Timeline for Which curves are boundary of pseudoholomorphic curves?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2020 at 21:40 | history | edited | Chris Gerig | CC BY-SA 4.0 |
deleted 1 character in body
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| Jun 30, 2020 at 0:04 | history | edited | Klaus Niederkrüger | CC BY-SA 4.0 |
improved another formulation after comments by Chris Gerig
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| Jun 29, 2020 at 17:08 | comment | added | Klaus Niederkrüger | @ChrisGerig I did not really think through all of the details needed to construct the sequence $\{L_k\}$, but any trivialization of $TM|_\gamma$ is isomorphic to any other one. A priori the only difference is that we have $S^1 \times \mathbb{C}^n$, with a certain section $\sigma$ given by the tangent direction of the loop. I think that you can split off $\mathbb{C}\cdot \sigma$ from $S^1 \times \mathbb{C}^n$, the remaining complementary subbundle is still trivial, and we can find then a totally real subbundle to obtain any desired Maslov index, don't we? :| | |
| Jun 29, 2020 at 16:58 | comment | added | Klaus Niederkrüger | @ChrisGerig to construct some $L$, I would simply trivialize the pull-back bundle $f^* TM$ (as a complex bundle). This way I have a trivialization of $TM$ over $\gamma$. Next, I choose a totally real subbundle of $TM|_\gamma$ that contains the direction of the loop. If I apply the exponential map to this subbundle, I get a submanifold, and this submanifold is still close to $\gamma$ totally real. (maybe I need to choose a Riemannian metric such that $\gamma$ is totally geodesic or so?) | |
| Jun 29, 2020 at 13:37 | history | edited | Klaus Niederkrüger | CC BY-SA 4.0 |
slight improvements of formulation
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| Jun 29, 2020 at 13:02 | history | edited | Klaus Niederkrüger | CC BY-SA 4.0 |
I realized that I can improve my explanation to show that there is generically no holomorphic curve.
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| Jun 29, 2020 at 11:49 | history | answered | Klaus Niederkrüger | CC BY-SA 4.0 |