Timeline for The norm-squared of a moment map behaves like a Morse-Bott function
Current License: CC BY-SA 4.0
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| Aug 29, 2022 at 19:53 | vote | accept | Mira | ||
| Aug 29, 2022 at 19:52 | vote | accept | Mira | ||
| Aug 29, 2022 at 19:53 | |||||
| S Aug 29, 2022 at 19:52 | history | bounty ended | Mira | ||
| S Aug 29, 2022 at 19:52 | history | notice removed | Mira | ||
| Aug 29, 2022 at 19:43 | answer | added | Gustavo Granja | timeline score: 4 | |
| Aug 29, 2022 at 19:27 | comment | added | Mira | Thank you very much @GustavoGranja for pointing out this paper! Could you please rewrite your comment in the answer section, since it answered my question ? | |
| Aug 28, 2022 at 20:03 | comment | added | Gustavo Granja | This is proved in Gradient flow of the norm squared of a moment map by Eugene Lerman who attributes the proof to Duistermaat. | |
| Aug 28, 2022 at 19:07 | history | edited | Mira | CC BY-SA 4.0 |
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| Aug 27, 2022 at 19:38 | history | edited | Mira |
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| S Aug 27, 2022 at 6:33 | history | bounty started | Mira | ||
| S Aug 27, 2022 at 6:33 | history | notice added | Mira | Canonical answer required | |
| Aug 26, 2022 at 14:35 | history | edited | Mira | CC BY-SA 4.0 |
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| Aug 26, 2022 at 14:33 | comment | added | Mira | @NickL Thank you for correcting me! I'll edit my question. | |
| Aug 26, 2022 at 10:45 | comment | added | Nick L | Ah ok, that is usually referred to as the norm squared. Square norm usually means $||\mu||$. Indeed the norm squared is differentiable. | |
| Aug 26, 2022 at 0:39 | comment | added | Mira | Unfortunately, I don't remember where I have found the statement, But I do remember that this result was proven in the paper Morse Theory of the moment map for representative of Quivers, in the case where $M$ is a symplectic vector space. | |
| Aug 26, 2022 at 0:34 | comment | added | Mira | @NickL, the definition of critical points that I'm using is the following: $\textbf{Def}$: we say that $x \in M$ is a critical point of $\vert \vert \mu \vert \vert ^2 $ if $d_x \vert \vert \mu \vert \vert ^2 =0$. | |
| Aug 25, 2022 at 11:53 | comment | added | Nick L | Also it may help if you could link to the place where you read it (if possible). | |
| Aug 25, 2022 at 11:46 | comment | added | Nick L | For a Hamiltonian $S^1$-action with Hamiltonian $H$, the square norm of the Hamiltonian may not be differentiable on the set $\{H=0\}$. For example when $\{H=0\}$ is disjoint from the fixed point set, then the square norm is not differentiable anywhere along it. So please clarify what you mean by critical points. | |
| Aug 25, 2022 at 3:40 | history | edited | Mira | CC BY-SA 4.0 |
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| Aug 25, 2022 at 2:54 | history | asked | Mira | CC BY-SA 4.0 |