The main reason to impose the Calabi-Yau condition is so that any special Lagrangian closed integral current is an absolute volume minimizer among all closed integral currents in the same homology class, by the calibration inequality (2). If we believe Thomas-Yau conjecture to be valid more generally for almost Calabi-Yau ambient structures, with

$$\frac{\omega^{n}}{n!}=(-1)^{n(n-1)}(\frac{\sqrt{-1}}{2})^{n}e^{2\rho}\Omega\wedge\overline{\Omega},$$

then we are naturally motivated to ask if Almgren’s regularity extends to special Lagrangians in this setting:

The following observations, left as easy exercises, are indications that almost Calabi-Yau manifolds behave similarly as Calabi-Yau manifolds.

• •

  By a variant of the calibration inequality (2), special Lagrangians minimize the weighted volume

  $$\int_{L}e^{-\rho}dvol_{L}\geq|\int_{L}\Omega|$$

  within its homology class.

• •

  Under the smoothness assumption, the mean curvature of a Lagrangian submanifold with phase function $\theta$ satisfies the formula

  $$\vec{H}=-\nabla^{\perp}\rho+J\nabla\theta,$$

  where $\nabla^{\perp}$ means the normal projection of the gradient, and $\nabla\theta$ is the derivative of $\theta$ along $L$. Thus for smooth special Lagrangians in a bounded region, we have the a priori bound $|\vec{H}|\leq C$.