Timeline for special Lagrangian n-Torus has Tubular neighbourhood?
Current License: CC BY-SA 3.0
12 events
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| Mar 6, 2013 at 13:14 | vote | accept | CommunityBot | ||
| Feb 22, 2013 at 15:53 | comment | added | user21574 | Yes, It was my mistake, But I have still this question in his proof that why L has Tubular neighbourhood ? because in MacLean theorem I don not see this fact | |
| Feb 22, 2013 at 15:40 | comment | added | Robert Bryant | @Hassan: OK, so I can see that you left out a crucial adjective: He is not assuming that all small Lagrangian deformations are flat, only the small special Lagrangian deformations. (I can see that you have just added the word 'special' in your question, so you have now realized your error.) Now his proof goes through, and I don't see why you are asking the question. Are you not able to follow the proof? Note that in the discussion that follows, he points out that this 'semi-flat' situation will almost never hold. (I'm not convinced it can happen at all outside the flat case.) | |
| Feb 22, 2013 at 15:32 | comment | added | user21574 | Thanks for your comment, I mean theorem 2.3.3, I had misprint in my previous comment, sorry | |
| Feb 22, 2013 at 15:31 | history | edited | user21574 | CC BY-SA 3.0 |
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| Feb 22, 2013 at 15:03 | comment | added | Robert Bryant | @Hassan: But the point I'm trying to make is that you are describing an impossible situation. When $n>1$ it cannot ever happen that all of the small Lagrangian deformations of $L$ are flat in the induced metric. That is why I suspect that either you are misreading something or the author has been careless. By the way, I just downloaded Baier's 2001 thesis (people.maths.ox.ac.uk/hitchin/hitchinstudents/baier.ps.gz) and there is no Theorem 2.2.3 in it, so I can't see what you are talking about. | |
| Feb 22, 2013 at 14:42 | comment | added | user21574 | Dear Robert Bryant, Yes, In fact the metric induced by the immersion into the Calabi-Yau manifold $X$. I am not sure that we need to use of Tubular neighbourhood Theorem or not? I am studying Patric Baier thesis student of Hitchin, theorem 2.2.3 | |
| Feb 22, 2013 at 14:29 | comment | added | Robert Bryant | @Hassan: By the way, I notice that you ask that "all small Lagrangian deformations" be flat. Does that mean 'flat in the metric induced by the immersion into the Calabi-Yau manifold $X$'? I assume you know that this cannot ever happen for $n>1$. Actually, when I wrote my answer below, I had ignored this assumption and assumed only that you meant for $L$ itself to be flat in the induced metric (which is all one needs for Maclean's theorem to give the local foliation result). I'm guessing that you are reading this somewhere and I suspect that the author of that article has been careless. | |
| Feb 22, 2013 at 11:38 | history | edited | user9072 |
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| Feb 21, 2013 at 18:14 | history | edited | user21574 |
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| Feb 21, 2013 at 16:55 | answer | added | Robert Bryant | timeline score: 6 | |
| Feb 21, 2013 at 16:47 | history | asked | user21574 | CC BY-SA 3.0 |