# Energy bounds for pseudoholomorphic curves without symplectic structure

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.