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.


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