In [40] the perturbations involved in the partial compactification and the genericity of the almost complex structure makes the holomorphic curves rather difficult to visualize.¹⁴¹⁴ 14 A typical feature of Floer theory, is that completely realistic examples about holomorphic curves are also non-explicit. We now present a heuristic way to see holomorphic triangles with the edges on $\bar{L}_{\phi,A}$, $\Pi_{0}\cup\{\infty_{0}\}$ and $\Pi_{\phi}\cup\{\infty_{\phi}\}$, by restricting attention to $\mathbb{C}^{n}$ with the standard complex structure, and we imagine the two vertices $\infty_{0},\infty_{\phi}$ as the intersection points at infinity.

We choose any $\vec{x}=(x_{1},\ldots x_{n})\in S^{n-1}$. Assume first that $x_{k}\neq 0$. Then coordinatewise, we have a real curve in $\mathbb{C}$ swept out by $z_{k}(y)x_{k}$ as $y$ varies from $-\infty$ to $+\infty$, and two straight rays emanating from the origin defined by $\text{sign}(x_{k})\mathbb{R}_{+}$ and $\text{sign}(x_{k})e^{i\phi_{k}}\mathbb{R}_{+}$. Inside $\mathbb{C}$, these three real curves enclose a noncompact holomorphic triangle, with one vertex at the origin, and two idealized intersection points at the infinity of $\text{sign}(x_{k})\mathbb{R}_{+}$ and $\text{sign}(x_{k})e^{i\phi_{k}}\mathbb{R}_{+}$. In the product space $\mathbb{C}^{n}$, this gives rise to a holomorphic triangle $\Sigma_{\vec{x}}$ with boundary on $L_{\phi,A},\Pi_{0},\Pi_{\phi}$, and corners at $0,\infty_{0},\infty_{\phi}$. Now in case some $x_{k}=0$, there is still a holomorphic triangle in the product space that makes sense; the $k$-th projection of this triangle is simply the origin. What happens when $x_{k}\to 0$, is simply that the $k$-th projection $\pi_{k}(\Sigma_{\vec{x}})$ becomes very thinly concentrated near the two rays $\text{sign}(x_{k})\mathbb{R}_{+}$ and $\text{sign}(x_{k})e^{i\phi_{k}}\mathbb{R}_{+}$, and its area shrinks to zero. Morever for any given $\epsilon$, the subset $\Sigma_{\vec{x}}\cap\pi_{k}^{-1}(|z|>\epsilon)$ disappears into the infinity of $\mathbb{C}^{n}$ as $x_{k}\to 0$. Thus when we restrict to any compact subset of $\mathbb{C}^{n}$, the holomorphic discs behave continuously as $x_{k}\to 0$.

Using any of the holomorphic triangles $\Sigma_{\vec{x}}$ parametrised by $S^{n-1}$, we can calculate its area cohomologically by

$$\int_{\Sigma}\omega=\int_{\partial\Sigma}\lambda=\int_{\partial\Sigma\cap L_{\phi,A}}df_{L_{\phi,A}}=A,$$

which is the intuitive explanation of why $A$ must be positive, an important ingredient of [40].

We want to draw attention also to a different aspect not explicit in [40]: that these holomorphic triangles naturally arise in an $(n-1)$-dimensional moduli, rather than as isolated triangles. Consequently, the universal family of such holomorphic triangles is naturally $(n+1)$-dimensional. A generic point on $\Pi_{0}\cup\Pi_{\phi}\cup L_{\phi,A}$ is swept out precisely once by some $\partial\Sigma_{\vec{x}}$. When the orientations are taken into account, then the total space of this universal family gives rise to an $(n+1)$-dimensional integration current, which provides a bordism current between the integration cycles of $L_{\phi,A}$ and $\Pi_{0}\cup\Pi_{\phi}$. Producing bordism currents via universal families of holomorphic curves will be essential to our proposals concerning the Thomas-Yau conjecture.