Timeline for Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 9, 2022 at 20:03 | vote | accept | user404153 | ||
| May 9, 2022 at 19:27 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added an example that is not even locally a product
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| May 9, 2022 at 19:08 | comment | added | Robert Bryant | @user404153: Just to be completely clear. That last sentence in my comment should have been "That torus is not metrically the product of two circles." Obviously, it is topologically (and smoothly) the product of two circles. | |
| May 9, 2022 at 18:36 | comment | added | user404153 | Hm, fair enough. Thanks again! | |
| May 9, 2022 at 18:27 | comment | added | Robert Bryant | @user404153: Actually, if the lattice $\Lambda$ is not a rectangular lattice, the torus is not a product (except locally). For example, suppose that $\Lambda$ is the hexagonal lattice, generated by three unit vectors that sum to zero. That torus is not a product of two circles. | |
| May 9, 2022 at 18:16 | comment | added | user404153 | Thanks! That's a good point. I suppose this counter-example is due to the product structure (your function is a product with a zero eigenvalue on one factor), and it question in the title still has some hope of a positive answer for non-product spaces. | |
| May 9, 2022 at 18:11 | history | answered | Robert Bryant | CC BY-SA 4.0 |