Let $(M,J)$ be a compact almost complex manifold and $C>0$ a constant.
Gromov's compactness theorem states that the space of $J$-holomorphic curves $u:S^2\to M$ with energy bounded above by $C$ is compact, modulo bubbling. The usual way of employing Gromov's theorem is by using a compatible symplectic form on $(M,J)$ and restricting to the space of $J$-holomorphic maps to a fixed homology class $A\in H_2(M;\mathbb{Z})$: all curves in this class have equal energy.
Question: Suppose $(M,J)$ has no compatible symplectic structure. How can we obtain energy bounds on $J$-holomorphic curves in a fixed homology class $A$?
In particular, suppose $g$ is a $J$-compatible Riemannian metric. Are there any conditions (e.g. curvature bounds) on $g$ which imply a uniform area bound for $J$-holomorphic curves $u:S^2\to M$ in $A$?
The reason I think a condition on $g$ might give uniform energy bounds is that the energy of a $J$-holomorphic curve is equal to the area of the curve with respect to a $J$-compatible metric.
The answer is no, since for $S^{6}$ the moduli spaces can be non-compact. See https://mathoverflow.net/questions/219193/pseudo-holomorphic-curves-in-the-six-sphere.