Timeline for Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 29, 2015 at 2:07 | comment | added | Renato G. Bettiol | @Otis: Good point; now that I'm thinking about it, I'm not so sure there's a GH limit either... | |
| May 28, 2015 at 23:34 | comment | added | Otis Chodosh | For some reason I remember that you should also be able to force the volume to be 1, but I couldn't find that statement in Lohkamp's paper. I think that Lohkamp is perturbing Colin de Verdière's metrics in $C^0$, so its probably just an issue of understanding the limit of the ones you mention. I guess it isn't even obvious without looking into Colin de Verdière's paper that there's a GH limit! | |
| May 28, 2015 at 22:08 | comment | added | Renato G. Bettiol | @Otis: Thanks for catching that typo, I've just fixed it. One curiosity I always had is what happens to these metrics as $k\nearrow +\infty$. I don't remember the details, but I think I somehow concluded that the volume of these metrics must go to zero (or the diameter has to blow up, depending what you want to normalize). The statement in Lohkamp's paper says one can arrange these metrics to also have Ricci bounded above by an arbitrarily negative number -- I wonder what the GH limit is, maybe some sort of graph? Do you know anything about that? | |
| May 28, 2015 at 22:02 | history | edited | Renato G. Bettiol | CC BY-SA 3.0 |
Fixed typo
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| May 28, 2015 at 21:26 | comment | added | Otis Chodosh | Hey Renato, nice answer! Two remarks: (1) there should be strict inequality for $a_1$. (2) An interesting extension of this, which I find very surprising, is due to Lohkamp which says that you can prescribe the first $k$ eigenvalues while forcing $Ric < - \alpha^2$ for any $\alpha$. ams.org/mathscinet-getitem?mr=1356779 Corollary 2.7 | |
| May 28, 2015 at 20:41 | history | answered | Renato G. Bettiol | CC BY-SA 3.0 |