The epsilon-regularity conjecture for the almost complex Calabi–Yau equation

Let $(M,\Omega)$ be a compact symplectic four-manifold equipped with an almost complex structure $J$ tamed by $\Omega$. Let $\tilde{\omega}$ be a symplectic form cohomologous to $\Omega$ and compatible with $J$, and let $\tilde{g}$ and $g$ be the metrics associated to $\tilde{\omega}$ and $\Omega$, respectively. Given a smooth volume form $\sigma$, suppose that

$$\tilde{\omega}^2=\sigma.$$

Epsilon-regularity conjecture. There exist constants $\varepsilon,C,r_0>0$, depending only on $\Omega$, $J$, and $\sigma$, such that if

$$\frac{1}{r^2}\int_{B_g(p,r)}\operatorname{tr}_{g}\tilde{g}\,dV_g\leq\varepsilon$$

for some $p\in M$ and $0<r<r_0$, then

$$\sup_{B_g(p,r/2)}\operatorname{tr}_{g}\tilde{g}\leq\frac{C}{r^4}\int_{B_g(p,r)}\operatorname{tr}_{g}\tilde{g}\,dV_g.$$

This is motivated by epsilon-regularity for harmonic maps and is intended to provide local control toward the global estimates. The source presents it as an expectation and gives no resolution.