Tag: contact

2010-04-14




2010-04-16

 




 

Symplectic polydisk and ellipsoid embeddings

Some explicit questions about polydisk and ellipsoid embeddings that Olga Buse and I hope to tackle:

  • Consider the map $ f \colon\thinspace B^4(1) \to B^2(1)$ defined in complex coordinates by $ f(w,z) = wz$ . Pick a regular value, say $ z_0 = 1/2$ . Let $ C = f^{-1}(z_0)$ . First of all, what is the area $ A=\int_C \omega_0$ ? Secondly, given any area $ a < A$ , let $ w(a)$ be the supremum over all $ w$ such that there is a polydisk $ B^2(a) \times B^2(w(a))$ embedded in $ B^4(1)$ with $ B^2(a) \times \{0\} \subset C$ . Calculate $ w(a)$ . It should obviously be a decreasing function of $ a$ . Then, for which $ a$ is $ a w(a)$ largest?
  • Same thing for any other Lefschetz fibration, or any complex polynomial for that matter.
  • In principle you might be able to use these same ideas to understand ellipsoid embeddings. Maybe the best way to say it is this: Given the symplectic surface $ C = f^{-1}(z_0)$ , find a smooth positive function $ r \colon\thinspace C \to (0,\infty)$ with a symplectic embedding of $ P(C,r)=\{(p,x,y) \in C \times \mathbb{R}^2 \mid \sqrt{x^2+y^2} < r(p)\}$ into $ B^4(1)$ mapping $ P \times \{(0,0)\}$ to $ P$ by the identity. Hopefully you can find such a function and such an embedding so that it ``fully fills'' $ B^4(1)$ . Then by trimming back $ C$ and trimming back $ r$ just right, we can get embeddings of ellipsoids, and by trimming back $ C$ and trimming $ r$ back to be constant, we can get embeddings of polydisks.
  • My original idea about using Lefschetz fibrations was a little different, but should probably amount to the same thing. Take your favorite topological Lefschetz fibration of $ B^4$ over $ B^2$ . Build it symplectically, with convex boundary, in some standard way, starting with $ F \times B^2$ (where $ F$ is the fiber) and then adding very thin Weinstein handles. You end up with a symplectic structure $ \omega$ on $ B^4$ with convex boundary, with induced contact form $ \alpha$ on $ S^3$ (contact form because you have kept careful track of the form, not just the underlying contact structure, as you attached the handles). And this nonstandard symplectic ball is almost fully filled with a polydisk. Now we know that there exists some function $ f \colon\thinspace S^3 \to [1,\infty)$ so that $ f \alpha = k \alpha_0$ , where $ \alpha_0$ is the standard contact form on $ S^3 = \partial B^4(1)$ , and $ k$ is some constant. I think that then you want to measure $ \int_{S^3} f$ , or something like that - this should be a measure of how much volume you have to add to change $ (B^4,\omega)$ into the standard round ball. Somehow this function is an interesting object attached to a word in Dehn twists.
  • The other idea for polydisks, which ultimately is still really the same basic idea, namely extending a symplectic surface out to a polydisk as much as possible, is to consider ribbon disks for interesting transverse ribbon knots.

 

P.S. I just gave a talk at MSRI on indefinite Morse 2-functions, my main project with Rob Kirby, and the pdf slides can be viewed here.

David Gay 2010-04-28






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