More on explicit correspondence between gradient trees in $\mathbb{R}$ and holomorphic convex quadrilaterals in $T^{*}\mathbb{R}$

Abstract

For given smooth functions $(f_1,\dots,f_n)$ on $M$, Fukaya and Oh showed that the moduli space of pseudoholomorphic disks in $T^*M$ which are bounded by Lagrangian sections $\{L_i^ε=\operatorname{graph}(εdf_i)\}$ is diffeomorphic to the moduli space of gradient trees in $M$ which consist of gradient curves of $\{f_i-f_j\}$. When the image of the pseudoholomorphic disk $w_ε$ is a polygon in $\mathbb{C}\simeq T^*\mathbb{R}$, we can describe $w_ε$ by a Schwarz-Christoffel map. In \cite{S25}, we proved that pseudoholomorphic disks $w_ε$ converge to the gradient tree in the limit $ε\to+0$ when the image of $w_ε$ is a generic convex quadrilateral. In this paper, we show such a convergence for any convex quadrilaterals by studying the non-generic case.