Explicit correspondences between gradient trees in $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$

Abstract

Fukaya and Oh studied the correspondence between pseudoholomorphic disks in $T^{*}M$ which are bounded by Lagrangian sections $\{L_{i}^{\epsilon}\}$ and gradient trees in $M$ which consist of gradient curves of $\{f_{i}-f_{j}\}$. Here, $L_{i}^{\epsilon}$ is defined by $L_{i}^{\epsilon}=$\,graph$(\epsilon df_{i})$. They constructed approximate pseudoholomorphic disks in the case $\epsilon>0$ is sufficiently small. When $M=\mathbb{R}$ and Lagrangian sections are affine, pseudoholomorphic disks $w_{\epsilon}$ can be constructed explicitly. In this paper, we show that pseudoholomorphic disks $w_{\epsilon}$ converges to the gradient tree in the limit $\epsilon\to+0$ when the number of Lagrangian sections is three and four.