4-dimensional aspects of tight contact 3-manifolds

In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: if a fibered link L induces a tight contact structure on Y then its fiber surface maximize Euler characteristic amongst all surfaces in Yx[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsv\'ath-Szab\'o contact invariant. We also show that any subsurface of a page of an open book inducing a contact structure with non-trivial invariant maximize "slice" Euler-characteristic for its boundary, and conjecture that this holds more generally for open books inducing tight contact structures.