Timeline for Multiplicity of Laplace eigenvalues
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2016 at 11:42 | comment | added | Liviu Nicolaescu | She does state the dimension condition explicitly, but she uses $\dim>1$ in the proof and, towards the end of the paper, she mentions that the result cannot be true in dim 1. | |
| Dec 17, 2016 at 0:20 | comment | added | Tom Price | Ok, thank you very much, this is important for a paper I'm writing. Are you referring to theorem 8 in that paper? It strongly resembles your original claim but I can't be sure what it's saying since I can't find the definition of $\mathfrak{M}_k$ (the set of $C^k$ metrics perhaps?). Oddly I can't seem to find a condition on the dimension of the underlying manifold. | |
| Dec 16, 2016 at 23:12 | comment | added | Liviu Nicolaescu | The dimension has to be $\geq 2$. Here's Uhlenbeck's original paper jstor.org/stable/2374041 | |
| Dec 16, 2016 at 22:33 | comment | added | Tom Price | "for a generic Riemann metric on a given compact manifold the spectrum of the Laplacian will not have multiple eigenvalues." I don't understand how this is possible. The metrics on $S^1$ are classified, up to isomorphism, by the total length, and so each is isomorphic to $\mathbb{R}/x\mathbb{Z}$ for some real $x$, right? And for these manifolds the eigenspaces are all 2-dimensional for nonzero eigenvalues, generated by a sine and cosine function. What am I missing? | |
| Oct 31, 2015 at 14:26 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
added 9 characters in body
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| Oct 31, 2015 at 0:10 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
edited body
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| Oct 30, 2015 at 17:17 | comment | added | Fan Zheng | do you mean will not have multiple eigenvalues? | |
| Oct 30, 2015 at 14:08 | history | answered | Liviu Nicolaescu | CC BY-SA 3.0 |