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Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is convex (in the contact geometry sense)?

If I can't do this rel boundary, what sort of things could I do to perturb $\Sigma$ so that it is convex? For example, I could perturb $\partial\Sigma$ so that it is Legendrian, and then I could pertub $\Sigma$ so that it is convex (assuming $tb(\partial\Sigma)<0$). Would any complications arise if I then perturb this boundary to be transverse after doing this?

I haven't found any explicit mention of these kinds of surfaces, and I am hoping there is just a quick answer to this.

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  • $\begingroup$ Check etnyres note on convex surfaces. If it is convex surface then you need consider the behavior of the dividing curves. $\endgroup$ – Anubhav Mukherjee May 29 at 4:24
  • $\begingroup$ I thought the statement was the surface $\Sigma$ with Legendrian boundary is convex iff it has dividing curves. So you could have a surface with transverse boundary this theorem doesn't say anything. $\endgroup$ – no_idea May 29 at 4:34
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    $\begingroup$ You can certainly fix the transverse boundary by modifying the typical proof: a $C^{\infty}$ generic perturbation rel boundary will make the characteristic foliation Morse-Smale, and having a Morse-Smale characteristic foliation implies convexity. $\endgroup$ – KSackel May 29 at 5:54

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