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 $J$.



Almost Complex Structures: 'Tame' versus 'Compatible'
If you assume that $M$ is compact and of dimension $4$ then Donaldson has conjectured that if $J$ is $\omega$-tame then $J$ is $\omega'$-compatible for some symplectic form $\omega'$, see Question 2 [here.](http://arxiv.org/abs/math/0607083) By classical results of Gromov and Taubes this is known to hold if $M=\mathbb{CP}^2$. It is also not hard [to verify this](http://arxiv.org/abs/0708.2520) when $J$ is integrable (so $(M,J)$ is a compact complex surface), using the classification of surfaces. It is [also known](http://arxiv.org/abs/1203.4331) to hold when $(M,J)$ is homogeneous.


In this same paper Donaldson proposed a way to attack this question by suitably extending the Calabi-Yau theorem to the symplectic case. Such an extension is still conjectural, but see [this survey](http://arxiv.org/abs/0901.1501) for some work that has been done in this direction. 


On the other hand Taubes has developed a different way of attacking Donaldson's question, and [has succeeded](http://projecteuclid.org/euclid.jsg/1309546043) in the case when $b^+(M)=1$ and $J$ is suitably generic.


In higher dimensions ($6$ or more) the analogous statement is false.


