Timeline for Laplacian spectrum asymptotics in neck stretching
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2019 at 15:05 | answer | added | macbeth | timeline score: 4 | |
| Mar 9, 2019 at 19:18 | comment | added | Neal | You could start investigating this with Cheeger's inequality. If $h_T$ is the Cheeger constant associated to $(M,g_T)$, then Cheeger-Buser inequalities show that $\lambda_1\sim \sqrt{h}$ and my guess is that for large $T$, you can realize $h_T$ with a (perturbation of a) cross section of $[-T,T]\times S$. | |
| Mar 9, 2019 at 16:36 | comment | added | Hadrian Quan | They crucially use the correspondence between Dirac operators with APS-boundary conditions on a manifold $Z$ with boundary, and the the $L^2$ boundary conditions on the prolongation $\hat{Z}$ of your manifold to one with infinite cylindrical ends. So it makes sense why you would consider this boundary condition, as it relates to the “limiting problem” of a neck with infinite length. | |
| Mar 9, 2019 at 16:36 | comment | added | Hadrian Quan | I think the standard reference for this type of question is Cappell-Lee-Miller (Self-Adjoint operators and Manifold decompositions I), where they consider the limiting behaviors of eigenmodes associated to a 1st order elliptic operator $D$. If you take your $D=d+\delta$ to be the de Rham operator, its square will be the Hodge Laplacian, so their analysis should help in your case. Roughly, small eigenvalues come from contributions of the 2 connected components of $M$, and their "interactions of eigenspaces along $S$." There is the subtlety that they impose APS-boundary conditions near $S$. | |
| Mar 9, 2019 at 15:13 | history | edited | YCor |
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| Mar 9, 2019 at 15:08 | history | edited | YCor | CC BY-SA 4.0 |
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| Mar 9, 2019 at 15:08 | history | asked | Guangbo Xu | CC BY-SA 4.0 |