Timeline for Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Current License: CC BY-SA 4.0
4 events
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| Dec 16, 2021 at 0:46 | comment | added | Otis Chodosh | You know the dimension of the space of conformal KVF is 6, right? Given this, $K+a^T$ generates a $6$ dimensional space of KVF, so the dimensions match up. By $Y^{-1}_1$ etc you mean 1-st spherical harmonics, right? If so, then any 1-st spherical harmonic is $a^T$ for some $a$, so the answer is yes. | |
| Dec 15, 2021 at 23:15 | comment | added | Laithy | @OtisChodosh Thank you :) How do we see that every conformal KVF is of the form $K + a^T$? It looks like the other way around is true too: for any $F$ that is a linear combination of $Y^{-1}_{1}, Y^0_1, Y^1_1$, there exists a unique conformal KVF with divergence equal $F$. Is that correct? | |
| Dec 15, 2021 at 18:15 | comment | added | Otis Chodosh | Any conformal-KVF is $K + a^T$ for $a\in \mathbb{R}^3$ constant and $K$ a KVF. You can check that $\textrm{Div}_{S^2}(K+a^T) = - 2 a \cdot x$ which is a first spherical harmonic. | |
| Dec 15, 2021 at 18:02 | history | asked | Laithy | CC BY-SA 4.0 |