Timeline for Global transversals of a codimension one foliation
Current License: CC BY-SA 4.0
17 events
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| Jul 24, 2022 at 11:55 | history | edited | user167131 | CC BY-SA 4.0 |
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| Jul 23, 2022 at 12:59 | history | edited | user167131 | CC BY-SA 4.0 |
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| Jul 20, 2022 at 22:27 | comment | added | Sam Nead | Ah, crud. I made a mistake in my previous comment. Let's take $n = 2$ and consider the foliation of $P$ by hyperplanes (line segments) orthogonal to $\Delta$. Each leaf is a graph over an open set (a precompact line segment in the $x$-axis). Note that each leaf is proper in the sense that the preimage of any compact set in $P$ are compact in $D$. | |
| Jul 20, 2022 at 22:12 | comment | added | user167131 | @Sam Nead: I appreciate the hint, and I WANT to do this as much as possible by myself, but I can't for the life of me see why the assumption that the leaf (graph) is proper would imply that d + (t,t,....,t) is in D whenever d is. Properness is satisfied (IIRC) if f goes to infinity at a point on the boundary of D (that is not in the boundary of P). Further, isn't d + (t,t,...,t) going to be parallel to the diagonal in Q, so will never meet it? I must be missing something. Also, your comments/answers have been extremely useful and I appreciate your willingness to help. | |
| Jul 20, 2022 at 18:26 | comment | added | Sam Nead | Ok, then I think you also want $D$, the domain of definition of $f$, to be open. Fine. Now prove that if $d$ is any point of $D$, then so is $d + (t,t,t, \ldots,t)$ (for all positive $t$). Next prove that $D$ meets the diagonal in $Q$. Finally, win as in my answer. | |
| Jul 20, 2022 at 11:34 | comment | added | user167131 | @Sam Nead: Thank you, I see. But is each leaf (flat disk) proper? For one thing, it doesn't separate P into two connected components; equivalently, f doesn't go to infinity at the boundary of D. Unless I'm missing something, which is entirely possible. | |
| Jul 20, 2022 at 7:06 | comment | added | Sam Nead | In that case, I take $D$ to be the closed round disk of radius one centred at $(2, 5)$ in the $xy$-plane. Also, I take $\mathcal{F}$ to be the product foliation - so the functions $f$ are all constant. In this case $\Delta$ misses all leaves. | |
| Jul 19, 2022 at 18:57 | comment | added | user167131 | @Sam Nead Each leaf is the image of a proper map. In this case, I believe that amounts to f being a proper map from D to R, which I take to mean the inverse image of every compact subset K in R is compact in D. Equivalently, for every sequence of points in D that approach the boundary the sequence of values of f must "escape to infinity". There is a result that says in this context a leaf is proper if and only if it is closed. | |
| Jul 19, 2022 at 18:35 | comment | added | Sam Nead | The only "propert[y] of the leaves" that can save us is "properness". What do you mean by "properness" in this context? | |
| Jul 19, 2022 at 11:29 | comment | added | user167131 | @Sam Nead Thanks for pointing this out, edits made. Yes, I guess really the question is is it possible for D to not intersect the diagonal given the properties of the leaves. In particular, it is known that each leaf L separates P into two connected components. That plus the fact we KNOW through each point of the diagonal passes a leaf and all leaves have positive normals might be enough to get the result. But I can't seem to prove it, which is unfortunate, since I need a way to index the leaves so I can obtain a "monotonic" projection onto the leaf space. | |
| Jul 19, 2022 at 11:22 | history | edited | user167131 | CC BY-SA 4.0 |
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| Jul 19, 2022 at 10:26 | history | edited | Sam Nead | CC BY-SA 4.0 |
fixed problem with boldface.
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| Jul 19, 2022 at 10:24 | comment | added | Sam Nead | You are using $D$ both for the "domain of definition of $f$" and for the diagonal. You might want to fix this. Also, with the new version of the question, isn't the answer trivially "no"? This is because there may be leaves whose domain of definition is far from the diagonal. | |
| Jul 19, 2022 at 10:19 | history | edited | Sam Nead | CC BY-SA 4.0 |
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| Jul 18, 2022 at 21:19 | history | edited | user167131 | CC BY-SA 4.0 |
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| Jul 18, 2022 at 20:50 | answer | added | Sam Nead | timeline score: 3 | |
| Jul 18, 2022 at 19:00 | history | asked | user167131 | CC BY-SA 4.0 |