Lotay and Pacini proved the convexity of their $J$-functional (cf. Prop. 2.13) through rather heavy calculations, so it is instructive to see that in the Calabi-Yau case, this result has a much simpler conceptual argument.

We interpret their ‘geodesic’ as a bordism current $\mathcal{C}$ between two Lagrangians $L,L^{\prime}$, constructed from universal families of holomorphic curves, such that automatic transversality and the positivity condition hold. In their highly idealized setting, only holomorphic strips $\Sigma\simeq\mathbb{R}_{s}\times[0,1]_{t}$ appear in the construction of $\mathcal{C}$. We define the holomorphic function $F$ as usual. The 1-parameter family of totally real submanifolds is given by the constant $t$-coordinate slices $\mathcal{C}_{t}$ of $\mathcal{C}$, whose $J$-volume functional is expressible through moduli space integrals

$$\text{Vol}_{J}(\mathcal{C}_{t})=\int_{\mathcal{C}_{t}}|\Omega|=\int_{\mathcal{M}}\int_{\mathbb{R}_{s}\times\{t\}}|\frac{\partial F}{\partial s}|ds.$$

Since $F$ is holomorphic, so is $\frac{\partial F}{\partial s}$, whence $|\frac{\partial F}{\partial s}|$ is subharmonic, which combined with the exponential decay at $s\to\pm\infty$ implies the convexity of the function in $t$

$$\int_{\mathbb{R}_{s}\times\{t\}}|\frac{\partial F}{\partial s}|ds.$$

Thus $\text{Vol}_{J}(\mathcal{C}_{t})$ is convex as a function of $t$, as Lotay and Pacini observed.