Ozsváth and Szabó proved Symplectic Thom conjecture [Annals of Mathematics, 151(2000), 93-124]. It states: An embedded symplectic surface in a **closed**, symplectic 4-manifold is genus-minimizing in its homology class.


Does a suitable relative version of above hold true ? More specifically, suppose we start with an embedded symplectic surface $\Sigma $ with boundary in symplectic 4-manifold with contact type boundary then is it true that $\Sigma $ is genus-minimizing in its (relative) homology class?



Relative version of Symplectic Thom conjecture. 
I think this must be a consequence of the version of the slice-Bennequin inequality proved by Mrowka and Rollin (but I might be wrong). Perhaps the argument also requires the boundary to have the "strong filling" property (Stein near the boundary).


Given a Legendrian knot $L$ in (to start with) the 3-sphere $S^3$, that inequality says
$$ 
 2 g\_\*(L) - 1 \ge tb(L) - r(L). 
$$ `
Every transverse knot $K$ is a push-off of a Legendrian approximation $L$, and the self-linking number of $K$ is related to the invariants of $L$ by
$$ sl(K) = tb(L) - r(L). $$
So the slice-Bennequin inequality says
$$
 2 g\_\*(K) -1 \ge sl(K).
$$
Unless I'm mistaken, this inequality is an equality for the case of a transverse knot bounding a symplectic surface in the 4-ball.


All this generalizes to the case of a symplectic 4-manifold $X$ that is Stein near its (contact) boundary. Given a Legendrian knot $L$ in the boundary, and a homology class $s$ of surfaces in $X$ with boundary $L$, one has invariants $tb(L,s)$ and $r(L,s)$. Then there is an inequality (as in Mrowka-Rollin),
$$
 2 g\_\*(L,s) - 1 \ge tb(L,s) - r(L,s),
$$
where $g\_\*(L,s)$ is the smallest possible genus of a surface in the class $s$ with boundary $L$. In terms of a transverse push-off $K$, one again has
$$
 2 g\_\*(K,s) - 1 \ge sl(K,s).
$$
If $K$ is actually the transverse boundary of a symplectic surface, then one has equality, this being (I think) just the adjunction formula in a relative version.


The Mrowka-Rollin version of the result can today be deduced from the existence of concave fillings (caps). We may assume from the outset that $tb(L,s)$ is positive: if it is not, we may sum in a bunch of Legendrian trefoils until it is. Now enlarge $X$ by first adding a 2-handle along $L$ (standard contact surgery) and then closing it up with a concave filling. The inequality one wants is just the adjunction inequality applied to the homology class formed from $s$ and the 2-disk in the core of the handle.


So with less notation, the answer to the original question is supposed to be: take a Legendrian approximation to the transverse knot and alter things so as to make $tb$ postive; then add a 2-handle and a cap to get a closed symplectic manifold. Then apply the adjuntion inequality to the homology class formed from the symplectic surface and the Lagrangian 2-disk.


