Timeline for Hodge decomposition in elliptic complexes
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 15, 2023 at 15:07 | comment | added | M.G. | @asv: Ah, I see now what you meant. Indeed one has to get the ingredients to establish (1) for concrete cases from elsewhere, so it's not self-contained in that regard. I'm sorry I couldn't be of more help. | |
| Mar 15, 2023 at 14:53 | comment | added | asv | I am not sure what you mean. I think that Chapter 1 is that book does not develop enough tools to prove Hodge decomposition. Exercise 1.3.15 might be potentially helpful to prove it as it might contain some intermediate steps, but that's all as far I understand. Thank you in any case. | |
| Mar 15, 2023 at 14:09 | comment | added | M.G. | @asv: well, it is, but you can ignore the inverse direction, if you are not interested in it. Though I find an equivalence to be a stronger statement and I haven't found anywhere else necessary and sufficient conditions for a Hodge decomposition to hold. | |
| Mar 15, 2023 at 8:53 | comment | added | asv | In seems that in the second reference you mentioned Exercise 1.3.15 is not about a proof of the Hodge decomposition, but just an equivalent reformation of it. | |
| Mar 14, 2023 at 19:50 | history | edited | M.G. | CC BY-SA 4.0 |
added the smooth case
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| Mar 14, 2023 at 19:26 | comment | added | M.G. | @LSpice: I think, that's fine. That way we have both links. | |
| Mar 14, 2023 at 19:23 | comment | added | LSpice | Re, if you prefer the link to the free version (I know some people don't like having links to paywalled content in their posts), then please feel free to overwrite my edit. I just like to have some link when possible. | |
| Mar 14, 2023 at 19:15 | comment | added | M.G. | @asv: I believe I have seen the Hodge decomposition stated for $\Gamma(E)$, where $E$ is the total vector bundle. Would that work for you? In fact, I think I even remember where. | |
| Mar 14, 2023 at 19:07 | comment | added | asv | Thank you. I would prefer to have Hodge decomposition for infinitely smooth sections rather than $L^2$. (Another less important remark is that in the mentioned lecture notes the differential operators have the first order.) | |
| Mar 14, 2023 at 19:06 | comment | added | M.G. | @LSpice: The notes are also freely available, the last version is here: www3.nd.edu/~lnicolae/Lectures.pdf | |
| Mar 14, 2023 at 19:03 | history | edited | LSpice | CC BY-SA 4.0 |
Link to book
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| Mar 14, 2023 at 18:47 | history | edited | M.G. | CC BY-SA 4.0 |
added reference
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| Mar 14, 2023 at 18:39 | history | answered | M.G. | CC BY-SA 4.0 |