# Convex surfaces with transverse boundary (contact geometry)

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.

• Check etnyres note on convex surfaces. If it is convex surface then you need consider the behavior of the dividing curves. – Anubhav Mukherjee May 29 at 4:24
• 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. – no_idea May 29 at 4:34
• 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. – KSackel May 29 at 5:54