TITLE: Almost Complex Structures: 'Tame' versus 'Compatible'
QUESTION [8 upvotes]: Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures. These are $J:TM\to TM$ with $J^2=-\text{id}$. A given $J$ is $\omega$-tame when $\omega(v,Jv)>0$, and $J$ is $\omega$-compatible when it is $\omega$-tame and $\omega(J\cdot,J\cdot)=\omega(\cdot,\cdot)$. The set of either such $J$ forms a contractible subspace. Note that in either scenario we can form a Riemannian metric $g_J$ by twisting $\omega$ in some way with $J$.
In practice, I sometimes see $J$ being tame, and more often I see $J$ being compatible, but in either case I am unsure of the restrictions that each imposes.  Is there any context in which it is necessary/helpful to use one condition over the other? This need not be related to pseudo-holomorphic curves in Floer theory. Do certain results fail when I relax "compatible" to "tame"?
A fundamental difference I see (quoted from McDuff-Salamon's big textbook) is that compatible $J$ minimize the energy of a J-holomorphic curve in its homology class, but not necessarily for tame $J$.

REPLY [5 votes]: A high dimensional situation, where one doesn't have the choice to work with a compatible almost complex structure is the case of a contact manifold $(V,\xi)$ weakly filled by a symplectic manifold $(W,\omega)$.  I do not want to give the precise definition of weak fillings here, but the important thing to understand is that to work in a sensible way with $J$-holomorphic curves on a manifold with boundary, we would like to choose a $J$ that is 

(at least!) tamed by the symplectic structure $\omega$ so that we can measure energy and hopefully use Gromov compactness,
furthermore we require $\xi$-convexity along the boundary so that the holomorphic curves cannot move "up" and escape from the symplectic manifold.

The correct definition of weak fillability is a compatibility condition between $\omega$ and $\xi$ that guarantees the existence of such almost complex structures satisfying both conditions.
Unfortunately, it is not always possible to find a $J$ that in this situation is both compatible with $\omega$ and with $d\alpha|_\xi$ (for a contact form $\alpha$), and we have to stick to tamed almost complex structures, see Remark 2.3 in 
Massot, Patrick; Niederkrüger, Klaus; Wendl, Chris, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192, No. 2, 287-373 (2013). ZBL1277.57026..