, 2021 at 14:03 | history | answered | emiliocba | CC BY-SA 4.0 |